Phonons, charge and spin in correlated systems Macridin

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Phonons, charge and spin in correlated systemsMacridin, Alexandru; Sawatzky, G.A

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Phonons, Charge and Spin inCorrelated Systems

Alexandru Macridin

MSC Ph.D.-thesis series 2003-07ISSN 1570-1530

The work described in this thesis was performed in the Surface Science Group (partof the Materials Science Center) at the Solid State Physics Laboratory of the Universityof Groningen, the Netherlands. The project was financially supported by the NetherlandsOrganization for Scientific Research (NWO) and the Spinoza Prize Program of NWO.

Printed by: Facilitair Bedrijf RuG, Groningen (2003)

Rijksuniversiteit Groningen

Phonons, Charge and Spin inCorrelated Systems

Proefschrift

ter verkrijging van het doctoraat in deWiskunde en Natuurwetenschappenaan de Rijksuniversiteit Groningen

op gezag van deRector Magnificus, dr. F. Zwarts,in het openbaar te verdedigen op

vrijdag 20 juni 2003om 16.00 uur

door

Alexandru Macridin

geboren op 5 april 1975te Timisoara, Romania

Promotor: Prof. Dr. G. A. Sawatzky

Beoordelingscommissie: Prof. Dr. M. JarrellProf. Dr. D. KhomskiiProf. Dr. D. van der Marel

ISBN: 90 367 1839 2

Contents

1 Introduction 51.1 Strongly Correlated Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 High Tc Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Electron-Phonon Interaction and Superconductivity . . . . . . . . . . . . . 111.4 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Diagrammatic Quantum Monte Carlo 232.1 Introduction to Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . 232.2 Diagrammatic Quantum Monte Carlo Simulation . . . . . . . . . . . . . . 26Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Holstein Polaron 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Analytical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.1 Weak-Coupling Perturbation . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Strong-Coupling Perturbation . . . . . . . . . . . . . . . . . . . . . 35

3.3 Diagrammatic Quantum Monte Carlo Algorithm . . . . . . . . . . . . . . . 373.4 Results of the QMC Calculation . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4.1 Ground State Properties . . . . . . . . . . . . . . . . . . . . . . . . 403.4.2 Momentum Dependent Properties . . . . . . . . . . . . . . . . . . . 46

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.6.1 Diagrams Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.6.2 Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Hubbard-Holstein Bipolaron 594.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Analytical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.1 Weak-Coupling Perturbation Theory . . . . . . . . . . . . . . . . . 614.2.2 Strong-Coupling Perturbation Theory . . . . . . . . . . . . . . . . . 62

4.3 Diagrammatic Quantum Monte Carlo Algorithm . . . . . . . . . . . . . . . 664.3.1 General Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

1

2 Contents

4.3.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4.1 Phonon Induced Attraction. U = 0 Case . . . . . . . . . . . . . . . 714.4.2 S0 Bipolaron to S1 Bipolaron Transition. U 6= 0 Case . . . . . . . . 744.4.3 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.6.1 Diagrams Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.6.2 Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 Dynamical Cluster Approximation 875.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.2 Dynamical Mean Filed Theory . . . . . . . . . . . . . . . . . . . . . . . . . 885.3 Dynamical Cluster Approximation . . . . . . . . . . . . . . . . . . . . . . . 89

5.3.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.3.2 Quantum Monte Carlo Algorithm . . . . . . . . . . . . . . . . . . . 915.3.3 Susceptibility Functions . . . . . . . . . . . . . . . . . . . . . . . . 93

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6 Multi-Band Hubbard Model 976.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.2 Reduction of Five-Band Hubbard Model to Two-Band Hubbard Model . . 99

6.2.1 Derivation of the Five-Band Hubbard Hamiltonian . . . . . . . . . 996.2.2 DCA Applied to the Five-Band Hubbard Model . . . . . . . . . . . 1046.2.3 Comparison of Five-Band and Two-Band Hubbard Models . . . . . 105

6.3 Two-Band Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.3.1 Phase Diagram and Other Properties . . . . . . . . . . . . . . . . . 1086.3.2 Electron-Hole Asymmetry . . . . . . . . . . . . . . . . . . . . . . . 114

6.4 Reduction of Two-Band Hubbard Model to Single-Band Hubbard Model . 1166.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7 Single-Band Hubbard Model 1257.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.2 Phase Separation in the Electron Doped Regime . . . . . . . . . . . . . . . 1297.3 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.4 Density of States and Pseudogap . . . . . . . . . . . . . . . . . . . . . . . 1327.5 Conclusions and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 134Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Samenvattig 139

Publications 143

3

Acknowledgements 144

4

Chapter 1

Introduction

1.1 Strongly Correlated Systems

The first striking success of the quantum theory of solids was the classification of crystalsinto metals and insulators as a function of the occupation of the electronic bands. Ac-cording to band theory a crystal is a metal if the last band is partially occupied and aninsulator otherwise. The band theory assumes that every electron is moving in a periodicpotential created by the positive ions and by the other electrons. In this one electronpicture, every electron is sensitive only to the average properties of the surrounding en-vironment, any correlation being neglected. The eigenstates of this one-particle problem,with the corresponding energies taking continuous values on some distinct intervals, con-stitute the bands.

Even though band theory has been successful in many respects, shortly after its ap-pearance, de Boer and Verwey [1] reported that many properties of transition-metal oxidesare in total disagreement with the band structure calculations. As a result of partiallyoccupied d type band, a metallic behavior was expected. In fact, these materials areinsulators with a conductivity gap an order of magnitude larger then the maximum gapderived even with the most sophisticated band structure calculation techniques. Mottand Peierls [2] were the first to suggest that the reason for the band theory failure mightbe the neglect of the repulsive Coulomb interaction between electrons on the d orbitals.Later on, Mott [3–5] and Hubbard [6, 7] showed that if the repulsion is larger than thebare electron bandwidth the Fermi level will be located inside a gap and the system willbe indeed an insulator.

In the last decades an impressive number of novel materials, poorly described by con-ventional techniques and displaying a rich variety of properties, were discovered. Amongthem, in order of both practical and theoretical importance, the transitional-metal oxidesand especially the cuprate superconductors occupy a leading position. A common char-acteristic of transitional-metal compounds is the essential contribution of the 3d orbitalsto the electronic properties. The 3d orbitals are special in the sense that they have asmall spatial extent (much smaller than the extent of the ligand p orbitals) and thereforeretain much of the initial atomic properties as the strong Coulomb repulsion between twoelectrons on the same ion. This unscreened short-ranged repulsion makes the correlations

5

6 Chapter 1. Introduction

play a predominant role in the underlying physics ∗, resulting in a large number of physi-cal properties, which are unfortunately not well understood from the theoretical point ofview.

The conventional treatment of systems with interacting electrons, valid in many cases,was developed by Landau. The main assumption of his Fermi-liquid theory is a oneto one correspondence between the interacting and the noninteracting ground state andlow-energy excitations. The interacting ground state is still assumed to have a sharpFermi surface, which results in well defined quasiparticles close to the Fermi level. This isdue to Pauli principle which blocks the scattering inside the Fermi sea (the only allowedscattering mechanism is by creating electron-hole pairs), resulting in a quasiparticle life-time which is much longer than the inverse of the excitation energy. The quasiparticlesbehave like electrons in a noninteracting system, and as a consequence, the Fermi-liquidproperties are described by the same expressions as the corresponding Fermi-gas ones.Thus it is possible to account for the effect of interaction only by renormalizing a fewparameters, such as the effective mass.

The experimental study of the correlated materials show that they exhibit “strange”behavior, which cannot be understood in the framework of Fermi-liquid theory. Atpresent, good theories which treat properly the correlations effects are missing. No oneknows a reliable way to deal with the problems where both the kinetic part and the inter-action part of the Hamiltonian are of the same order of importance. Besides the strongelectronic interaction, another common characteristic of many strongly correlated mate-rials is the low dimensionality (quasi-two or quasi-one dimensional). This implies strongquantum fluctuations, making the theoretical treatment even more difficult.

In the last years, a large number of analytical approaches, which use more or lessuncontrollable approximations, were applied to strongly correlated systems, giving some-times contradictory results. Besides those, numerical algorithms, as unbiased methodsto solve the model Hamiltonians, acknowledged a huge development. However these nu-merical techniques have several shortcomings too. They are not able to deal with largesystems, making the extrapolation of the results to the thermodynamic limit unreliable.Nevertheless, we believe that the new cluster mean-field methods, which treat numeri-cally exact the short-range correlations inside a cluster and at the mean field level thephysics on longer length scales, should give good results provided only the short-rangecorrelations are important.

1.2 High Tc Superconductors

In 1986 Bednorz and Muler [9] discovered a new type superconductor, based on Cu andO, with a surprisingly then high critical temperature (≈ 30 K). Their discovery hasbeen one of the most important discovery in condensed matter physics of the secondhalf of the last century. Soon after their discovery, a large number of similar materials(called generic cuprates) were synthesized, having the critical temperature above the

∗By definition everything beyond the Hartree-Fock approximation (i.e. beyond the one-particle pic-ture) is a correlation effect.

1.2. High Tc Superconductors 7

Although a large number of compounds have been dis-covered as high-Tc cuprate superconductors since then,they all have a common structure of stacking of CuO2layers sandwiched by block layers, as shown schemati-cally in Fig. 117. The superconductivity itself has beenan attractive and extensively studied subject. The sym-metry of the pairing order parameter of the high-Tc cu-prates now appears to be settled as the dx22y2 symmetryafter enormous combined effort involving magneticresonance experiments, Josephson junctions, etc. (see

for a review Scalapino, 1995; Annett et al., 1996). How-ever, in this review article, we focus on the unusualnormal-state properties of high-Tc compounds; proper-ties in the superconducting state are beyond the scope ofthis article.

Before going into a detailed discussion of experimen-tal data, we first outline the basic electronic structureand the phase diagram in the plane of doping concentra-tion x and temperature T . Because of the two-dimensional anisotropy of this lattice structure, CuO6octahedra are slightly elongated along the direction per-pendicular to the CuO2 layer. This means that the dis-tance from Cu to the apex oxygen is slightly longer thanthe distance to the in-plane oxygen, which may beviewed as a Jahn-Teller-type distortion. This distortionlifts the degeneracy of the eg orbitals of Cu 3d electronsto the lower-lying dz22r2 and higher-lying dx22y2 orbit-als. Because the parent compound La2CuO4 has the va-lence of Cu21, and hence as the nominal d9 state theabove crystal-field splitting leads to fully occupied t2g

orbitals and d3z22r2 orbital, while the dx22y2 orbital re-mains half-filled, the Fermi level lies in a band con-structed mainly from the dx22y2 orbital, relatively closeto the level of the 2ps oxygen orbital, compared withother transition-metal oxides. In La2CuO4, since the 2ps

band lies within the Mott-Hubbard gap, it is a charge-transfer (CT) insulator, as discussed in Secs. II.A andIII.A. La2CuO4 is a typical insulator with antiferromag-netic long-range order. Up to the present time, all thecuprate superconductors with high critical temperaturesare known to be located near the antiferromagneticMott insulating phase. La2CuO4 is, in fact, a Mott insu-lator with the Neel temperature TN;300 K. Becausethe dx22y2 band is half-filled, it is clear that the insulat-ing behavior experimentally observed even far above TN

is due to strong correlation effects.La22xMxCuO4 with M5Sr and Ba provides a typical

example, with a wide range of controllability of carrierconcentration, from the antiferromagnetic Mott insulat-ing phase to overdoped good metals, by the doping ofSr, Ba, or Ca. Therefore La22xMxCuO4 is a good ex-ample of an FC-MIT system with 2D anisotropy. Thebasic electronic structure of the cuprate superconductorsin all regions, from insulator to metal, is believed to bedescribed by the d-p model defined in Eqs. (2.11a)–(2.11d). The parameters deduced from photoemission(Shen et al., 1987; Fujimori, Takekawa, et al., 1989),LDA calculations (Park et al., 1988; Stechel and Jenni-son, 1988; Hybersten, Schluter, and Christensen, 1989;McMahan, Martin, and Satpathy, 1989), and s(v) giveoverall consistency with Udd.6 – 10 eV, tpd.1 – 1.5 eV,D.1 – 3 eV, Upp.1 eV, and Vpd;1 – 1.5 eV. The de-tails of methods for determining the model parametersare given in Sec. III.B.

With increasing x , La22xSrxCuO4 very quickly under-goes a transition from an antiferromagnetic insulator toa paramagnetic metal with a superconducting phase atlow temperatures. The phase diagram of La22xSrCuO4is shown in Fig. 118. At sufficiently low temperatures, itis believed that the MIT with x is actually a

FIG. 116. The layered perovskite structure of La22xMxCuO4;large open circles, La(M); small open circle, Cu; large filledcircle, O.

FIG. 117. Conceptual illustration of stacking of CuO2 layerssandwiched by block layers.

1174 Imada, Fujimori, and Tokura: Metal-insulator transitions

Rev. Mod. Phys., Vol. 70, No. 4, October 1998

Figure 1.1: Conceptual illustration of staking of CuO2 layers sandwiched by block layers.

boiling point of liquid nitrogen (77 K) ∗ and therefore opening enormous opportunitiesfor technological applications. The common belief is that the understanding of thesematerials could provide the knowledge for growing room temperature superconductors.Thus, since their discovery, the cuprates have been under an extraordinary intensive, bothexperimental and theoretical, investigation.

However, the results so far are modest comparative to the invested effort. After morethan fifteen years of intensive research, the cuprates are far from being understood. Asveritable strongly correlated systems, they exhibit many “anomalous” properties, impos-ing severe constrains on any reasonable theory.

Despite of their different chemical composition, the cuprates superconductors havemany common characteristics, and it is natural to assume that these characteristics de-termine the basics physics. The high Tc superconductors are layered materials, containingquasi-two-dimensional CuO2 planes sandwiched between block layers. The generic struc-ture is illustrated in Fig. 1.1, but it also should be said that some materials have more

∗Nowadays the highest Tc under normal pressure conditions is around 130 K and, under high pressure,around 160 K.


Figure 1.2: Generic phase diagram of cuprates superconductors [8].

than one CuO2 layer in between the block layers.The electronic states close to the Fermi level are contained in the CuO2 planes. The

block layers play the role of charge reservoirs, controlling the number of charge carries inthe conduction planes upon chemical doping. Consequently, theoretical models designedto describe the physics of a single CuO2 plane for different electronic concentrations wereproposed for the cuprates. However, there is no general agreement on what should be theminimal model which captures the basic physics of a CuO2 plane. The photoemissionexperiments show that the first electron removal states have O p origin, unlike the firstelectron addition states which have Cu d character, placing the cuprates in the charge-transfer regime of Zaanen-Sawatzky-Allen scheme [10]. It is believed that the only Cuorbital which participates to low-energy physics is the dx2−y2 one. It couples to the in-paneoxygen p orbitals, and therefore the first models contained all these degrees of freedom (dand p). However, based on the strong Cu-O hybridization, many physicists believe thatone step further can be taken and a reduction to a one-band Hamiltonian which containsonly the dx2−y2 orbitals can be done. But there is no general agreement on that, and

1.2. High Tc Superconductors 9

some authors consider the oxygen orbitals as essential for explaining the properties of thephase diagram, such as the pseudogap for example [11, 12].

The common physical properties of cuprates can be summarized in the generic phasediagram presented in Fig. 1.2 [8]. The undoped materials are antiferromagnetic insula-tors. Upon doping the antiferromagnetism is quickly destroyed. For the doping rangeapproximatively in between 5% and 30%, the cuprates become superconductors at lowtemperature. The doping value where the Tc is maximum (around 15%) is called optimaldoping. The lower and higher doping regime are called underdoped and respectively over-doped regimes. Unlike in the conventional superconductors, the superconducting gap inthe cuprates has d-wave symmetry, with nodes along the diagonal directions and changingthe sign upon a π/2 rotation [13–16]. In the overdoped region, above Tc the system is aFermi-liquid metal. But the most intriguing physics occurs in the underdoped region, inthe proximity to the antiferromagnetic phase. The majority of physicist believe that theunderstanding of the underdoped normal state would provide the key for elucidating thesuperconductivity mechanism.

Almost all the quantities measured on underdoped samples exhibit “anomalous” be-havior. The most remarkable anomaly is the pseudogap, and many other “strange” prop-erties are believed to be related to it. The pseudogap is a depletion in the density of statesclose to the chemical potential, noticed bellow a characteristic temperature, T ∗. The ear-liest indications of the pseudogap were found in NMR experiments [17–19] where the spinsusceptibility measured by the Knight-shift was seen to decrease strongly below T ∗. Manyother measurements like heat capacity [20], resistivity [21, 22], infrared absorption [23–25],neutron scattering [26], and tunneling [27, 28] show also evidences of pseudogap. However,the most direct and unambiguous evidence comes from the angle-resolved photoemissionspectroscopy (ARPES) [29, 30]. At least for the hole doped systems, the ARPES spectrashow that the pseudogap is well developed at the zone boundary and vanishes along thediagonal directions, suggesting the same symmetry as the superconducting gap. There-fore, in many scenarios suggested by the theorists the pseudogap is considered as theprecursor of the superconducting gap.

At present, with respect to the cuprates physics in general and pseudogap in particular,there is a great disagreement between theorists, and perhaps there are as many theoriesas there are the groups investigating the problem. However, in the following we introducebriefly some of the most important ideas and scenarios which have appeared since thehigh Tc discovery.

Based on the vicinity of the underdoped region to the antiferromagnetic phase, somescenarios assume that the pseudogap appears as a result of coupling of quasiparticles tosharp antiferromagnetic spin fluctuations peaked at (π, π) [31]. With reducing the doping,the pseudogap will evolve into the full spin-density-wave gap characteristic to undopedantiferromagnets with nested Fermi surface.

There are another scenarios which assume that in the pseudogap region the pairs arealready formed. The formation of pairs starts at the pseudogap temperature, T ∗, and Tc isthe temperature where the phase coherence occurs [32]. The bipolaron superconductivitytheory, which we discuss in the next section (Section 1.3), should also be included in thiscategory.

Theories based on resonating valence bond (RVB) and spin-charge separation are also


Figure 1.3: Phase diagram of electron doped (left) and hole doped (right) cuprates [8].

popular in the field [33, 34]. Here the spins pair into short-ranged singlets when T < T ∗,resulting in a gap in the spin excitation spectrum which is assimilated to the pseudogap.The elementary excitations are not quasiparticles but spinons which are charge neutralfermions with spin 1/2 and holons which are charged bosons with spin 0. At Tc theholons condensate and recombine with the spinons giving rise to superconductivity.

Other authors relate the pseudogap to the existence of a quantum critical point closeto the optimal doping and identify the pseudogap as a region characterized by circulatingcurrents and time-reversal broken symmetry [11].

Because the physics of the normal and of the superconducting state is so unconven-tional, it is rather widely believed that phonons do not play an essential role in the su-perconductivity mechanism. Nevertheless, there is strong experimental evidence showing

1.3. Electron-Phonon Interaction and Superconductivity 11

that the electron-lattice coupling is strong [35–38], beyond the applicability of the con-ventional theories which describe the electron-phonon interacting systems. Consequently,other authors [35, 39] claim that the superconductivity is driven by the phonons as inthe conventional superconductors, the difference being only the strength of the couplingbetween charge carriers and phonons.

Before ending this section, we would like to stress another important characteristic ofcuprates, namely the asymmetric behavior of the electron and hole doped samples. Theasymmetry can be seen from the experimentally determined phase diagram presented inFig. 1.3. In the electron doped materials the antiferromagnetism survives up to a largercritical doping while the opposite thing can be said about the superconductivity. However,the recent angle-resolved photoemission data on electron-doped materials [40], which ex-hibit a pseudogap around (π/2, π/2) and well defined quasiparticles at the zone boundaryshow that this asymmetry is more profound. In the case that the superconducting gapin the electronic doped systems has still x2 − y2 symmetry (as our calculations suggest),these results would question the assumption that the pseudogap is a precursor of the su-perconducting gap. Nevertheless, there is no agreement so far about the symmetry of thesuperconducting gap in electron doped materials, some experiments indicating a s-wavesymmetry and others a d-wave symmetry. Nevertheless, the electron doped materialsare much less studied than the hole doped ones, due to the difficulty encountered in thepreparation of good samples.

1.3 Electron-Phonon Interaction and Superconductiv-ity

The interaction between electrons and lattice degrees of freedom plays a crucial role inunderstanding the properties of many materials and results in a multitude of physicalphenomena. Structural transitions like Jahn-Teller distortion in perovskite or Peierlsdimerization in one-dimensional systems, the pairing and the condensation of the chargecarriers resulting in superconductivity are some of the most spectacular effects which orig-inate from electron-lattice interaction. Even in situations where no real phase transitionstake place the transport properties in crystals are largely influenced by the interactionof the charge carriers with the lattice. Not only does the inelastic scattering of the elec-trons with the impurities or with the thermally activated phonons determine the carriersmobility but also the formation of low-energy coherent states which result in a fundamen-tal change of the charge carriers properties. These states are called polarons, a conceptintroduced long time ago by Landau (1933) [41] which describes a charge carrier whichcarries with it a lattice deformation. Besides dressing the charge carriers, the phononsalso induce an effective attraction between them. As Bardeen, Cooper and Schrieffer(BCS) showed [42], even a weak attraction is enough to make a Fermi liquid state insta-ble towards electron pairing. The electrons form pairs in momentum space close to theFermi level (Cooper pairs). The condensation of the Cooper pairs leads the system tothe superconducting phase. However if the electron-phonon attraction is too strong theresult may be different, now two electrons forming a strong bound pair of small size inreal space. Such an entity is called a small bipolaron and can be identified with a hard


core charged boson. If the density of charge carries is small the physics can be consideredas being well described by a liquid of bipolarons which suffers Bose-Einstein condensationresulting also in superconductivity.

In the last years there is growing experimental evidence that in the fashionable stronglycorrelated materials like manganese and copper oxides, aside from the unscreened Coulombrepulsion, the electron-lattice interaction is also extremely important [43–50]. The electron-phonon coupling strength is usually of intermediate strength so that neither the weak-coupling nor the strong-coupling theories work. Therefore numerical solutions for modelswhich address the electron-phonon interaction in this difficult region of parameter spaceare highly desirable. Aside from a realistic (and difficult) multi-electron problem solution,for a basic understanding of the underlying physics special interest should be paid to sim-pler problems which address the effect of a polarizable medium (lattice) on a single chargecarrier (polaron problem) and the phonon induced pairing mechanism of two electrons (orholes) in a lattice (bipolaron formation problem).

Electron-phonon interaction In general, the lattice is considered as a set of linearoscillators, being modeled by

Hph =∑q,ν

ων(q)b†q,νbq,ν (1.1)

where b†q,ν (bq,ν) is the creation (annihilation) operator of a phonon with momentum qand ν is the vibration branch index. The electrons interacting with the lattice potentialare described by

H1 = Hel +Hel−ph =Ne∑i=1

p2i

2me

+ V (ri) (1.2)

where Ne is the total number of electrons and

V (ri) =∑

l

v(ri −Rl) (1.3)

is the crystal field potential. Rl is the position of the “l”-th ion, and in general itsdeviation, xl, from the equilibrium position, R0l, is small, so we can write

v(ri −Rl) ≈ v(ri −R0l) +∂v

∂R(ri −R0l) xl (1.4)

The contribution of the first term of Eq. 1.4 in Eq. 1.2 results in a Hamiltonian whichdescribes the interaction of electrons with a periodic potential, thus producing noninter-acting Bloch states. In the second quantization formalism it can be written as

Hel =∑

kσ

ε(k)c†kσckσ (1.5)

where c†k,σ (ck,σ) is the creation (annihilation) operator of a electron state with momentumk and spin σ. Taking into account that the Fourier transform of the displacement xl is

xq ∝∑

ν

√1

ων(q)(b†q,ν + bq,ν) (1.6)


the contribution of the last term of Eq. 1.4 in Eq. 1.2 results in the electron-phononinteraction part of the Hamiltonian

Hel−ph =1√N

∑

kσ,qν

g(k, q, ν)c†k+qσckσ(b†−q,ν + bq,ν) (1.7)

The general form of a Hamiltonian which describes both the electrons and the latticedegrees of freedom should be

H = Hel +Hph +Hel−ph +Hel−el (1.8)

where Hel−el is a electron-electron interaction term.The electron-phonon coupling is given by the general matrix g(k, q, ν). Several sim-

plified models which considerg(k, q, ν) = g(q) (1.9)

like Frohlich model with g(q) ∝ 1q which is expected for a free or nearly free electron gas

and Holstein model with g(q) = g = constant which is expected if only the on-site energyin a tight binding model is affected, have been under investigation in the course of time.

Migdal’s approach to normal state The treatment of systems characterized by alarge Fermi surface and a weak to intermediate electron-phonon coupling is greatly sim-plified following an observation made by Migdal [51] which shows that the renormalizationof the electron-phonon interaction is negligible in all orders of perturbation theory. Thereason is the Pauli exclusion principle which blocks the scattering inside the Fermi sea,resulting in corrections of the order ω/εF , where ω is the phonon characteristic frequencyand εF is the Fermi energy ∗.

Within Migdal’s approximation solving the problem means solving the following sys-tem of equations

Σ(k, ω) =∫dq

∫dΩg(q)2D(q,Ω)G(k − q, ω − Ω) (1.10)

Π(q,Ω) =∫dk

∫dωg(q)2G(k, ω)G(k + q, ω + Ω) (1.11)

where G(k, ω) is the electron propagator,

G−1(k, ω) = G−10 (k, ω)− Σ(k, ω) (1.12)

and D(q,Ω) is the phonon propagator

D−1(q,Ω) = D−10 (q,Ω)− Π(q,Ω) (1.13)

The analysis of the real and of the imaginary part of the self-energy shows that welldefined quasiparticles exist close to the Fermi surface (when |ε − µ| < ω) and far fromit (when |ε − µ| >> ω), but not at the excitation energies of the order of the phonon

∗This is not true at long wave-length scattering in the systems with optical phonons which arecharacterized by ω(q = 0) 6= 0 [52].


frequency (|ε− µ| ≈ ω). The quasiparticle close to the Fermi level is an electron carryinga phononic cloud (which results in an increase of its effective mass, m∗) being similar toa polaron and the high energy quasiparticle is a free electron like [35, 52]. The effectivemass ratio

m∗

m= 1 + λ (1.14)

which can be determined by measuring the quasiparticles dispersion provides a means ofcalculating the electron-phonon coupling strength, λ, in materials. Here λ is the BCS like

defined electron-phonon coupling, i.e. λ = V N(0) ≈ 2g2

ω N(0) where N(0) is the densityof states at the Fermi level and V the effective electron-electron attraction induced by thephonons.

The poles of the phonon propagator D give for small momentum phonon the renor-malized dispersion [35, 53]

ω(q) = ω(q)√

1− 2λ (1.15)

which shows that Migdal’s theorem fails when the electron-phonon coupling λ > 1/2. Astrong electron-phonon interaction changes the ions equilibrium position (as the strongcoupling perturbation theory shows) driving a phonon vacuum instability resulting in thebreakdown of the Migdal approximation.

Migdal-Eliashberg’s approach to superconductivity The Eliashberg’s theory ofsuperconductivity [54, 55] is an extension of Migdal’s treatment for the normal state ofelectron-phonon coupled systems to the superconductivity phase. It uses the same approx-imation which neglects the electron-phonon vertex corrections and therefore it is expectedto work only at small and intermediate coupling values. From another perspective it isa generalization of the BCS theory which allows a proper treatment of the retardationeffects and to include realistic phonon dispersion and electron-phonon interaction matrix.

The difference from the normal state treatment is that now (as in the BCS approach)the anomalous averages 〈c†c†〉 are allowed to take a nonzero value. In order to do thatthe electron Green’s function is defined as a 2× 2 matrix

G(k, τ) =

(Tτ 〈ck↑(τ)c†k↑〉 Tτ 〈ck↑(τ)c−k↓〉Tτ 〈c†−k↓(τ)c

†k↑〉 Tτ 〈c†−k↓(τ)c−k↓〉

)(1.16)

In zero order the off-diagonal elements (the anomalous averages) are zero. The Green’sfunction is calculated by summing the same set of diagrams as in the Migdal’s normalstate approach (Eq. 1.10) but now a 2 × 2 matrix corresponds to every propagator andalso to the electron-phonon vertex [56, 57]. The phonon propagator

D(k, ω) = D(k, ω)

(1 00 −1

)(1.17)

and the electron-phonon vertex

g(q) = g(q)

(1 00 −1

)(1.18)


are diagonal matrices. The Green’s function propagator is the Fourier transform ofEq. 1.16. The self-consistent solution of (Eq. 1.10) shows that at low temperature theoff-diagonal elements of the Green’s function are nonzero. An important result is that thishappens only when all the noncrossing diagrams are summed up. In any finite order ofthe perturbation theory the Green’s function is a diagonal matrix, thus the superconduc-tivity is a nonperturbative effect. The Eliashberg’s approach reproduces the BCS resultsif the same approximation is made for the phonon propagator (i.e. D(q, ω) = −1 forω < ωD and zero otherwise). The critical superconductivity temperature is defined as thetemperature where the anomalous averages vanish. The BCS value is

Tc = 1.13ωDe−1λ (1.19)

One of the great advantage of Eliashberg’s formulation is the possibility to include theCoulomb repulsion in the same framework. Even though there is no adiabatic parameterfor the Coulomb interaction, the neglect of vertex corrections corresponds to RandomPhase Approximation (RPA) and provides good results for a qualitative analysis. Aremarkable result is obtained. At the Fermi surface the relevant parameter introduced bythe screened Coulomb repulsion, V (q), will be µc defined as

µc = V (qF )N(0) (1.20)

The electron-electron repulsion competes with the phonon induced attraction, but thecritical temperature will be

Tc =2ωD

πe− 1λ− µ∗c (1.21)

where µ∗c is the Coulomb pseudopotential, defined as

µ∗c =µc

1 + µc ln(εF/ωD)(1.22)

The remarkable thing is that the Coulomb repulsion effect enters only through the stronglysuppressed pseudopotential µ∗c . It means that even a strong repulsion does not destroythe superconductivity or implicitly the pairing. This happens due to the fact that asidefrom the sign the two competing interactions are fundamentally different. The attractiveinteraction is retarded, being defined in a small frequency window of the order ωD, whichmeans that it still acts for a period of time after two electrons meet one another. On theother hand the Coulomb repulsion acts on a much larger frequency window, of the orderof εF , being more or less instantaneous. When two electrons meet they separate quicklyat a distance where the repulsion becomes small but the retarded attraction delayed intime will be still effective there, making the pairing possible.

Nonadiabatic superconductivity When applied to high Tc materials, there are inprinciple two reasons for Migdal-Eliashberg’s theory to fail. One is the large value of theelectron-phonon coupling which leads to phonon instability (see Eq. 1.15), and we aregoing to discuss about it in the next paragraph. The other reason is the small Fermienergy (0.1 − 0.3 eV ) [58, 59] which does not allow the Migdal’s “adiabatic” parameter


εF/ω to be small. L. Pietroniero et al. [60, 61] pointed out that even for a small electron-phonon coupling the vertex corrections are important, and in some cases can lead toa strong enhancement of Tc. In their work they generalized the Eliashberg’s theory toinclude the first order vertex corrections.

Alexandrov’s bipolaronic theory of superconductivity The experimental data [35–38] show that the electron-lattice interaction in high Tc materials is large, being above theapplicability range of Migdal’s approximation. Therefore Alexandrov and co-workers [35,39] proposed a theory based on the strong electron-phonon coupling expansion for ex-plaining the physics of high Tc superconductors.

It is known that when the electron-phonon coupling is large the phonon mediatedattraction between electrons leads to the formation of strong bound pairs localized in thereal space (bipolarons). Alexandrov argues that the low-energy physics is described by aliquid of bipolarons. At low density, where the overlap of the bipolaron wave functionshas a negligible effect, the bipolarons can be treated as charged 2e− bosons which Bose-Einstein condensate and suffer a transition to a superfluid state at the critical temperatureTc. The superfluid state of a charged Bose-gas is equivalent to the superconductivityphase.

However some authors raised objections against the bipolaron theory. Chakravertyet al. [62] argued that the bipolarons are much too heavy states to account for the highvalue of Tc found experimentally in the cuprate materials. From our calculations (seeChapter 4) we also found an exponentially large bipolaron mass for the on-site bipolaron(i.e. a bound state with the electrons residing on the same site) even in the intermediateelectron-phonon coupling region. Nevertheless as the strong coupling theory and numeri-cal calculations show [63], the inter-site bipolarons (i.e. bound states with the electrons onneighboring sites) are much lighter states. These are states resulting from a non-localizedelectron-phonon interaction (unlike the on-site interaction considered in our model). Infact Alexandrov’s proposed bipolaron is an inter-site state with one hole in a CuO2 planeon a oxygen plaquette (formed by the four O atoms around a Cu ion) and with the otherhole on the apex O situated above the plaquette. Aside from being a light particle andtherefore able to provide a high Tc, Alexandrov argues that this particular bipolaron statecan also explain another important characteristic of the cuprates superconductors, the d-wave symmetry of the order parameter [13–16]. The apex oxygen participates to thebipolaron formation with two orbitals, px and py, resulting in two degenerate bipolaronbands.

Ex(k) = −t cos(kx) + t′ cos(ky) (1.23)

Ey(k) = t′ cos(kx)− t cos(ky) (1.24)

with t, t′ > 0 and tÀ t′. The bipolaron containing the apical px hole has a large dispersionalong the X direction because it involves the inter apical oxygens ppσ hopping integraland a small dispersion along the Y direction where the ppπ hopping integral is involved.It is orthogonal to the bipolaron formed with the py apical oxygen orbital. The relevantthing is that the energy bands minima are not at (0, 0) but at the Brillouin zone boundarypoints, (0,±π) for the “along x”band and respectively at (±π, 0) for the “along y” band.


When the bipolaron liquid Bose-Einstein condenses the states situated at these pointsin the Brillouin Zone will be macroscopically occupied. Alexandrov has proposed thefollowing real-space order parameter function for d-wave cuprates superconductors [64],

Ψ(rx, ry) = (x

2)1/2(cos(rxπ)− cos(ryπ)) (1.25)

constructed by taking a linear combination of the zone boundary condensate states. Thesymmetry of the order parameter is a result of the bipolaron energy dispersion with theminima at the zone boundary and has nothing to do with the symmetry of the internalwave function of a single bipolaron∗.

Because of the low-dimensionality, the bipolarons localize easily in the presence ofrandom potential wells. Only the non-localized bipolarons participate in the condensatestate. The critical temperature Tc in a quasi-two-dimensional superfluid (but still ananisotropic 3D system) is proportional to the particle density. With increasing the dopingthe number of localization wells is increasing too and as a result the number of non-localized bipolarons decreases at large doping. The maximum number of non-localizedbipolarons (and implicitly the maximum Tc) will be somewhere around 15% doping [39]in agreement with the experimental phase diagram. Another characteristic feature ofcuprates physics, the normal state pseudogap is explained in Alexandrov’s theory as beinga result of bipolaron state formation and therefore the pseudogap temperature T ∗ shouldbe proportional to the bipolaron binding energy.

The solution to the question whether the high Tc superconductivity mechanism is aconsequence of large electron-phonon interaction or of purely electronic correlation effectsis still controversial. More physicists are in favor of the later, believing that the intriguingphysics of the cuprates has its origin in the unscreened short range Coulomb repulsionbetween electrons. However, even if true, there are no priori reasons to neglect thelattice polarization effects and presumably models which include both electron-phononand electron-electron interaction should be considered. The calculations presented inthe last two chapters show that the main characteristics of cuprates superconductorscan be obtained neglecting the phonons, but also show that for a better agreement withexperiment something else, and we believe that it would be the phonons, should be addedto the model.

∗However we believe that strong objections can be raised against Alexandrov’s proposal [65]. First,Eq. 1.25 describes a charge density wave (CDW) of bipolarons, with a 2 × 2 supercell, rather than ad-wave superconductor. The state proposed by Alexandrov is an interesting novel state of matter, but itsproperties are completely different from superconductors ones. For example we don’t believe that a statedescribed by Eq. 1.25 would exhibit the Andreev reflection phenomenon characteristic to superconductors.Second, we don’t see any reason why the condensation should be in a standing superposition of thecondensate states at the zone boundary. A general wave function, with N bipolarons at (π, 0) and Mbipolarons at (0, π) should have the form

|Ψ >= (Ψ(π, 0)† + Ψ(−π, 0)†)N (Ψ(0, π)† + Ψ(0,−π)†)M |0 > (1.26)


1.4 Scope

Since the discovery of high Tc superconductors, the computation tools used for dealingwith strongly correlated materials, both analytical and numerical, have developed in animpressive manner. Among them, a new Quantum Monte Carlo method able to calculateexactly the Green’s functions, based on the summation of the corresponding Feynman’sdiagrams, proved to be very powerful for solving polaron like (interaction of a quantumparticle with a bosonic environment) and bipolaron like (pairing of two fermion as aresult of exchange of bosonic particles) problems. Related to the large electron-latticecoupling encountered in the majority of the correlated materials and especially in highTc superconductors, the solution to the polaron and bipolaron problems are of a extremeimportance in elucidating the properties of quasiparticles and the pairing mechanism,and should be considered as a first step in the understanding of the more complex multi-electron plus phonons problem.

In the last three years, a powerful method for dealing with interacting electrons systemswas developed. The technique is called Dynamical Cluster Approximation (DCA) and isan extension of the previous popular Dynamical Mean Field Theory (DMFT). The DMFTmaps the lattice problem to an impurity embedded self-consistently in a host and thereforeneglects spatial correlations. The DCA maps the lattice to a finite-sized cluster embeddedin a host. Non-local correlations up to the cluster size are treated explicitly, while thephysics on longer length scales is treated at the mean field level. The rapid developmentof the computational resources available at the main Supercomputers Centers aroundthe world results in the possibility to increase both the cluster size (i.e. the correlationrange) and the number of relevant interactions and degrees of freedoms inside the clusteraccordingly, making us to believe that in the very next future many of the intriguingproperties of correlated materials will be elucidated through the DCA approach.

Our calculations, which take advantage of the above mentioned techniques, addressmainly the physics of high Tc superconductors, even though the polaron and bipolaronconcepts have a much larger range of applicability. This thesis can be divided in moreor less two independent parts, the first dealing with electron-phonon interaction andthe second addressing the Hubbard type Hamiltonians used for modeling the electronicinteraction in cuprates superconductors. We consider both parts as absolute necessarysteps in solving and understanding of the more realistic problem which consider bothelectron-lattice and electron-electron interaction at the same time. As we are going tomention at the end of this thesis, this complex problem can be addressed within the DCAframework too, and we plan it as a research project for the next future.

The manuscript is organized as follows. In Chapter 2 we describe the general princi-ples of the Diagrammatic Quantum Monte Carlo technique used in the next two chaptersto solve the polaron and respectively the bipolaron problem. Chapter 3 studies the inter-action of a single electron with sharp on-site phonons, addressing both the ground stateand the momentum dependent properties of the system. In Chapter 4 the pairing mecha-nism of two electrons (bipolaron formation) as a function of both electron-lattice couplingand Coulomb repulsion is studied. Chapter 5 gives a short description of the DynamicalCluster Approximation technique. Using DCA with a 2×2 cluster, in Chapter 6 we studya five-band Hubbard Hamiltonian, which we consider as a realistic starting point for the

1.4. Scope 19

description of the cuprates CuO2 plane. The importance of the different oxygen degrees offreedom is studied, as well as the possibility of Hamiltonian reduction from the five-bandto a simpler effective one-band model. The electron-hole asymmetry seen in the phasediagram and in the ARPES spectra is discussed too. In Chapter 7 we consider the single-band Hubbard model, both with and without the next-nearest-neighbor hopping integralfound previously to be responsible for the electron-hole asymmetry. The calculations aredone now on a larger cluster, with 8 sites. The results are compared to the corresponding4 site cluster ones. We also discuss here the effects introduced by the periodic boundaryconditions when are imposed on very small clusters.


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Chapter 2

Diagrammatic Quantum Monte Carlo

2.1 Introduction to Monte Carlo Simulation

Monte Carlo is a numerical method used in various fields of interest, which proved to bevery powerful in solving different statistical and quantum physics problems. In this thesiswe present two numerical algorithms, both of them relaying on Monte Carlo simulations.Therefore we believe it will be appropriate to introduce here some basic notions of thistechnique.

This method provides an efficient means to solve multidimensional integrals and sums.In order to exemplify, let’s start with a n dimensional integral over a domain Γ of volume1, which we desire to solve numerically

I =

∫

Γ

f(x)dx (2.1)

The Monte Carlo way to compute this integral is to generate a large number of pointsuniformly distributed over the domain Γ and to approximate

I =1

M

M∑i=1

f(xi) (2.2)

where M is the number of generated points. The estimated error will be

∆I ∝ 1

M1/2(2.3)

which for a large dimension turns out to be better than any other numerical method’s esti-

mated error (for example the trapezoidal quadrature has an estimated error of 1M1/n ) [1].

For the same number of points generated (i.e. for fixedM) the numerical error dependson the behavior of the function f(x). In the extreme case when the function is a constant,one point will be enough for the exact evaluation. For a smooth function the errors aresmaller. In general the function f(x) is not smooth. One way to improve the accuracy isbased on the idea of importance sampling introduced by Metropolis et al. [2]. Supposewe find a distribution function w(x)

∫

Γ

w(x)dx = 1, w(x) > 0 (2.4)

23

24 Chapter 2. Diagrammatic Quantum Monte Carlo

which can mimic the drastic changes in f(x). The Eq. 2.1 can be written as

I =

∫

Γ

w(x)f(x)

w(x)dx (2.5)

The importance sampling idea is to generate the points according to the distribution w(x)instead of generating them uniformly. In this case the evaluation of Eq. 2.1 is

I =1

M

M∑i=1

f(xi)

w(xi)(2.6)

and a much faster convergence is expected.The expression in Eq. 2.5 is similar to the one used in statistical physics for calculating

the averages of different observables.

〈A〉 = 1ZTr〈e

−βHA〉 (2.7)

= 1Z

∑n e−βEn〈n|A|n〉 (2.8)

where the sum can be an integral in case that the eigenstates |n〉 are continuouslydistributed. A Monte Carlo simulation will consist in generating points (configurations)with the distribution

wi =1

Ze−βEi (2.9)

and to calculate

〈A〉 =1

M

M∑i=1

〈i|A|i〉 (2.10)

As it can be seen, in the hart of Monte Carlo method is the procedure to generateconfigurations with a certain nonuniform distribution. The way this is usually done isknown as Metropolis algorithm [2] and consists by the following steps:

• choose randomly a starting point i

• choose randomly another point j and accept the transition from i to j with aprobability which satisfies the equation (detailed balance condition)

wi p(i→ j) = wj p(j → i) (2.11)

Here wi (wj) is the weight of the i (j) configuration and p(i→ j) (p(j → i)) is thetransition rate from i to j (from j to i).

• From the new configuration (which is j if the previous transition was accepted or i ifit wasn’t) go in the same fashion (i.e. through transitions which satisfy the detailedbalance condition) to another one and so on.

2.1. Introduction to Monte Carlo Simulation 25

In the process described above every point is generated from the previous one (Markov pro-cess). The essential condition is the equilibrium one, which is achieved imposing Eq. 2.11.We give here two practical ways in which the Eq. 2.11 is usually implemented in theMonte Carlo codes. Being in the configuration i we accept the transition to configurationj with the probability p given by:

• Metropolis criteria

p = min wj

wi

, 1 (2.12)

• heath bath criteriap =

wj

wi + wj

(2.13)

To accept a transition with probability p means to compare p with a uniform randomnumber, r, generated in the interval [0, 1]. The transition is accepted if p ≥ r andrejected otherwise.

Of course Eq. 2.8 can be applied only if we already knew the eigenstates, but this meansthat we already solved the problem. For real problems we do not know the eigenstatesand different basis can be used for expanding Eq. 2.7. The calculation will imply farmore complicated expressions. The challenge in Monte Carlo coding is to bring theseexpressions in the form

I =

∫w(x)f(x)dx∫w(x)dx

(2.14)

and to generate in an efficient way configurations according to the distribution w(x).This is not always easy. It can happen that the distribution has pronounced maximaand minima and as a result the time to go from one maximum to the other will becomeextremely large, or in another words the generated configurations will stay for a verylong time around one maximum. As a consequence the result will be dependent of thestarting point, which of course can not be correct. Depending on the particular problemthe correlation time can be reduced using some tricks. An useful one is to notice that

〈f〉w =

∫w(x)f(x)dx∫w(x)dx

=

∫ w(x)g(x)

f(x)g(x)dx∫ w(x)

g(x)g(x)dx

=

∫ w(x)g(x)

f(x)g(x)dx∫ w(x)

g(x)dx

∫ w(x)g(x)

dx∫ w(x)

g(x)g(x)dx

=〈fg〉w/g

〈g〉w/g

(2.15)

which means that the average of f(x) over the distribution w(x) can be written as theaverage of f(x)g(x) over the distribution w(x)/g(x) divided by the average of g(x) overthe distribution w(x)/g(x). Now the configurations will be generated according to thedistribution w(x)/g(x). g(x) should be chosen to smooth the generating distribution andas a result the correlation time will be reduced.

A much more serious problem with the Monte Carlo simulations is the sign problem.Very often in quantum problems the distribution function w(x) is not positive definite.One way to proceed is to generate configurations with the distribution |w(x)|, which is


equivalent to choose g(x) = sgn[w(x)] in Eq. 2.15. The problem is that the average signof w(x) becomes extremely small and because the final result should be divided by it, theerrors are going to be strongly amplified making any calculation impossible. No solutionhas been found yet for the sign problem. Sometimes expanding Eq. 2.7 in a different basiscan help to reduce or even to eliminate the sign problem.

2.2 Diagrammatic Quantum Monte Carlo Simulation

The one-particle and the multi-particle Green’s functions provide important informationabout a physical system such as its eigenstates, the spectral and response functions, whichare the most common quantities measured in experiments. Therefore the Green’s functioncalculation provides a direct means of comparison between theory and experiment.

In many cases of interest the Green’s function can be written as an expression of thefollowing form:

Q(y) =∞∑

m=0

∑

ξm

∫dx1...dxmDm(ξm, x1, ..., xm; y) (2.16)

The problems we address in Chapter 3 and Chapter 4 are going to provide examples ofsuch calculations. Usually every term can be associated with a corresponding Feynmandiagram. The quantity Q(y) is defined by an (or a set of) external parameter(s), y (forexample the time and the momentum of the Green’s function) and it is expressed as aseries of integrals with an ever increasing number of integration variables. The integrationvariables are due to the interaction vertices and internal propagators which typicallyappear in the expansion of the Green’s function. If for example an internal propagator isdefined by a momentum and a time the integration over x1, ..., xm in Eq. 2.16 will be theintegration over all the internal propagators times and momenta. Usually there are manydifferent terms with the same integration order m and the label ξm is used to distinguishformally between them.

Diagrammatic Quantum Monte Carlo algorithm calculates Eq. 2.16 by generating theterms stochastically, according to their weight Dm(ξm, x1, ..., xm; y), using the Metropolisprocedure described in Section 2.1. The characteristic of the present method that dis-tinguishes it from other Monte Carlo algorithms consists in the fact that the integrationmultiplicity in Eq. 2.16 is varying. This will imply two qualitatively different kinds ofprocedures during the stochastic generation of the configurations: (i) one which preservesthe integration multiplicity and (ii) one which generates configurations with a differentintegration multiplicity.

The general structure of the code is the following. Several subroutines which executeelementary updates of the configurations (of both type (i) and (ii)) are defined. Everypoint in the configuration space must have a finite probability to be reached via a chainof such updates (ergodicity condition). Starting from a randomly chosen initial configu-ration the Metropolis procedure of generating configurations (the importance sampling)is achieved by calling the update subroutines one after another. Every update subroutineis called with an arbitrary nonzero probability. In practice these probabilities are tunedto assure a good efficiency of the code. The average of the generated configurations will

2.2. Diagrammatic Quantum Monte Carlo Simulation 27

converge to Q(y) provided the updating procedure obeys the detailed balance condition(see Eq. 2.11 and below).

Updates of type (i) These updates are the standard ones used in the Monte Carlosimulations. Being in a configuration Dm(ξm, x1, ..., xi, ..., xm; y) the transition to a ran-domly proposed configuration Dm(ξm, x1, ..., x

′i, ..., xm; y) is accepted with the probability

p = minW, 1 (2.17)

where

W =Dm(ξm, x1, ..., x

′i, ..., xm; y)

Dm(ξm, x1, ..., xi, ..., xm; y)(2.18)

As an example, these are the updates used to change the internal propagators time ormomentum.

Updates of type (ii) This kind of update is the distinguishing characteristic of thisMonte Carlo technique and implies transitions to configurations with a different inte-gration multiplicity. Suppose we are considering a transition from the configurationDm(ξm, x1, ..., xm; y) to the configuration Dm+n(ξm+n, x1, ..., xm, xm+1, ..., xm+n; y). Thenew integration variables xm+1, ..., xm+n are chosen according to an arbitrary distributionP(xm+1, ..., xm+n), in practice taken so as to maximize the acceptance ratio of the update.The update can be accomplished via a subroutine called from the main program with theprobability pa. In order to satisfy the detailed balance condition the counterpart subrou-tine which performs the inverse transformation (i.e. the transformation from Dm+n(...) toDm(...)) should also be considered. Suppose it is called from the main program with theprobability pb. Then the detailed balance condition reads

paP(xm+1, ..., xm+n)Dm(ξm, x1, ..., xm; y) p(Dm → Dm+n) =

= pbDm+n(ξm+n, x1, ..., xm, xm+1, ..., xm+n; y) p(Dm+n → Dm) (2.19)

which implies that the transition from Dm(...) to Dm+n(...) would be accepted with theprobability

p = minW, 1 (2.20)

and the transition from Dm+n(...) to Dm(...) with the probability

p = min 1

W, 1 (2.21)

where

W =pb

pa

Dm+n(ξm+n, x1, ..., xm, xm+1, ..., xm+n; y)

Dm(ξm, x1, ..., xi, ..., xm; y)P(xm+1, ..., xm+n)(2.22)

As an example, a typical update of this kind is one which adds an internal propagatorand its counterpart is the one which removes it. In time-momentum representation thenew integration variables would be the (q, τ1, τ2), i.e. the propagator momentum and itsinitial and final time.


The Diagrammatic Quantum Monte Carlo technique was developed by Prokof’ev etal. [3–5]. It has proved to be very efficient in solving polaron type problems (Frohlich po-laron [4, 5], Holstein polaron [Chapter 3], spin polaron [6], Rashba-Pekar polaron [7]) andinteracting two-body systems (exciton [8] and Hubbard-Holstein bipolaron [Chapter 4]).As presented in Ref. [3] the algorithm is universal and can be applied to any generallattice Hamiltonian as long as the sign problem can be avoided.

2.2. Diagrammatic Quantum Monte Carlo Simulation 29

References

[1] T. Pang, Introduction to Computational Physics (Cambridge University Press, 1997).

[2] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller,Journal of Chemical Physics 21, 1087 (1953).

[3] N. Prokof’ev, B. Svistunov, and I. Tupitsyn, JETP 87, 310 (1998).

[4] N. Prokof’ev and B. Svistunov, Phys. Rev. Lett. 81, 2514 (1998).

[5] A. Mishchenko, N. Prokof’ev, A.Sakamoto, and B. Svistunov, Phys. Rev. B 62, 6317(2000).

[6] A. Mishchenko, N. Prokof’ev, and B. Svistunov, Phys. Rev. B 64, 0033101 (2001).

[7] A. Mishchenko, N. Nagaosa, N. Prokof’ev, A.Sakamoto, and B. Svistunov, Phys. Rev.B 66, 020301 (2002).

[8] E. Burovski, A. Mishchenko, N. Prokof’ev, and B. Svistunov, Phys. Rev. Lett. 87,168402 (2001).

30

Chapter 3

Holstein Polaron

3.1 Introduction

The concept of a polaron was introduced by Landau [1] to describe an electron moving in apolarizable lattice carrying the lattice deformation with it. In a more general context, theterm of polaron describes a quantum particle interacting with a bosonic environment. Thebosonic modes dress the particle, changing its properties like the energy and the effectivemass. Recently, the environments and particles encountered in condensed matter physicshave become more diverse, so for example the polaron concept is also used now to describethe interaction of a hole or of an electron with spin or orbital excitations.

The polaron properties are dependent on both the particle-environment couplingstrength and polaron momentum. At zero momentum and at small coupling the par-ticle is lightly dressed, polarizing weakly the environment over a large spatial extentaround itself. Upon increasing the coupling, the particle changes its properties more orless abruptly depending on the environment modes (phonons) characteristic frequency,becoming extremely heavy and deforming the environment strongly over a small spatialextent around itself at large coupling. In the cuprates, manganates and most of the othercorrelated materials the electron-lattice interaction was found to be in the intermediateregion where the transition from light to heavy polaron takes place. The analytical cal-culations based on perturbation theory fail in this region. Therefore, recently a largevariety of non-perturbative approaches like exact diagonalization on small clusters [2–6],Quantum Monte Carlo simulations [7–11], dynamical mean field theory [12], variationaltechniques [13–16], density matrix renormalization group [17] and exact diagonalizationon a variational determined Hilbert subspace [18] have been applied, giving more or lessaccurate results for different regions of parameter space. However the transition region isnot well understood, and aside from the ground state properties little is known about thestable excited states which exists at intermediate coupling.

At small and intermediate coupling the polaron at zero momentum is fundamentallydifferent from the one at large momentum. If at zero momentum the free electron con-figuration has the most significant contribution, at large k the configuration with thehighest weight is an electron plus a phonon which carries almost the entire momentum.Consequently at large momentum the quasiparticle weight is extremely small and the dis-persion becomes flat, having the phonon characteristic shape. Perturbation theory fails

31

32 Chapter 3. Holstein Polaron

to describe the large momentum polaron even at small coupling (although it works wellfor the zero momentum state).

In this chapter we address the polaron problem using a Quantum Monte Carlo algo-rithm. We calculate the electron and the electron plus an arbitrary number of phononimaginary time Green’s functions by expressing them as a sum of integrals with an everincreasing multiplicity number and applying the general approach introduced in Chap-ter 2. Every term in the sum can be represented by a Feynman diagram, this being thereason why the algorithm is called the Diagrammatic Quantum Monte Carlo. We extractthe polaron properties from the long time behavior of the Green’s functions, where theground state is projected out. The method was developed by Prokof’ev et al. [7, 8] andapplied to the Frohlich polaron. It addresses the problem in the thermodynamic limit(infinite lattice) and it proves to give accurate results (better than 1% accuracy) for allregions of the parameter space provided that the binding energy is not extremely small.Coupled with the analytic-continuation spectral analysis algorithm [8] information aboutthe excited states can be obtained too.

The model Hamiltonian we consider is the popular Holstein one which assumes anon-site coupling of the electrons with a dispersionless lattice vibration mode

H = −t∑

〈ij〉(c†icj +H.c) + ω0

∑i

b†ibi + g∑

i

ni(b†i + bi) (3.1)

Here c†iσ(ciσ) is the creation (annihilation) operator of an electron at site i, and analoguesb†i , bi are phonon creation and annihilation operators. The first term is the electrontight-binding kinetic energy. The second term describes the lattice degrees of freedomconsidered as a set of independent oscillators at each site, with frequency ω0. The electronscouple through the density niσ = c†iσciσ to the local lattice displacement xi ∝ (b†i + bi)with the strength g and this interaction is described by the last term of Eq. 3.1. In themomentum representation Eq. 3.1 becomes

H =∑

k

ε(k)c†kck + ω0

∑q

b†qbq +g√N

∑

k,q

c†k+qck(b†−q + bq) (3.2)

with

ε(k) = −2tz∑

δ=1

cos(kδ) (3.3)

where z is the dimensionality of the problem. The model is characterized by two rela-tive parameters, the dimensionless coupling constant α = g2/ztω0 (see Sec. 3.2.2 for thedefinition) and the adiabaticity ratio ω0/zt.

3.2 Analytical Approaches

Despite its apparent simplicity the model can not be solved analytically. A frequentlyused approach is the adiabatic approximation which considers the ion mass infinitelylarge (which would result in ω0 = 0), thus reducing the problem to an electron interacting

3.2. Analytical Approaches 33

one phononone electron and

k

E(k)

free electron

ω0

K

Figure 3.1: Schematic plot of the Holstein Hamiltonian states when g = 0. The full linerepresent the free electron dispersion. The dashed and the dotted lines are states withone phonon and one electron. There are N (i.e. a infinity in the thermodynamic limit)such states, one for every phonon, shifted on the horizontal axis with the correspondingphonon momentum. There are also (not shown here) states with more phonons whichwould be shifted on the vertical axis by ω0 times the number of phonons.

with a static field produced by a deformed lattice. The lattice configuration is deter-mined self-consistently by minimizing the system energy. Unrealistic results, like a firstorder transition from a mobile to a localized self-trapped state, appear as a result of suchapproximation. We are not going to discuss the adiabatic approximation. The perturba-tive calculations are analytical approaches which consider dynamical phonons but theirvalidity is restricted to the extreme cases of weak and strong electron-phonon coupling.

3.2.1 Weak-Coupling Perturbation

When the electron-phonon interaction is weak the last term of Eq. 3.2 can be treated assmall perturbation. A schematic representation of the situation is presented in Fig. 3.1.Here the dispersion of a few states is shown for g = 0. When g is switched on, thefree electron state will mix with a continuum of states situated vertically (momentumconservation) above the energy (see for example the relevant states for K marked inFig. 3.1) ≈ ω0 − ε(k) + ε(0).

At small momentum the electron will be lightly dressed by the phonons and its effectivemass will increase slightly. The lattice is deformed weakly on a large extent around theelectron and this state is called large polaron. The best perturbation theory schemedescribing the large polaron at small momentum is the ordinary Rayleigh-Schrodinger


+ +

+ . . . .

Σ =

+ =

Figure 3.2: Self-Consistent-Born calculation of the self-energy.

one, where the first order energy correction is

ε(1)RS(k) = − 1

N

∑q

g2

ω0 + ε(k − q)− ε(k)(3.4)

The reason why this approach works so well is related to the fact that there alwaysis a continuum of states (corresponding to zero momentum polaron plus a free phonon)starting at the energy E0+ω0. E0 is the zero momentum polaron energy. Eq. 3.4 takes thisenergy separation properly in the denominator unlike the Wigner-Brillouin (or Green’sfunction) perturbation theory scheme

ε(1)WB(k) = − 1

N

∑q

g2

ω0 + ε(k − q)− ε(k)− ε(1)WB(k)

(3.5)

which always overestimates the denominator and therefore produces a smaller energycorrection [19].

However when the momentum is increased the Rayleigh-Schrodinger perturbation the-ory fails, strongly overestimating the energy correction term. This will result in an un-physical maximum in the energy dispersion as it is shown with dashed line in Fig. 3.12.The failure of perturbation theory at large momenta happens in other polaron modelstoo, as for example in the Frohlich polaron model [7, 20, 21]. At large momentum theelectron energy is very close to or degenerate with the electron plus phonon continuum.This will cause the Rayleigh-Schrodinger perturbation theory suitable for discrete levels tofail. The Green’s function technique is more appropriate for describing interactions withcontinuum states. With this scheme the unphysical downturn in the energy dispersiondisappears. However in first order the results are still not satisfactory. Now the stateswith more than one phonon are not considered and as a result the continuum energywill start at ε(0) + ω0, containing one electron (instead of one polaron) plus one phonon.Higher order calculations are required for acceptable results. From this point of viewthe best analytical approximation is the Self-Consistent-Born-Approximation (sometimesalso called Hartree-Fock or Non-Crossing approximation) which consists in solving thefollowing system of equations:

Σ(k, ω) = 1N

∫dΩ

∑q g

2D0(q,Ω)G(k − q, ω − Ω) (3.6)

G−1(k, ω) = ω − ε(k)− Σ(k, ω) (3.7)

where G(k, ω) and D0(q,Ω) are the full electron and the bare phonon propagators. Here aninfinite set of particular diagrams (the non-crossing ones, see Fig. 3.2) which can contain


from one to an infinite number phonons is summed up. As a result the continuum ofstates will have the right energy.

At small momenta the Self-Consistent-Born-Approximation results are poorer thanthe Rayleigh-Schrodinger perturbation theory ones, still underestimating the energy cor-rection. One should expect a similar underestimation of the Self Consistent Born Approx-imation energy correction at large momenta and therefore an exact solution is desirable.

3.2.2 Strong-Coupling Perturbation

If the electron-phonon interaction is strong, the hopping term in the Eq. 3.1 can be treatedas a perturbation. The last two terms can be diagonalized using the Lang-Firsov canonicaltransformation [22]. This is obtained via the unitary operator eS, where

S = − g

ω0

∑i

ni(b†i − bi) (3.8)

Using formula

A = eSAe−S = A+ [S,A] +1

2[S, [S,A]] + .. (3.9)

the transformed operators become

bi = bi +g

ω0

ni (3.10)

ci = ci egω0

(b†i − bi) (3.11)

The physical meaning of this canonical transformation is a shift of the ions equilibriumposition at the sites where the electron is present.

〈xi〉 = 〈b†i + bi〉 = 〈b†i + bi +2g

ω0

ni〉 = 〈xi〉+2g

ω0

〈ni〉 (3.12)

The Hamiltonian written in the new basis is

H = Ht +H0 (3.13)

with

H0 = ω0

∑i

b†i bi −g2

ω0

∑i

ni, (3.14)

Ht = −t∑

〈ij〉,(c†i cjX

†iXj +H.c) (3.15)

and

Xi = e− g

ω0(b†i − bi) (3.16)

As can be seen from the second term of Eq. 3.14, the lattice deformation energy gaineddue to the electron presence is

Ep = g2/ω0 (3.17)


The dimensionless electron-phonon coupling constant should be defined as the ratio be-tween this energy and the bare electron kinetic energy which is proportional to the hoppingt and with the lattice dimensionality z. We define it as ∗ †

α =g2

zω0t(3.18)

The electron hopping is accompanied by the changing of ions equilibrium position (seethe term X†

iXj in Eq. 3.15).The Hamiltonian H0 produces a N -degenerate ground state, every state consisting of

a localized electron at a particular site. In the zeroth order the translational symmetry isbroken, the electron being “trapped” by the lattice deformation. The first order correctionlifts the degeneracy, resulting in an exponentially reduced nearest-neighbor hopping

teff = t 〈i|c†i cjX†iXj|j〉 = t e

− g2

ω20 = t e

−αztω0 (3.19)

The second order perturbation theory has a much stronger effect. It corresponds to avirtual transition of the electron without carrying its lattice deformation to a nearest-neighbor location. The intermediate state will have an energy equal to 2Ep, because itcontains a site with deformation and without the electron and a site with the electronand without the deformation. Therefore the second order correction will be

E(2) = −2zt2

2Ep

= − t

α(3.20)

Explicit calculation of the matrices involved in the first and the second order perturbationtheory can be found in [24]. Summing up Eq. 3.19 and Eq. 3.20 the polaron dispersionresults in

E(k) = −αzt− t

α− 2teff

z∑i=1

cos(ki) (3.21)

The exponentially reduced teff implies an exponentially large effective mass

m∗ = m eαztω0 (3.22)

The number of phonons (in the initial basis), Nph, and the quasiparticle weight (the frac-tion of free electron configuration in the polaron state), Z0, can also be easily calculated.In zeroth order we get

Nph = 〈b†ibi〉 = 〈(b†i −g

ω0

)(bi − g

ω0

)〉 =g2

ω20

=αzt

ω0

(3.23)

∗If the approximation N(0) ≈ 1/2W where W is the electron bandwidth is considered, the relationto the electron-phonon coupling constant defined in Section 1.3 will be α = 2λ.

†Other authors define the dimensionless coupling constant as the ratio between the deformationenergy and phonon frequency [23], which indicates the number of phonons in the polaronic cloud (seeEq. 3.23).

3.3. Diagrammatic Quantum Monte Carlo Algorithm 37

and

Z0 = |〈Xi〉|2 = e− g2

ω20 = e

−αztω0 (3.24)

To conclude, for large electron-phonon coupling, the polaron is a heavy particle withan exponentially large mass and an exponentially small quasiparticle weight. Within avery small range around the electron the lattice is strongly deformed, for this reason thestrong-coupling polaron is also called small polaron.

3.3 Diagrammatic Quantum Monte Carlo Algorithm

Let’s consider the equation

〈ψ|e−τH |ψ〉 =∑

ν

|〈ψ|ν〉|2e−τEν (3.25)

where |ψ〉 is a whatever state and |ν〉 form the complete set of the eigenstates withenergy Eν . We see that at large τ time Eq. 3.25 converges to

|〈ν0|ψ〉|2e−τEν0 (3.26)

where |ν0〉 is the ground state of the system. Suppose the ground state is separated fromthe first excited state by a gap ∆. We can obtain the ground state energy and the overlapof the ground state with |ψ〉 with an accuracy better than 1% (for example) calculatingEq. 3.25 at a time τ ≈ 5/∆.

If we take |ψ〉 ≡ |k〉, where |k〉 is the free electron with momentum k state, thecalculation of Eq. 3.25 will result in the calculation of the zero temperature MatsubaraGreen’s function

G(k, τ)def= 〈0|c†k(τ)ck|0〉 = 〈k|e−τH |k〉 =

∑ν

|〈k|νk〉|2e−τEν(k) (3.27)

which at large time will provide the energy E(k) (≡ Eν0(k), the lowest energy in the kirreducible channel) and the quasiparticle weight of the polaron with momentum k

Z0(k) = |〈ν0k|k〉|2 (3.28)

As we are going to show, aside from the electron Green’s function, our algorithm allowsas to calculate at the same time the n-phonon correlation functions

P n(k, τ) =∑

q1,q2,...,qn

〈0|ck−q1−q2−...qn(τ)bq1(τ)bq2(τ)...bqn(τ)b†qn...b†q2

b†q1c†k−q1−q2−...qn

|0〉

(3.29)from which we can extract

Zn0 (k) =

∑q1,q2,...,qn

|〈ν0k|b†qn...b†q2

b†q1c†k−q1−q2−...qn

|0〉|2 (3.30)

i.e. the contribution of the n-phonon configurations to the polaron state.


τ1τ2τ=0 τ

q

kk k−q

Figure 3.3: A typical diagram which represents a term in in Eq. 3.37. The weight ofthis diagram determined with the rules presented in Appendix 3.6.1 is D(q, τ1, τ2; k, τ) =

(g dτ)2 dq e−ε(k)(τ1) e−ε(k − q)(τ2 − τ1) e−ω0(τ2 − τ1) e−ε(k)(τ − τ2).

The total number of phonons in the polaronic cloud will be

Nph(k) = 〈ν0k|∑

q

b†qbq|ν0k〉 =∑

n

nZn0 (k) (3.31)

In principle, calculating the polaron energy for many values of momentum around k =0 the effective mass (1/m∗ = ∂2E(k)/∂k2) can be determined. A more effective wayto compute the polaron mass is by making use of the energy estimator introduced inAppendix 3.6.2.(Eq. 3.61).

Now we are going to show that the one-electron Green’s function, Eq 3.27, (and similarthe n-phonon Green’s function, Eq 3.29) can be written in the general form of Eq. 2.16,therefore allowing us to apply the general Diagrammatic Quantum Monte Carlo techniquediscussed in Chapter 2. Let’s start with the Hamiltonian (3.2) and consider

H = H0 +H1 (3.32)

with H1 being the electron-phonon interaction term

H1 =g√N

∑

k,q

c†k+qck(b†−q + bq) (3.33)

The imaginary time evolution operator can be written as

e−τH = e−τH0S(τ) (3.34)

with

S(τ) =∞∑

n=0

(−1)n

n!

∫ τ

0

...

∫ τ

0

dτ1...dτnT [H1(τ1)...H1(τn)]

(3.35)

where H1 in the interaction picture is

H1(τ) = eτH0H1e−τH0 (3.36)

Using Eq. 3.34 and Eq. 3.35 the Green’s function (Eq. 3.35) will be

G(k, τ) = e−ε(k)τ∞∑

n=0

(−1)n

n!

∫ τ

0

∫ τ

0

...

∫ τ

0

dτ1...dτnT [〈k|H1(τ1)H1(τ2)...H1(τn)|k〉](3.37)


q1

q1

k+q1

k−q+

q1

k

q

τ=0 τ

kk−q

Figure 3.4: A diagram which contributes to the 1-phonon Green’s function, P 1(k, τ).

which is formally similar to Eq. 2.16, i.e. we reduced the calculation of the Green’s functionto the calculation of a series of integrals with an ever increasing number of integrationvariables.

It can be easily shown that every term in Eq. 3.37 can be represented by a diagram. Asimple set of rules can be derived to determine the value (the weight) of every particulardiagram. An example is given in Fig. 3.3. The algorithm generates stochastically allthe diagrams, according to their probability. The rules used for the determination ofdiagrams weight and the practical implementation of the Diagrammatic QMC updatesare presented in Appendix 3.6.1.

In order to determine the polaron energy we need to calculate the behavior of theGreen’s function at large imaginary time, therefore we need knowledge of G(k, τ) inmany τ points. If a Monte Carlo run is necessary for every τ point the algorithm is notvery efficient. However we can generate in the same run diagrams at all times (length)between zero and a τmax chosen large enough to project out the ground state properties.Further, we notice that the one-electron Green’s function (Eq. 3.27) is a particular casecorresponding to n = 0 of n-phonon Green’s function, P n(k, τ) (Eq. 3.29). Therefore,we generate in one run all the possible diagrams with length in the interval (0, τmax) andwith the number of external phonons between 0 and Nmax

ph . An example of a diagramwith one external phonon is given in Fig. 3.4.

When generating the diagrams, the momentum of the internal and external phononpropagators is chosen randomly from a continuum set of values situated in the BrillouinZone. Thus our code addresses the problem in the thermodynamic limit. The length ofthe generated diagrams can also take a continuum set of values in the interval (0, τmax).The Green’s functions, the energy, the effective mass and the phonon distribution arecomputed using the estimators introduced in Appendix 3.6.2.

In Fig. 3.5 we show a typical result of the Diagrammatic QMC algorithm applied toour problem. The sum of all possible diagrams with length τ gives

P (k, τ) =∞∑

n=0

P n(k, τ) =∑

ν

e−τEν(k) τ→∞−→ e−τE(k) (3.38)

Because of the exponentially small probability of large time diagrams, it is more efficientto generate diagrams multiplied by a factor eτµ, where µ is chosen close to the value ofthe polaron energy, E(k). We plotted logarithm of P (k, τ)eτµ and of G(k, τ)eτµ versusimaginary time. Notice that at long time convergence is reached. The extrapolation ofG(k, τ) at zero gives the quasiparticle weight Z0(k). An important remark should bemade about the strong drop seen in P (k, τ) at short time. This is due to the fact that we


0 10 20 30 40 50Imaginary time

-2

-1.5

-1

-0.5

0ln(P( )*exp( ))

ln(G( )*exp( ))

ln(Z

) 0ω = 0.5 tα = 2.645µ = -3.156 tE = -3.155 tZ = 0.1190

µ∗τ

µ∗τ

τ

τ

1 dimensional case

Figure 3.5: A typical example of the Diagrammatic QMC computed Green’s functions.

generate only connected diagrams (diagrams where the phonon propagators are alwaysglued to the electron propagator). The disconnected diagrams have an exponentiallysmall contribution at large time, therefore they can be neglected, but at small time theiromission will result in a strong potential drop which will not allow an efficient samplingfor both long and short time diagrams. This problem can be eliminated using a fictitiouspotential renormalization [8], which means a proper choice of function g(x) in Eq. 2.15.The neglect of the disconnected diagrams makes also possible the normalization

P (k, τ) = G(k, τ) = 1 (3.39)

3.4 Results of the QMC Calculation

Related to low-dimensional physics of strongly correlated materials we study both theone-dimensional and the two-dimensional case. We study the polaron properties as afunction of both electron-phonon coupling and its momentum and look at the influenceof phonon frequency ω0 on these properties. In the subsequent calculations we take thehopping t to be the unit of energy.

3.4.1 Ground State Properties

In this section we focus on the intermediate electron-phonon coupling physics which ischaracterized by the transition from the weak-coupling light state to the strong-couplingheavy one. It was proved mathematically [25, 26] that this transition is analytic, thus noabrupt changes in the polaron properties are expected to happen in the transition region.

3.4. Results of the QMC Calculation 41

0 1 2 3 4 5α

-10

-8

-6

-4

-2

Ene

rgy

(eV

)

1D, = 0.1 t1D, = 0.5 t2D, = 0.5 t2D, = 1.0 t

ωω

ωω

second order PT

second order PT

Figure 3.6: Ground state polaron energy versus electron-phonon coupling in the one-dimensional (diamonds) and two-dimensional (circle) cases. The solid line correspondsin both cases to smaller adiabaticity ratio ω0/zt. The dotted line is the second orderperturbation theory result (Eq. 3.21).

0 1 2 3 4 5α

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Z

1D, =0.1 t1D, =0.5 t2D, =0.5 t2D, =1.0 t

ωωωω

0

Figure 3.7: The quasiparticle weight Z0 (Eq. 3.28) versus electron-phonon coupling inthe one-dimensional (diamonds) and two-dimensional (circle) cases. The solid line corre-sponds in both cases to smaller adiabaticity ratio ω0/zt.


0 1 2 3 4 5α

0

2

4

6

8

10

12

14

16

18

20

1D, =0.1 t1D, =0.5 t2D, =0.5 t2D, =1.0 tω

ωωω

Nph strong-coupling PT

strong-coupling PTstrong-coupling PT

Figure 3.8: The average number of phonons in the phononic cloud versus electron-phononcoupling for the one-dimensional (diamonds) and two-dimensional (circle) cases. Thedotted line are the strong coupling perturbation theory results (Eq. 3.23)

0 1 2 3 4 5α

0

1

2

3

4

5

6

7

8

ln(

m*/

m )

1D, =0.1 t1D, =0.5 t2D, =0.5 t2D, =1.0 t

ωωωω

strong coupling PT

Figure 3.9: Logarithm of the polaron effective mass versus electron-phonon coupling inthe one-dimensional (diamonds) and two-dimensional (circle) cases. The dotted line is thestrong coupling perturbation theory result (Eq. 3.22) corresponding to both (1D,ω0 = 0.5)and (2D,ω0 = 1) cases.


A popular interpretation of the large to small polaron transition is the following [14, 15,27–29]. There are two variational kinds of polaron states, one having the characteristicsof the large polaron described by the weak-coupling perturbation theory and the otherbeing the heavy state which distorts the lattice strongly around itself and correspondingto the strong-coupling regime. In the intermediate coupling regime the two proposedstates are close in energy. The hybridization matrix element between these two statesis different from zero, thus the states mix with each other. Therefore the transitionphysics is a crossover between the large and the small polaron states. One consequenceof this interpretation is the existence of a second polaron state in the crossover region.For the one dimensional case one more polaron state in the intermediate coupling regionwas obtained by Bonca et al. [18] using a technique based on exact diagonalization ofa variational determined Hilbert space, and by A. Mishchenko analyzing our QMC datawith a novel imaginary-time analytic continuation method [8]. The agreement of V.Cataudella et al. ground state polaron variational calculation [14], which assumes thisscenario, with other methods seems to indicate that the crossover assumption is good.However, as we discuss below, this is not the whole story, and, in order to explain thetransition region, one should either consider a crossover of more than two states or evenabandon the crossover interpretation and find something else.

The spectral calculation of the similar Rashba-Pekar model [30] shows more than onestable excited state in the crossover region. A similar situation is encountered even forthe one-dimensional Holstein polaron case when the adiabaticity ratio is small [31], thusdisputing the two states only crossover supposition. Bonca et al. [18] showed that the firstexcited state of the transition region approaches asymptotically the continuum when theelectron-phonon coupling is increased. In the strong-coupling limit this state is a weaklybound state of a localized small polaron and one more phonon. Thus, the phonon numberof this first excited state at large electron-phonon coupling is (see also Eq. 3.23)

N1ph = 〈b†ibi〉 = 〈b†i bi〉+

αzt

ω0

= 1 +αzt

ω0

(3.40)

On the other hand, if the crossover assumption is correct this state should be the con-tinuation of the light polaron to large α and thus should be characterized by a relativelysmall number of phonons and it is evident that this is not happening.

One solution to this problem is to assume a crossover of more than two states. Besidesthe large polaron and the small polaron states some other states above the continuumthreshold can be involved too. The mixing with each other and with the large polaronand small polaron states can be strong in the transition region, resulting in more than onestable excited states. However the transition region physics is not well understood andfurther investigation is required, especially with respect to the excited states properties.One efficient way to determine them is to do the spectral analysis of the data obtainedwith Diagrammatic Quantum Monte Carlo. Nevertheless, besides showing one particularcase which proves the existence of another stable polaron state, our results are strictlyrelated to the ground state properties only.

The ground state energy of the Holstein polaron model as a function of electron-phonon coupling α for different values of ω0 is shown in Fig. 3.6. Notice that the slopechanges around αc = 2. Thus the transition takes place when the lattice deformation


0 5 10 15 20n

0

0.2Z(n

) =2.102 =2.25 =2.89

0 5 10 15 20

0

0.2

0.4Z

(n)

=2.00 =2.645 =3.92

1D, =0.5 t

2D, =0.5 tααα

α

α

ω

α

ωa)

b)

Figure 3.10: Phonon distribution, Z(n) (Eq. 3.30), in the transition region for 1-dimensional (a) and 2-dimensional (b) case at different electron-phonon couplings.

0.0 0.5 1.0 1.5 2.00.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

=2.645α= −3.145

= 0.42−

1D

ω

=0.5 tω

E

EE1

0arbr

itrar

y un

its 0

0

0

Z =0.28

Z = 0.119

10

Figure 3.11: Spectral function (Eq. 3.41) calculated by analytical continuation of QMCdata. One-dimensional case in the transition region. Two stable states exist below thecontinuum which begins at 0.5 t.


energy becomes approximatively equal to the bare electron bandwidth. The smaller isthe adiabaticity ratio the sharper is the transition, but it is always smooth, as it shouldbe. The quasiparticle weight Z0 (Eq. 3.28) is shown in Fig. 3.7. The one-dimensional andthe two-dimensional cases behave differently. In the one-dimensional case the electrongets dressed rapidly when α is increasing and the smaller is ω0 the faster Z0 decreases.In the two-dimensional case, at small α, the dressing is much smaller and independent ofω0. In the transition region Z0 drops rapidly. We notice again that when ω0 is small thetransition is very sharp (resembling a first order transition). Similar conclusions can bedrawn from Fig. 3.8 where the number of phonons in the polaronic cloud is presented. InFig 3.9 the polaron effective mass on a logarithmic scale is shown. Again, the sharpesttransition happens in the two-dimensional case at small phonon frequency. Notice thateven for our largest α the effective mass is not very close to the strong-coupling value,showing that we are still in the intermediate region. However the other properties (energy,average number of phonons) are much closer to the perturbation theory value.

In Fig 3.10 we show how the phononic cloud evolves with the coupling strength inthe transition region. Before the transition, the largest contribution to the polaron stateis the free electron configuration. After transition, at large α, the phonon distributionis Gaussian centered on Nph (Eq. 3.31). A fundamental difference is observed betweenthe one-dimensional and the two-dimensional case. Unlike the 1D case, in the transitionregion the 2D polaron exhibits a phonon distribution with two maxima, one at zero andthe other corresponding to a large phonon number. We observed the same feature evenat higher phonon frequency (at ω0 = 1, not shown), but less pronounced. In the anti-adiabatic limit (large ω0) the two-peak structure disappears. Similar characteristics werealso noticed in other polaron models [30]. This peculiar distribution of phonons in thepolaronic cloud suggests that the transition region polaron is a mixture of a state wherethe free electron configuration has the most significant contribution and another one witha large number of phonons Gaussian distributed. In V. Cataudella et al.’s calculation [14]the variational space consists of two kinds of states, states with phonon distributioncharacteristic of the large polaron and states with phonon distribution characteristic ofthe small polaron. The agreement of their one-dimensional and two-dimensional resultswith the exact ones is remarkable, proving that the polaron at the intermediate couplingis a mixture of large and small polarons. This is true even in the one-dimensional caseand the absence of the zero-maximum in the phonon distribution we believe is due tothe fact that the one-dimensional large polaron state has a small Z0 at the intermediateelectron-phonon coupling, according to the results shown in Fig. 3.7. However the abovestatement does not say anything about the excited states which cannot be described onlyas a superposition of large and small polarons presumably because they are made-up alsowith states from the continuum.

Preliminary calculations show that the number of the stable excited states is stronglydependent of the adiabaticity ratio and lattice dimension. We did not study this problemsystematically. In Fig. 3.11 we show only one example of such calculation, for the one-dimensional case, at the adiabaticity ratio ω0/t = 0.5. The quantity plotted is the spectralfunction at k = 0 defined as:

S(k, ω) =∑

ν

|〈ν|c†k|0〉|2δ(ω − Eν(k)) =∑

ν

Z0νδ(ω − Eν(k)) (3.41)


Two polaron states exist below the continuum, both of them having a pretty large quasi-particle weight (Z0

0,1 > 0.1).

3.4.2 Momentum Dependent Properties

Even for weak electron-phonon interaction the polaron problem is far from being trivial.The difficulty occurs at large momenta where the perturbation theory fails. The best an-alytical results at large momenta are obtained by summing all the non-crossing diagramsin the calculation of the self-energy. However this approximation is still poor underesti-mating the energy correction. Therefore, the exact solution obtained with DiagrammaticQMC is of extreme importance.

In Fig. 3.12 we present the energy dispersion for the one-dimensional polaron at asmall electron-phonon coupling. When the polaron momentum is small, the Rayleigh-Schrodinger perturbation theory works well, even in first order. Close to the momentumvalue where the bare electron energy reaches the phonon energy (ε(k) − ε(0) = ω0), theperturbation theory fails, producing an unphysical peak in the polaron dispersion (seethe dashed line). The QMC results show that, in fact, the polaron energy continues toincrease and the closer it gets to the threshold value E(0) + ω0, corresponding to thecontinuum, the flatter the dispersion becomes. The dispersion is weakly renormalized atsmall momenta until the polaron energy gets close to the continuum threshold from wherethe dispersion becomes suddenly flat. The polaron at large k is a weakly bound state.In the intermediate coupling region the same flattening of the polaron dispersion at largek is seen but now the binding energy is large everywhere in the Brillouin Zone (see theinset).

Not only the dispersion but also the quasiparticle weight dependence on k, presentedin Fig 3.13-a, shows that the polaron at large momentum is fundamentally different fromthe one at small momentum. At small coupling (solid line) and small k the quasiparticleweight is large, the polaron being a lightly dressed electron. From Fig 3.13-b it can beseen that at large k the number of phonons in the polaronic cloud is larger by almost one.Here the quasiparticle weight is vanishing small. The reason is that the polaron becomesmostly a one electron plus one phonon state which is characterized by a zero Z0(k) and aflat dispersion, now the momentum being entirely carried by the dispersionless phonon.However for larger α (dashed line), the number of phonons at large k is larger by morethan one than the corresponding number at k = 0, showing that states with two and morephonons also participate in the electron dressing process.

The two-dimensional polaron behavior is similar. The situation is presented in Fig. 3.14.At the zone boundary the polaron is weakly bound and the dispersion is flat. The quasi-particle weight is large at the zone center and is decreasing rapidly with increasing k.Because we are not in the very small coupling regime as in the previous one-dimensionalcase, the number of phonons at large k is larger by more than one than the correspondingk = 0 number.

In both the one- and the two-dimensional cases we noticed that the number of phononsand the dispersion at large momenta are related. In the extremely weak coupling regimethe dispersion is close to the perfectly flat shape and the number of phonons equalsthe number of phonons at k = 0 plus one. When the coupling is increased the large k


0 0.2 0.4 0.6 0.8 1k/

-2.25

-2

-1.75

-1.5

E(k

)

-2.6

-2.4

1D, =0.5 t

=0.5

ω

α

π

-2 t cos(k)

RS pertu

rbati

on

QMC

E +

0

0ω

=2.0α

π

E(k

)

k/

Figure 3.12: The one-dimensional polaron dispersion at small electron-phonon coupling.The intermediate coupling case is shown in the inset.

0 0.2 0.4 0.6 0.8k/

0

0.2

0.4

0.6

0.8

Z (

k)

=0.5 =2

0 0.2 0.4 0.6 0.8 1k/

0

2

4

6

8

N

(k)

=3.92

π π

0 ph

αα

1D, =0.5 tω

a) b)

α

Figure 3.13: One-dimensional case. The quasiparticle weight (a) and the number ofphonons (b) versus momentum at small (solid line) and intermediate (dashed line)electron-phonon coupling. The phonon number versus k at large coupling is shown withthe dotted line in b).


-4.5

-4.2

-3.9E

(k)

0

0.3

0.6

Z (

k)

0

1

2

3

N

(k)

0ph

(0,0) (π,π) (0,0)(π,0)

E (0) + ω00 a)

b)

c)

2D, =1.69=0.5 t αω

Figure 3.14: a) The two-dimensional polaron dispersion along the principal symmetryaxes of the Brillouin Zone. b) The two-dimensional polaron quasiparticle weight versusmomentum. c) The phonon number of the two-dimensional polaron versus momentum.

0 2 4 6 8n

0

0.2

0.4

0.6

0.8 k=0k=

0 2 4 6 8 10n

0

0.2

0.4

0.6

0.8k=(0,0)k=(0.2 , 0.2 )k=( , )

πππ π

π

1D, 2D,=0.5 t =0.5 tω ω

αα

=0.5=1.69

a) b)

Figure 3.15: Phonon distribution for different values of k. a) One-dimensional case. b)Two dimensional case.


polaron dispersion departs from the perfectly flat shape and the number of phonons keepincreasing with k.

The phononic cloud evolution with k is shown in Fig. 3.15 for one and for two-dimensional case. Notice that the phonon distribution at large k has a strong maximumat one, showing that the one-electron-and-one-phonon states are the most important con-figurations. For the very weak coupling case, which characterizes the one-dimensionalexample, the distribution at large k can be obtained by adding one more phonon to thezero momentum one. It shows that the large k polaron is a bound state between a zeromomentum polaron and a phonon which carries the momentum. In the two-dimensionalexample the phonon distribution still has a maximum at one but now the configurationswith more than one phonon participate also significantly to the large k polaron state for-mation. The qualitative differences noticed between the one-dimensional case (Fig. 3.15-a)and two-dimensional case (Fig. 3.15-b) results from the electron-phonon coupling strengthand not from the dimensionality. We should mention that for the one (two) -dimensionalcase the phonon distribution at other large k values not shown in the figure is practicallythe same with the shown distribution at k = π (k = (π, π)).

In the strong coupling regime, the momentum dependence of the polaron is a con-sequence of the exponentially reduced hopping parameter (Eq. 3.19). Therefore we didnot expect significant changes as in the weak and intermediate coupling cases. The dot-ted line plot in Fig 3.13-b, where the phonon number versus momentum at large α isshown, confirms that. Unlike the weak-coupling regime, the number of phonons is weaklyk dependent here.

For the Frohlich polaron model, analytical calculations [20, 21] confirmed by numericalresults [7] show that, unlike in our case, there is a critical momentum kc above which thepolaron state does not exist. We found that for all k in the Brillouin Zone there is astable polaron state. However this statement can not be proved numerically for the entireparameter space (α, ω0) because, even if true, at very small electron-phonon coupling thepolaron binding energy at large k will be so small that our method can not resolve it. TheFrohlich model is special because the electron band, k2/2m, is infinite. Other differencefrom our model is the electron-phonon coupling matrix form, V (q) ∝ α/q, which is smallwhen large momentum phonon scatter (and they enter with predilection in the formationof large momentum polarons), but presumably this is not so important for the existenceof the kc end point. We believe a relevant fact is that in the extreme case of an infinitebare electron bandwidth the derivative ∂E(k)/∂k is zero at kc. The Frohlich polaron nearkc is a weakly bound state of the zero momentum polaron with one more phonon [8] .It can be said that for k larger than kc the polaron band is perfectly flat and has zerobinding energy (which indicates a zero momentum polaron and a free phonon). Based onour results, we believe that when the infinite band condition is released, the zero bindingenergy at large momenta will become finite and the perfectly flat dispersion will transforminto a very slowly increasing one. Of course, for a real proof an analytical approach shouldbe taken.


3.5 Conclusions

In this chapter we have calculated the Holstein polaron properties using the DiagrammaticQuantum Monte Carlo. The algorithm which we have discussed in detail here is anexample of the general technique introduced in Chapter 2. It has been proved to bevery efficient in calculating both the ground state (k = 0) and the momentum dependentpolaron properties.

We have studied the one-dimensional and the two-dimensional polaron transition fromthe weak electron-phonon coupling regime to the strong electron-phonon coupling one.The transition is always continuous, but sharp when the phonon frequency ω0 is small.It takes place around the critical electron-phonon coupling αc = 2 at which the latticedeformation energy becomes equal to the bare electron bandwidth. The weak couplingpolaron is a light state carrying a small number of phonons which results in a weak latticepolarization extended over a large distance around the electron. The strong couplingpolaron is a heavy state with an exponentially large effective mass and a large numberof phonons. The electron polarizes the lattice strongly over a small distance aroundits position. The dimensionality has a strong influence on the weak coupling polaronproperties. In the one-dimensional case, the electron gets dressed very rapidly as thecoupling increases, resulting in a small quasiparticle weight before the transition. In thetwo-dimensional case where the polaron “before the transition” is characterized by a largequasiparticle weight which drops abruptly when the transition takes place. The evolutionof the phononic cloud with α shows that the transition region polaron is a mixture of largepolaron like states with small polaron like ones. The involvement of more than two statesis required to explain the number and the properties of the excited stable states whichappear in the transition region. However for a complete understanding of the transitionregion physics a systematic study of the excited states properties is needed, and a possibleand promising way is the spectral analysis of Diagrammatic Quantum Monte Carlo data.

In the weak-coupling region the polaron at large momentum is fundamentally differentfrom the one at small momentum. At small α the polaron changes from the almost freeelectron state at k = 0 to the weakly bound zero momentum polaron with one phononstate at large k. Consequently the large momentum polaron will be characterized by aflat dispersion, an almost vanishing quasiparticle weight and a number of phonons largerby one than the corresponding k = 0 polaron. At larger α (intermediate coupling regime)the large momentum polaron contains a significant fraction of states with two and morephonons, therefore the number of phonons will be larger by more than one than thenumber of phonons at k = 0 and the the dispersion will deviate from the perfectly flatshape, but the phonon distribution at large k still has a strong maximum at one. Thedimensionality does not play an important role on the momentum dependent polaronproperties. In the strong-coupling regime the polaron properties are weakly momentumdependent as a consequences of the exponentially small effective hopping integral.

Unlike the Frohlich polaron where a critical momentum kc was found, above whichthe polaron state cannot exist, in the Holstein model we found stable polaron states atall k in the Brillouin Zone. Even though numerically it is impossible to prove the abovestatement at extremely small α and ω0, our calculation suggests that this is indeed trueand the kc end point in Frohlich model is a consequence of the infinite electron bandwidth.

3.6. Appendix 51

τ τ’0 τ

k

τ1

τ1

0

k

21

Figure 3.16: Changing the diagram length

3.6 Appendix

3.6.1 Diagrams Updates

The weight of every diagram can be determined using a simple set of rules. These rulescan be found by inspecting directly Eq. 3.37. This calculation is easy but tedious, andwill result in the following: (i) every electron-phonon vertex corresponds to a term g dτ ,(ii) every phonon propagator corresponds to a term dq, (iii) every electron propagator of

length τ and moment k corresponds to a term e−ε(k)τ and (iv) every phonon propagatorof length τ corresponds to a term e−ω0τ . An example is given in Fig. 3.3.

In order to cover all the possible configurations, we need at least three types of updateprocedures: updates which change the diagram length (time), updates which add/removeinternal phonons and updates which add/remove external phonons.

Changing the diagram length These updates do not change the integration multi-plicity. We propose the following procedure which changes the length of the last (the oneat the right end) interval. An interval is a part of the diagram between two successiveinteraction vertices or between the last (first) vertex and the final (initial) end. The situ-ation is illustrated in Fig. 3.16. If we propose the end time of last interval to change fromτ to τ ′ the ratio between the final and the initial diagram weight will be

D2

D1

= e−(ε(k) + nω0 − µ)(τ ′ − τ) (3.42)

where n is the number of external phonons. We have more options. One is to chooseτ ′ uniformly random between τ1 and τmax and accept the change with the probabilityp = D2/D1, but a more efficient one is to choose τ ′ with the probability

P(τ ′) = e−(ε(k) + nω0 − µ)(τ ′ − τ)dτ ′ (3.43)

between τ1 and τmax which results in an acceptance ratio of 1.

Adding/Removing internal phonons These updates change the integration mul-tiplicity. The situation is illustrated in Fig. 3.17. According to the general discussionfrom Chapter 2 two distinct related subroutines are necessary: one for adding phononpropagators and the other for removing them.

Adding one phonon propagator (1 −→ 2). 1) Choose an initial τ1 and afinal τ2 time for the phonon propagator, with a probability P(τ1, τ2)dτ1dτ2. 2) Choose


a momentum q for the phonon propagator, with a probability P(q)dq. 3) The diagramsweight ratio will be

D2/D1 = g2dτ1dτ2dq∑

i

e−(ε(ki − q) + ω0 − ε(ki))∆τi (3.44)

where a sum over all intervals i of length ∆τi between τ1 and τ2 is performed. Accordingto Eq. 2.20 and Eq. 2.22 the change will be accepted with probability

p =prem

padd

g2∑

i

e−(ε(ki − q) + ω0 − ε(ki))∆τi niph + 1

P(τ1, τ2)P(q)(3.45)

where niph is the number of internal phonon propagators (which will become niph + 1 ifthe update is accepted) and padd (prem) is the calling probability for the internal phononaddition (removal) subroutine.

Removing one phonon propagator (2 −→ 1). 1) Choose a phonon propagatorwith a probability 1/niph. 2) The diagrams weight ratio is

D1/D2 =1

g2dτ1dτ2dq

∑i

e−(ε(ki + q)− ω0 − ε(ki))∆τi (3.46)

where τ1 (τ2) is the initial (final) phonon with momentum q propagator time. Accordingto Eq. 2.21 and Eq. 2.22 the change will be accepted with probability

p =padd

prem

1

g2

∑i

e−(ε(ki + q)− ω0 − ε(ki))∆τiP(τ1, τ2)P(q)

niph

(3.47)

The probability functions P(τ1, τ2) and P(q) are arbitrary and can be tuned to max-imize the code efficiency. We made the following choices. First we choose τ1 randomlybetween 0 and τ and afterwards we choose τ2 > τ1 with a probability ∝ e−ω0(τ2−τ1). Withrespect to the phonon momentum q, we chose it uniformly random in the Brillouin Zone.

Adding/Removing external phonon The procedures of adding and removing ex-ternal phonon (see Fig. 3.18) are similar to the ones regarding the internal phonon. Thedifference is that now τ1 (τ2) represents the connection time of the left (right) end externalphonon propagator and the summation over i in Eq. 3.44 and in the similar subsequentequations means summation over all intervals from left end (zero time) to τ1 and from τ2to τ . The number of internal phonons niph should also be replaced with the number ofexternal phonons n.

1 2

p k p kq

τ τ1 2τ τ

Figure 3.17: Adding/Removing an internal phonon

3.6. Appendix 53

1τ 2τ

1 2

k k−p

p q

k

ττk−p k−q

q

k

p

k−q

Figure 3.18: Adding/Removing an external phonon

The updates introduced above constitute a minimal set for ensuring the ergodicitycondition. Other updates can be imagined. For example we used three more subroutines,one which changes the time of the interaction vertices, one which interchanges two verticesand another which stretches the diagram. We did not present them here. However,introducing them turns out to be important, the convergence being improved by oneorder of magnitude.

3.6.2 Estimators

Our algorithm generates all the possible diagrams with the length between 0 and τmax

and with the external phonon number between 0 and Nmaxph . Suppose we want to measure

a certain quantity A. Every diagram (i) generated according to its weight P ni(k, τi)contributes to A with a certain amount given by the estimator a(i), such as in the end

A =1

M

M∑i=1

P ni(k, τi)a(i) (3.48)

We used estimators to compute the Green’s function, the polaron energy, the phonondistribution in the polaronic cloud, the polaron phonon number and the polaron effectivemass.

Green’s function estimator The n0-phonon Green’s function at time τ0 is

P n0(k, τ0) =1

M

1

Γ

M∑i=1

P ni(k, τi)δ(τi − τ0)δni,n0 (3.49)

where Γ is

Γ =1

M

M∑i=1

δ(τi − τ0)δni,n0 (3.50)

Therefore it seems that we should define the estimator for P n0(k, τ0) as (1/Γ) δ(τi −τ0)δni,n0 . However the diagram time can take continuous values, which means a infinitelysmall probability for the diagram time, τi, to be exactly τ0. Consequently the statistics(and also Γ) will be infinitely small and the measurements are impossible. In order toovercome this difficulty we write the Green’s function in a different way

P n0(k, τ0) =1

M

1

Γ

M∑i=1

P ni(k, τi)P n0(k, τ0)

P ni(k, τi)Θ(a− |τi − τ0|)δni,n0 (3.51)


where

Γ =1

M

M∑i=1

Θ(a− |τi − τ0|) = 2a (3.52)

is a constant equal to the area of the (arbitrary chosen) radius a time window in whichthe statistics are collected. Θ(x) is the Heaviside step function. The Green’s functionestimator will be

pn0(k, τ0) =1

2a

P n0(k, τ0)

P ni(k, τi)Θ(a− |τi − τ0|)δni,n0 (3.53)

This estimator allows us to collect statistics at all the times in the window (τi− a, τi + a)from a particular diagram with time τi, making the algorithm very efficient.

Energy estimator In order to determine an estimator for the energy, let’s notice thatat large τ

P (k, (1 + λ)τ)

P (k, τ)−→ e−λE(k)τ (3.54)

The estimator for the left hand side of Eq. 3.54 is

Q(i, λ) =P ni(k, (1 + λ)τi)

P ni(k, τi)= (1 + λ)Nv

∏

l

e−λεl(k)∆τl∏

j

e−λω0∆τj (3.55)

where l (j) counts the electron (phonon) propagators of time ∆τl (∆τj), and Nv = 2(n+niph) is the total number of vertices. Up to the first order in λ, Eq. 3.55 becomes

Q(i, λ) = 1 + λ(Nv −∑

l

εl(kl)∆τl −∑

j

ω0∆τj) (3.56)

The right hand side of Eq. 3.54 is, up to first order in λ, equal to 1−λE(k)τ which makesthe polaron energy estimator to be

e(i) =1

τi(−Nv +

∑

l

εl(kl)∆τl +∑

j

ω0∆τj) (3.57)

The measurements should be taken only on the large time diagrams where the asymptoticregime is reached.

Inverse of the effective mass estimator Analogously, let’s notice that at large timeand for small λ

P (λ, τ)

P (0, τ)−→ e−(E(λ)− E(0))τ = e−λ2τ/2m∗

(3.58)

The estimator for the left hand side of Eq. 3.58 is

Q(i, λ) =∏

l

e−(εl(kl + λ)− εl(kl))∆τl (3.59)

3.6. Appendix 55

Expanding Eq. 3.55 up to second order in λ we get

Q(i, λ) = 1− λ2(z∑

δ=1

∑

l

(t cos(klδ)− 2t2 sin2(klδ))∆τl) . (3.60)

The inverse of the effective mass estimator will result in

me

m∗ (i) =1

τi

z∑

δ=1

∑

l

(2t cos(klδ)− 4t2 sin2(klδ)) , (3.61)

where me is the free electron mass.

n-phonon probability estimator The estimator for Zn0 (k) is

zn(i) = δni,n (3.62)

For n = 0 Eq. 3.62 gives the quasiparticle weight estimator.

Total number of phonon estimator is

nph(i) = nδni,n (3.63)

In order to determine the total number of phonons accurately, Nmaxph should be taken

large enough such that the contribution of diagrams with a larger than Nmaxph number of

phonons is exponentially small and therefore negligible.


References

[1] L. Landau, Z. Phys 3, 664 (1933).

[2] E. de Mello and J. Ranninger, Phys. Rev. B 55, 14872 (1997).

[3] M. Capone et al., Phys. Rev. B 56, 448 (1997).

[4] A. S. Alexandrov, Phys. Rev. B 49, 9915 (1994).

[5] G. Wellein and H. Fehske, Phys. Rev. B 56, 4513 (1997).

[6] G. Wellein and H. Fehske, Phys. Rev. B 58, 6208 (1998).

[7] N. Prokof’ev and B. Svistunov, Phys. Rev. Lett. 81, 2514 (1998).

[8] A. Mishchenko, N. Prokof’ev, A.Sakamoto, and B. Svistunov, Phys. Rev. B 62, 6317(2000).

[9] H. De Raedt and A. Langedijl, Phys. Rev. B 27, 6097 (1982).

[10] P. E. Kornilovitch, Phys. Rev. Lett. 81, 5382 (1998).

[11] P. E. Kornilovitch, Phys. Rev. B 60, 323 (1999).

[12] P. E. Kornilovitch, Phys. Rev. B 56, 4494 (1997).

[13] A. H. Romero, et al., Phys. Rev. B 59, 13728 (1999).

[14] V. Cataudella, G. D. Filippis, and G. Iadonisi, Phys. Rev. B 62, 1496 (2000).

[15] V. Cataudella, G. D. Filippis, and G. Iadonisi, Phys. Rev. B 60, 15163 (1999).

[16] G. Iadonisi, V. Cataudella, G. D. Filippis, and D. Ninno, Europhys. Lett 41, 309(1998).

[17] E. Jeckelmann, et al., Phys. Rev. B 57, 6367 (1998).

[18] J. Bonca, S. A. Trugman, and I. Batistic, Phys. Rev. B 60, 1633 (1999).

[19] J. W. Negel and H. Orland, Quantum Many-Particle Systems (Addison-Wesley,1988).

[20] G. D. Whitfield and R. Puff, Phys. Rev. 139, A338 (1965).

[21] D. M. Larsen, Phys. Rev. 14, 697 (1966).

[22] I. G. Lang and Y. A. Firsov, Sov. Phys. JETP 16, 1301 (1963).

[23] C. Kittel, Introduction to solid state physics (New-York, 1996).

[24] F. Marsiglio, Physica C 224, 21 (1995).

3.6. Appendix 57

[25] H. Lowen, Phys. Rev. B 37, 8661 (1988).

[26] B. Gerlach and H. Lowen, Rev. Mod. Phys. 63, 63 (1991).

[27] D. M. Eagles, Phys. Rev. 145, 645 (1966).

[28] D. M. Eagles, J. Phys. C 17, 637 (1984).

[29] D. M. Eagles, Phys. Rev. 181, 1278 (1969).

[30] A. Mishchenko, N. Nagaosa, N. Prokof’ev, A.Sakamoto, and B. Svistunov, Phys.Rev. B 66, 020301 (2002).

[31] A. Mishchenko, private communications.

58

Chapter 4

Hubbard-Holstein Bipolaron

4.1 Introduction

Aside form dressing the charge carriers, the electron-phonon interaction also introduces aneffective attraction between them. As the BCS theory shows [1], even a small attractionmakes a Fermi liquid instable towards electron pairing, resulting in superconductivity.The pairs (Cooper pairs) are formed in momentum space, having a large spatial extent(of the order of several thousand lattice constants). On the other hand, when the electron-phonon interaction is large, two electrons (or holes) will form a small size pair in real space,called bipolaron.

Bipolaronic effects (real space pairing) were found in many materials like transitionmetal-oxides [2–5], superconducting materials [6] and conjugated polymers [7]. The dis-covery of high Tc superconductors renewed the interest in the study of electron-phononinteraction at intermediate and large coupling and special theoretical attention was givento the scenarios where electrons form light bipolaron states which suffer Bose-Einsteincondensation, leading to superconductivity [8].

In correlated systems both electron-electron and electron-phonon interactions are im-portant. The first is repulsive and instantaneous and the second induces a retarded at-traction between electrons. The retarded nature of the phonon mediated attraction is ex-tremely important, the time delay stabilizing the pair formation even when the Coulombrepulsion is large. The strength of the electron-phonon coupling in high Tc and otherstrongly correlated materials is situated in the intermediate region, where the applicabil-ity of weak or strong coupling perturbation theories is not valid. Therefore solutions forHamiltonian models which consider both the electron-electron and the electron-phonon in-teraction in this difficult region of parameters space are desirable. The Hubbard-HolsteinHamiltonian, which we address in this chapter, is one of the most popular model studiedin the literature. Even if it is not the best candidate to explain the physics of cuprates,producing bipolarons with extremely large effective mass, its solution is important forunderstanding the competition between the phonon induced electron-electron attractionand the electron-electron Coulomb repulsion. For the one dimensional case the problemwas satisfactory solved by Bonca et al. [9] with a method which uses the exact diagonal-ization technique on a variationally determined Hilbert subspace. Other one-dimensionalcalculations were based on variational methods [10] and density-matrix combined with

59

60 Chapter 4. Hubbard-Holstein Bipolaron

Lancszos diagonalization technique [11, 12]. The two-dimensional case was investigatedin the adiabatic approximation [13, 14] and with variational methods [15].

In this chapter we present the calculation of the two-dimensional Hubbard-Holsteinbipolaron based on a Diagrammatic Quantum Monte Carlo algorithm. To our knowledge,this is the first two-dimensional calculation which considers dynamical phonons and doesnot imply any artificial truncation of the Hilbert space. The idea of the algorithm isto compute the imaginary time two-particle Green’s function and to extract informationabout the bipolaron state from the Green’s function behavior at long imaginary time,where the ground state of the system is projected out. An algorithm based on the sameidea was used in [16] for exciton problem calculation, but in contrast to that situationwhere the calculation was done in momentum space, we work in direct space (site) rep-resentation, the basis consisting of Wannier orbitals and phonons at each site. In thisway we manage to avoid the sign problem which would appear in the momentum repre-sentation when the Coulomb repulsion is introduced. The code can be easily adapted toinclude longer range electron-phonon or electron-electron interaction and to study modelsmore suitable to the cuprates, as for example the extended Hubbard-Holstein model [17].The disadvantage is that in this basis the momentum dependence of different quantitiesis difficult or sometimes impossible to compute. We work on a square lattice of 25 × 25sites with periodic boundary conditions which is large enough for having negligible finitesize errors. There are no other truncations of the Hilbert space.

The Hubbard-Holstein Hamiltonian is

H = −t∑

〈ij〉,σ(c†iσcjσ +H.c) + U

∑i

ni↑ni↓ +

+ω0

∑i

b†ibi + g∑i,σ

niσ(b†i + bi) (4.1)

Here c†iσ(ciσ) is the creation (annihilation) operator of an electron with spin σ atsite i, and analogues b†i , bi are phonon creation and annihilation operators. The firstterm describes the nearest-neighbor hopping of the electrons, and the second the on-siteCoulomb repulsion between two electrons. The lattice degrees of freedom are describedby a set of independent oscillators at each site, with frequency ω0. The electrons couplethrough the density niσ = c†iσciσ to the local lattice displacement xi ∝ (b†i + bi) with thestrength g. This Hamiltonian is a tight-binding model together with an on-site Coulombrepulsion term and an on-site electron-phonon interaction term. The Holstein and theHubbard models are limiting cases for U = 0 and g = 0 respectively.

We are going to address the pairing problem as a function of both Coulomb repulsionand electron-phonon interaction by studying two electrons on a two-dimensional lattice.

4.2 Analytical Approaches

Similar to the Holstein polaron problem, perturbation theory can be used to calculate theweak and the strong electron-phonon interaction regimes. In this section we present theresults for these two cases.


k

p

q

p+q

k−q

Figure 4.1: Effective phonon induced interaction in second order perturbation theory.

4.2.1 Weak-Coupling Perturbation Theory

For g = 0, the ground state will be formed by two electrons with zero momentum movingfreely through the lattice. When the electron-phonon interaction is switched on, twothings happen. First, the electrons (at the bottom of the band) get lightly dressed whichresults in an increase of their effective mass, and second the electron-phonon interactionintroduces an effective attraction between the electrons. Up to second order in g theeffective interaction induced by the phonon is proportional to the phonon propagator

V effph (ω) = g2D(q, ω) = − 2g2ω0

ω20 − ω2

(4.2)

as can be seen from Fig. 4.1. This is a retarded interaction and attractive at small fre-quency (for ω < ω0). In a two-dimensional lattice an attractive interaction will cause theelectrons to form a bound pair. A similar situation happens even in a three-dimensionalcase with a Fermi surface present, the instability towards electron pairing leading to thesuperconductivity [1].

In our model the attractive phonon induced interaction competes with the Coulombrepulsion, which results in a total effective interaction

V eff (ω) = U − 2g2ω0

ω20 − ω 2

(4.3)

When the electron-phonon coupling is increased the situation gets rapidly complicated,and any calculation should consider the renormalization of the electron-phonon interactionvertex. Migdal’s theorem [18] applied in classical superconductivity theory is not validhere because of the absence of the Fermi sea ∗.

In the antiadiabatic limit (ω0 −→∞) where the ions are considered light and able tofollow instantaneously the motion of the electrons, the effective interaction (Eq. 4.3) isinstantaneous too

V eff = U − 2g2

ω0

(4.4)

and the situation can be described by a pure Hubbard model.

∗Migdal’s theorem shows that the vertex corrections are of order ω0/εF due to Pauli exclusion principlewhich blocks the electron-phonon scattering inside the Fermi sea.


4.2.2 Strong-Coupling Perturbation Theory

When g is large, one can apply perturbation theory with respect to the hopping partof the Hamiltonian. The last three terms in Eq. (4.1) are diagonal in the rotated basis

obtained by applying the unitary operator eS [19] where

S = − g

ω0

∑i,σ

niσ(b†i − bi) (4.5)

The transformed phonon operators are

bi = bi +∑

σ

g

ωniσ (4.6)

This is the same transformation we applied for Holstein polaron model (Eq. 3.8), theonly difference being the summation over the spin index σ. All the other transformedoperators are identical with the ones defined in Section 3.2.2.

The Hamiltonian written in the new basis is

H = Ht +H0 (4.7)

with

H0 = ω0

∑i

b†i bi −g2

ω0

∑i,σ

ni,σ + (U − 2g2

ω0

)∑

i

ni↑ni↓ (4.8)

andHt = −t

∑

〈ij〉,σ(c†iσ cjσX

†iXj +H.c) (4.9)

The dimensionless electron-phonon coupling constant is defined in the same way as inthe Holstein polaron model (Eq. 3.18),

α =g2

2ω0t(4.10)

For identical reasons with the ones given for Holstein polaron model (Section 3.2.2) theelectron hopping implies a change of the ions equilibrium position and it is exponentiallyreduced (see Eq. 3.19). The effective on-site interaction between electrons is

U1 = U − 2g2

ω0

≡ U − 2Ep (4.11)

(the same as in Eq. (4.4)), and in the antiadiabatic limit (when ω0, g −→∞, g/ω0 −→ 0and 2g2/ω0 is finite) Xi = 1 and the model can be mapped again in a pure Hubbard one.

It is clear that for negative U1 the electrons form a bound state. But even for positiveU1 a stable bipolaron state can exist, as second order perturbation theory shows.

Let’s consider the case of large U which results in U1 > 0. The ground state of H0 isformed by the degenerate states

|ai〉 = c†i↑c†i+a↓|0〉, a 6= 0 (4.12)


HtHtEp

Ht HtEp

Ht HtHt

2Ep

Ht HtHt

HtHt

U

a)

b)

c)

d)

e)

pc1

c2

e1

e2

U−E

Figure 4.2: Second order processes which determine the elements of matrix T. The smallhorizontal lines represent the lattice sites. If an electron is on a particular site then theequilibrium position of the ion at that site will be changed. In the large g limit the corre-

sponding matrix elements are: a) − t2

Epe−g2/ω2

0 . b) − t2

Epe−g2/ω2

0 . c) − t2

U − Epe−g2/ω2

0 .

d) − t2

2Ep. e) − t2U .

The meaning of this notation is that the electron with spin ↓ is sitting at the distance“a” from the residing at site “i” electron with spin ↑. “a” can take all the possible valuesexcept 0. In first order perturbation theory, the calculation of the matrix 〈ai | Ht | bj〉is similar to the calculation of Eq. 3.19 and results in an exponentially reduced nearest-neighbor hopping. Second order perturbation theory stabilizes the bipolaron states. Itreduces to diagonalizing the operator

T = Ht1

E0 −H0

Ht (4.13)

on the subspace spanned by all the degenerate states of H0. The processes which cantake place are shown schematically in Fig. 4.2. We can classify them in two classes. First,processes like in Fig. 4.2-a, -b and -c, where the final lattice configuration is different fromthe initial one. This results in an exponential reduction of the matrix elements, so that


we can neglect them in first approximation. Second, processes like the ones in Fig. 4.2-d and -e where the initial and the final lattice configuration is unchanged. They arenot exponentially reduced. In Fig. 4.2-d an electron hops on a neighboring site withoutcarrying the lattice deformation around it and afterwards comes back. The energy of theintermediate state is 2Ep, because it contains a site with deformation and without electron,and a site with an electron and without deformation (see also Eq. 3.20). The gain in energyis −t2/2Ep. This process has contribution to the diagonal elements only. In Fig. 4.2-eone electron hops without carrying the lattice deformation on the neighboring site whichis occupied by the other electron. The intermediate state contains a doubly occupied sitewhich has a deformation corresponding to only one electron and a site with deformationand without electron, therefore the energy of this state is U1 + 2Ep = U . The final statecan be identical with the initial state (Fig. 4.2-e1), and the process contributes to thediagonal elements Tδ,δ, or the final state can have the electrons interchanged (e.g.: fromthe initial |↑, ↓> to the final |↓, ↑> - Fig. 4.2-e2), and this process contributes to the non-diagonal elements Tδ,−δ. δ means nearest-neighbor here. The energy gain correspondingto each of the two processes shown in Fig. 4.2-e is −t2/U .

Neglecting the exponentially reduced terms, what remains is the diagonal terms andthe off-diagonal ones which connect the states | δ〉 and | −δ〉.

Ta,a = 8× (− t2

2Ep) a 6= 0, 1

Tδ,δ = 6× (− t2

2Ep) + 2× (− t2U )

Tδ,−δ = 2× (− t2U ) δ = 1

(4.14)

Solving the secular equation, we find the condition for the bipolaron existence to be

U < 4Ep (4.15)

and the bipolaron binding energy

∆b = − t2

Ep

+4t2

U(4.16)

Notice that even though the bipolaron exists up to a large value of U it is a weakly boundstate when U1 > 0. The energy given by Eq. 4.16 corresponds to a double degeneratesinglet state formed by two electrons siting on nearest-neighbor sites. One state is a singletalong the X direction and the other along the Y direction. The physical interpretation isstraightforward. In distinction to Hubbard model where the exchange energy can neverwin over the kinetic energy and therefore cannot bind two electrons, here the interactionwith phonons results in a strong reduction of the electron bandwidth but not of theexchange energy because it implies virtual transitions of electrons on double occupied siteswithout carrying the lattice deformation with them. Therefore now the exchange energycan easily win and produce singlet bound states. However there is another effect whichintroduces an effective repulsion between two nearest-neighbor electrons and wins over theexchange energy when U ≥ 4Ep. A virtual transition of an electron to an empty nearest-neighbor site without carrying the lattice deformation will lower its energy by −t2/2Ep


(Fig. 4.2-d). But if the nearest-neighbor site is occupied by the other electron this processis not possible resulting in an effective repulsion of t2/Ep between two nearest-neighborelectrons. Therefore Eq. 4.16 reflects the competition between this effective repulsion andthe exchange attraction equal to −4t2/U .

When the processes shown in Fig. 4.2 -a and -b are taken into account the degeneracyof the two singlets is lifted, and two states are formed. This results in a ground state withs-wave (A1g) symmetry and another state with d-wave (B1g) symmetry. It should also bementioned that if a positive next-nearest-neighbor hopping t′ is introduced in the modelthe d-wave symmetry state will be stabilized and it can become the ground state.

Let’s summarize the strong coupling regime physics, neglecting at the beginning theexponentially reduced terms. When U is small the ground state energy is U − 4Ep andconsists by two electrons siting on the same site. The first excited state contains one morephonon and has an energy U −4Ep +ω0 (this is a N degenerate state because the phononcan be at any site). When U is increased and U − 4Ep + ω0 becomes larger than −2Ep

(which is the zero order energy of two electrons staying on different sites), the first excitedstate is a double degenerate nearest-neighbor singlet. When the hopping is switched onthe low-energy physics can be described by the Hamiltonian

H = −teff

∑

〈i,j〉,σ(c†iσcjσ +H.c)+J

∑

〈ij〉(SiSj−ninj

4)+V

∑

〈i,j〉ninj +U1

∑i

ni↑ni↓+H ′ (4.17)

with the hopping teff = t e−g2/ω20 , the exchange J = 4t2

U , the nearest-neighbor repulsion

V = t2

Epand the on-site interaction U1 = U − 2Ep. H ′ describes the processes shown

in Fig. 4.2 -a, -b and -c. Their magnitude is eitherteff

Epor

teff

U − Ep(see the caption of

Fig. 4.2), thus much smaller than teff . In the literature [9, 13, 14] the bipolaron withthe electrons located on the same site is called S0, and the one with the electrons locatedon nearest-neighbor sites S1. The Hamiltonian (4.17) describes the transition from S0to S1 bipolaron in strong coupling regime. For small (negative) U1 the order of thelowest energy states is: s-wave S0, s-wave S1, d-wave S1. When U1 increases the S0state starts mixing with s-wave S1 state. The mixing between the S0 and the s-waveS1 states is of order of teff , and the splitting between the s-wave and the d-wave S1states is given by H ′, thus being much smaller. This makes the order of low-energy statesto become: linear combination of s-wave S0 and S1, d-wave S1, linear combination ofs-wave S0 and S1. For larger U only two bound states exists: s-wave S1 and d-waveS1. In conclusion, the ground state evolves analytically from S0 bipolaron to s-wave S1bipolaron with increasing U . The situation is different for the first excited state. Hereat a critical value of U , a nonanalytical transition takes place, and the first excited statechanges from s-wave symmetry to d-wave symmetry.


4.3 Diagrammatic Quantum Monte Carlo Algorithm

4.3.1 General Approach

The general idea is the same as the one introduced for Holstein polaron, namely to projectout the ground state of the system using the evolution operator in imaginary time repre-sentation (see Eq. 3.25 and Eq. 3.26).

The total momentum K is a quantity which is conserved in our problem. Thereforewe can obtain the lowest energy in the K channel by calculating

P n(K, τ) =∑

k,q1,...qn

〈(K − k − q1 − ...− qn)↓, k↑; q1, ..., qn|e−τH |(K − k − q1 − ...− qn)↓, k↑; q1, ..., qn〉

−→∑

k,q1,..qn

|〈(K − k − q1 − ...− qn)↓, k↑; q1, ..., qn|ν0K〉|2e−τE(K)

(4.18)

at large τ . Here |k1↓, k2↑; q1, q2, ..., qn〉 is a state with two electrons, one with momentumk1 and spin down and the other with momentum k2 and spin up, and with n phonons withmomentum q1, q2,... and qn respectively. |ν0K〉 is the ground state (the state with thelowest energy) in the K channel. The calculation of P n(K, τ) gives both the bipolaronenergy and the n-phonon configuration probability in the bipolaron state.

For reasons related with the sign problem (which are going to be explained in Sec-tion 4.3.3), we choose to calculate P n(K, τ) in real space representation

P n(K, τ) =1

N

∑

i,x,l1,l2,...ln

eiKx〈i|e−τHTx|i〉 (4.19)

where|i〉 ≡ |i↓, (i+ a)↑; i+ l1, i+ l2, ..., i+ ln〉 (4.20)

is a state with a spin down electron at site i, a spin up electron at site i+ a and phononsat sites i+ l1, i+ l2, ... and i+ ln, and

Tx|i〉 =

|i+ x↓, (i+ x+ a)↑; i+ x+ l1, i+ x+ l2, ..., i+ x+ ln〉(4.21)

is the state |i〉 translated with the vector x. Another quantity which is conserved is thetotal spin. We have two spin channels, the singlet and respectively the triplet. For thesinglet we calculate

P ns (K, τ) =

1

N

∑

i,x,l1,l2,...ln

eiKx〈is|e−τHTx|is〉 (4.22)

with|is〉 ≡ |(i, i+ a)s; i+ l1, i+ l2, ..., i+ ln〉 (4.23)


where (i, i + a)s is the singlet state with electrons at sites i and i + a. Similar equationscan be written for the triplet channel.

The Diagrammatic Quantum Monte Carlo algorithm will generate stochastically termsof the form

Gij(τ) = 〈i|e−τH |j(i, x)〉 (4.24)

where |i〉 is a general state as in Eq. 4.20 with two electrons at an arbitrary distancefrom each other and with an arbitrary number of phonons. The state |j(i, x)〉 can bea translation of |i〉 with an arbitrary vector x, i.e. |j〉 = Tx|i↓, (i + a)↑; phonons〉, or atranslation with the vector x of a state obtained from |i〉 when the electrons positions areinterchanged, i.e. |j〉 = Tx|(i+ a)↓, (i)↑; phonons〉.

The value of an observable A in a particular K and S channel is

A(K) =1

M

∑m

eiKx(m)gS(m)a(m) =

=

∑m e

iKx(m)gS(m)w(m)a(m)∑mw(m)

(4.25)

In Eq. 4.25 we sum over all m generated configurations with the weight w(m). M is thetotal number of measurements, x(m) is the translation vector which correspond to theconfiguration m, a(m) is the estimator of A and gS(m) is the factor which separates thetriplet from the singlet. For singlet gS(m) is 1 when electrons are on the same site and1/2 otherwise. For triplet gS(m) is zero when the electrons are on the same site, andotherwise, 1/2 when |j〉 is a translation of |i〉 and −1/2 when |j〉 is a translation of a stateobtained from |i〉 by interchanging the electrons position.

4.3.2 Implementation

We start with the Hamiltonian (4.1) and consider

H0 = ω0

∑i

b†ibi + U∑

i

ni↑ni↓ (4.26)

as the noninteracting part of the Hamiltonian. Notice that H0 is diagonal in real spacerepresentation.

Writing the evolution operator as in Eq. 3.34, Eq. 4.24 becomes:

Gij(τ) = e−εiτ∞∑

n=0

(−1)n

n!

∫ τ

0

∫ τ

0

...

∫ τ

0

dτ1...dτnT [〈i|H1(τ1)H1(τ2)...H1(τn)|j〉] (4.27)

As in the Holstein polaron case, we reduced the calculation of Gij(τ) to a series ofintegrals, with an ever increasing number of integration variables. The general ideas ofDiagrammatic Quantum Monte Carlo (Section 2) can be applied. It is easy to showthat every term in Eq. (4.27) can be represented by a diagram and a set of simple rulescan be derived to determine the diagrams weight. Typical examples of such diagrams arepresented in Fig. 4.3. Aside from a translation, the electronic configuration at the diagram


Tx

Tx

b)

0a)

τ

Figure 4.3: a) A typical diagram which represents a term in Eq. (4.27). The solid line(dashed line) represents a spin up (down) electron. The wavy line is a phonon propagator.The end at time τ is a translation with the vector x of the end at time 0. b) The finalstate is a translation of the state obtained from the initial state by interchanging theelectrons position. We generate only diagrams where the final state is a translation of theinitial state as in a) or where the final state is a translation of the state obtained fromthe initial state by interchanging the electrons position as in b).

t

t t

t1 2

3 4

Figure 4.4: The weight of this diagram, determined according to the rules given in Ap-

pendix 4.6.1 is (tdτ)2 e−U(t2 − t1) (gdτ)2 e−ω(t4 − t3)

ends must be either identical (Fig. 4.3-a) or with the electrons position interchanged(Fig. 4.3-b). The rules for determining the diagram weight are given in Appendix 4.6.1.An example is presented in Fig. 4.4.

We generate all the possible diagrams with an arbitrary number of phonons, and withthe length between 0 and a τmax chosen long enough to project the ground state. Thediagram update procedures are presented in Appendix 4.6.1. Estimators for the Green’sfunction, energy, effective mass, phonon distribution, phonon number and electrons posi-tion correlation function can be easily found. We introduce them in Appendix 4.6.2. The


0 5 10 15 20 25 30

-2

0

2

4

6

ln(P

(0,

)*ex

p(

))

0 15 30 45 60 75 90

τ0

1

2

3

4

5

6

U=4α=2.42

U=9.68α=2.42

a)

b)

µτµ=−16

µ=−11.25

Ε=−15.91

Ε=−11.28

τ

Figure 4.5: ln(Ps(0, τ) ∗ eµτ ) versus τ . a) For the given parameters a large bipolaronbinding energy results. b) For the given parameters a small bipolaron binding energyresults. The linear asymptotic behavior starts at smaller time in case a). Notice that thetime scale is different for the two cases presented.

measurements are taken only at large time where the ground state is projected out.As an illustration, in Fig. 4.5 we show ln(Ps(0, τ) ∗ eµτ ) versus τ where

Ps(0, τ) ∗ eµτ =∑

n

P ns (0, τ) ∗ eµτ =

∑ν

e−(Eν−µ)τ (4.28)

with P ns defined in Eq. (4.22) for the K = 0 channel . µ is an arbitrary parameter which is

chosen close to the bipolaron energy to avoid the exponentially small weight of the largetime diagrams. It can be seen that at long imaginary time ln(Ps(0, τ)) becomes linear inτ , the slope being proportional to the ground state energy. Similar to the polaron casewe are generating only the connected diagrams. As a consequence Ps(0, τ) suffers a fastdrop at small time, which will not allow an efficient sampling for both long and shorttime diagrams. We use (as in polaron calculation) the fictitious potential renormalizationtrick to eliminate this problem.

4.3.3 Discussions

Our code calculates the Green’s functions in the real space representation (Wannier basis).A similar expression to Eq. 4.27 can be written in the momentum space (Bloch basis),and similar rules for determining the diagrams weight can be found. This approach wasconsidered in [16] for the exciton model calculation. The problem in our model is that,unlike in the exciton case where the conduction-electron valence-hole interaction is attrac-tive, we have a repulsive interaction. This will make all the diagrams with an odd number


of electron-electron interaction vertices to be negative, which implies a very severe signproblem. In the real space representation the sign problem is avoided, all the diagramsbeing positive definite. However, this representation introduces other problems, for ex-ample it makes the study of the bipolaron at large momentum difficult and inaccurate. Ina more general sense, problems appear in all the irreducible channels aside from the onewhich contains the system ground state (i.e. K = 0, singlet). The difficulty is two-fold.First there is the sign problem. Aside from the singlet and K = 0 channel where all theterms in Eq. 4.25 are positive definite, at K 6= 0 or/and in the triplet channel the factors

eiKx(m) and gS(m) can take negative values. Second, in the real space representationwe generate all the possible configurations with all the possible symmetries. To projectout the lowest energy state of the channel “γ” we have to calculate Pγ(τ) up to a timeproportional to 1/∆γ, where ∆γ is the γ channel gap. Therefore we have to simulate upto a larger imaginary time for a channel characterized by a smaller gap. But because allthe symmetry channels configurations are generated in the same run, the statistics for the

channel γ is proportional to e−(Eγ − E0)τ (here E0 is the ground state energy). Thus ifthe imaginary time is increased, the statistics for other than the channel which containsthe ground state will be exponentially reduced.

Another problem occurs if there are more than one bound state in the same symmetrychannel, quasi-degenerate in energy. In this case, at large imaginary time we project out allthese states. The results obtained in this case are going to be the average of the propertiesof all the projected states. We encounter this problem at large electron-phonon coupling,where the difference of the s-wave and d-wave bipolaron energies is exponentially small,as the strong coupling theory predicts. However in the intermediate coupling regime wemanaged to separate these states and we always found a s-wave ground state.

Before to talk about the necessary modifications for adapting our algorithm to otherbipolaron models, we want to mention that, even for the present Hubbard-Holstein bipo-laron, the code can be improved. The momentum K and the spin S are not all thequantum labels which can be used to distinguish between the different symmetry chan-nels. We also have the point group symmetries which break the Hilbert space in differentirreducible representations. We have already talked about s-wave (A1g) and d-wave (B1g)bipolaron states. In principle we can look for all the symmetries given by the representa-tions of the D4h point group. These symmetries can be separated in a similar way to whatwe did when separating the singlet and the triplet channels. What we have to do is togenerate diagrams where the time τ end is obtained after a translation and a point groupoperation applied at the time 0 end. In another words the electronic configurations ofthe diagrams ends should be connected by a space group operation∗. For every operationthere is a certain factor tD(m) which separates the different representations, and the valueof an observable is calculated analogues to Eq. 4.25

A(K) =1

M

∑m

eiKx(m)gS(m)tD(m)a(m) =

=

∑m e

iKx(m)gS(m)tD(m)w(m)a(m)∑mw(m)

(4.29)

∗In fact the separation of singlet and triplet obeys the same idea, the permutation group operationsbeing used.

4.4. Results 71

For example, at K = 0, tD is always 1 for A1g representation. In general tD should beproportional to the characteristic of the representation. The sign problem can intervenefor other representations than A1g. The above approach applied to our present modelwill improve the accuracy of the results which describe the properties of the s-wave S1bipolaron. However the study of the d-wave symmetry bipolaron will still be difficultand not very accurate, because of the smallness of the binding energy of this state. Webelieve that no new physics will appear to justify the large coding effort necessary for theabove approach implementation. Nevertheless the separation of the different point grouprepresentations can be essential for other models which include longer electron-phononinteraction and where strongly bound bipolarons with electrons residing on different sitesexist [8].

The code can be easily modified to include longer range electron-electron and electron-phonon interaction. For an electron-phonon interaction term of the form

∑i,j

gijni(b†j + bj) (4.30)

and for an electron-electron interaction of the form

∑i,j

Vijninj (4.31)

there will be no sign problem. The diagrams are similar to the ones shown in Fig. 4.3aside from the possibility that now a phonon propagator at site “j” can be created ordestroyed by an electron at site “i”.

4.4 Results

In all the subsequent calculations the phonon energy and the electron hopping term arechosen to be one (ω0 = 1, t = 1). In real materials ω0 is smaller than t (with almostone order of magnitude), but calculations with such small ω0 are very computer timeconsuming and besides that we find that they do not bring any new qualitative results.Our calculation is done on a 25× 25 square lattice with periodic boundary conditions.

4.4.1 Phonon Induced Attraction. U = 0 Case

If U = 0 the effective interaction will be always attractive. The situation is shown inFig. 4.6. Increasing α the system evolves from a very weakly bond bipolaron (almost twofree polarons) to a strongly bound bipolaron. The transition is sharp. It can be seen fromFig. 4.6-c, where the electrons relative position correlation function C(r) defined as

C(r) =1

N

∑i

〈nini+r〉 (4.32)

is shown at r = 0, that in a very short interval around αc = 1 the system ground statechanges from almost two free polarons to a state where the electrons are practically on the


0 1 2 3 4 5 6α

-25

-20

-15

-10en

ergy

bipolaron2 x polaron

0

5

10

15

20

25

30

phon

ons

0 1 2 3 4 5 6α

0

0.2

0.4

0.6

0.8

C(0

)

a) b)

c)

Figure 4.6: U = 0, ω0 = 1, t = 1. a) The bipolaron energy versus electron-phononcoupling (circles). The dashed line (squares) is 2× free polaron energy. b) The bipolaronaverage number of phonons (circles) and 2× free polaron number of phonons (squares).c) The probability to have the electrons on the same site, C(0), in the bipolaron state.The dotted line in -a ( -b) represents the two free polarons energy (number of phonons)versus the effective coupling, αeff = 2 α.

same site. In comparison to the polaron case, the critical electron-phonon coupling wherethe transition takes place is approximately two times smaller. This can be understood bynoticing that (see Eq. 4.8) the deformation energy at a particular site is proportional tothe square of the number of electrons on that site. Therefore for the on-site bipolaron theeffective α is two times larger than the corresponding polaron one. Thus, the bipolaronenergy at a particular coupling α (in the strong coupling regime) is equal to two timesthe free polaron energy corresponding to a double α. The same is true for the averagenumber of phonons in the bipolaronic cloud (which is also proportional to the square ofthe number of electrons). This is shown with dotted line in Fig. 4.6-a and -b. From thesame figures it can also be seen that the bipolaron transition is very sharp compared tothe large to small polaron transition.

In Sec. 4.2 we showed that in the antiadiabatic limit (i.e when ω0 −→ ∞) the ef-fective attraction induced by phonons becomes instantaneous and as a consequence theHubbard-Holstein model is equivalent to a an attractive Hubbard model. The attractiveHubbard Hamiltonian was considered as a realistic model to explain the properties ofsystems like amorphous semiconductors [20–22] or high Tc superconductors [8] and wasunder investigation in the past [23]. However, when the phonon frequency is finite, theinteraction becomes retarded and the Hubbard-Holstein physics will differ from the at-tractive Hubbard one. Therefore, we think that a comparison of the Hubbard-Holsteinmodel and the attractive Hubbard model is necessary.

4.4. Results 73

0 5 10 150

0.2

0.4

0.6

0.8

1

C(0

)

Holstein bipolaronatractive Hubbard

0

2

4

6a)

b)

U /t effeff

∆E

t/ef

f

| |

Figure 4.7: A comparison between Hubbard-Holstein bipolaron (solid line) and attractiveHubbard model (dashed line). On the horizontal axis is Ueff/teff , where Ueff = −4αtand teff = teff (α) is the polaron effective hopping. a) The binding energy in terms ofteff . b) Probability to have two electrons on the same site.

Aside from inducing a retarded effective attraction, the other main effect of theelectron-phonon interaction is to dress up the electrons, increasing their effective mass orequivalently reducing their effective hopping. Consequently, in Fig. 4.7 we compare theHubbard-Holstein Hamiltonian (with U = 0) with the corresponding attractive Hubbardmodel, defined by Ueff = −4αt ≡ −2Ep (see Eq. 4.4 and Eq. 4.8) and polaron effectivehopping teff = teff (α). The effective polaron hopping as a function of electron-phononcoupling was calculated numerically in Chapter 3. Increasing Ueff/teff the system evolvesin both cases from two free electrons state to a state where the electrons are mainly onthe same site (S0 bipolaron). At small Ueff/teff , namely when α is smaller than thetransition critical αc, the effective attraction induced by phonons is weaker than the cor-responding Hubbard attraction. The transition to S0 bipolaron is very sharp for theHubbard-Holstein model, unlike the attractive Hubbard model where it is rather smooth.We found (not shown) that the smaller ω0 the sharper the transition is. In the adiabaticlimit (ω0/t −→ 0) the transition becomes of first order [13, 14]. Not only the transitionis different but also the properties of the bipolaron at large coupling are different in thetwo cases. The Hubbard-Holstein S0 bipolaron has a large effective mass proportional to

t2eff , thus with a factor of e4α/ω0 larger than the free electron mass [9]. On the otherhand, the effective mass of the attractive Hubbard bipolaron is increasing linearly with|Ueff | = 4αt in the large U regime, as it can easily be shown analytically.

From both Fig. 4.6 and Fig. 4.7 we can conclude that for Hubbard-Holstein model there


is a very narrow transition region where the system evolves from almost two free lightpolarons to a very heavy S0 bipolaron. Before the transition, the effect of the electron-phonon interaction is small, especially when the phonon frequency is small. Increasingthe phonon frequency results in increasing the effective attraction too. The physics afterthe transition is well described by the strong coupling theory. Now the energy and thenumber of phonons is proportional to α (E = −8αt and Nph = 8αt/ω0), as it can be seen

in Fig. 4.6-a and -b, and the effective mass is proportional to e4α/ω0 . Unlike the weakcoupling regime, here a smaller ω0 results in a heavier bipolaron state.

4.4.2 S0 Bipolaron to S1 Bipolaron Transition. U 6= 0 Case

The weak coupling regime is characterized by the formation of a weakly bound state evenwhen U = 0, as can be seen from Fig. 4.6. The binding energy is extremely small, situatedat the resolution limit of our algorithm. We expect the bipolaron to be destroyed by avery small Coulomb repulsion.

In the strong coupling regime, according to our discussion of Sec. 4.2.2, when U isincreasing the system evolves from a strongly bound S0 to a weakly bound s-wave S1bipolaron. The S0-S1 transition takes place around U = 2Ep. At the critical valueU = 4Ep the bipolaron state ceased to exist, and the system becomes two polaronsmoving freely on the lattice. The binding energy of S0 bipolaron decreases linearly withU . The S1 bipolaron binding energy is proportional to 1/U (Eq. 4.16).

The intermediate coupling regime situation is presented in Fig. 4.8. When U = 0the bipolaron is a S0 state. While U is increasing, first the binding energy is decreasinglinearly with U . After that, the binding energy behavior as a function of U is changing,decreasing much slower as a function of U . The bipolaron state disappears much before Ureaches the values 4Ep. The proportionality to U of the binding energy is a characteristicof the strong bound S0 bipolaron. From Fig. 4.9 it can be seen that this is correlatedwith the probability to have the electrons on the same site, C(0) being close to one.At the value of U where the binding energy behavior changes, there is a transition to aweakly bound state (with the binding energy smaller than the one given by the strongcoupling theory, Eq. 4.16) where the probability to have the electrons on neighboring sitesis enhanced and where simultaneously C(0) drops to small value. For large couplings, likein Fig. 4.9-a, this state is a small S1 bipolaron, the electrons residing practically only onthe nearest-neighbor sites. For smaller couplings as in Fig. 4.9-d, this state is a “large”S1 bipolaron, the wave function being spread over many sites, but still with an enhancedprobability for the electrons to be nearest-neighbors. This can be seen from Fig. 4.10,where the correlation function, C(r) (Eq. 4.32), as a function of the electrons relativedistance, r, is shown∗.

For the strong electron-phonon coupling regime we know that a d-wave S1 bipolaronquasi-degenerate in energy with the s-wave S1 ground state exists. Our code project outboth of these states at long imaginary time. Therefore the results we obtained represent

∗Notice that the total probability to have the electrons at the distance r from each other is C(r) timesthe number of sites situated at the relative distance r. Therefore, in Fig. 4.9, we plotted 4 × C(1) and4 × C(2) which represent the probability to have the electrons on nearest and respectively next-nearestneighbor sites.

4.4. Results 75

0 5 10 15 20

U

-25

-20

-15

-10

ener

gy α=3.125α=2.42α=2α=1.62α=1.125

8 10 12 14 12 14 16 182 4 6 8

0

0.2

0.4bi

ndin

g en

ergy

a)

b) c) d)

Figure 4.8: a) The bipolaron energy versus U for different values of the electron-phononcoupling α. b), c) and d) The binding energy of the bipolaron in the transition region,for α = 1.62, α = 2.42 and respectively α = 3.125, versus U . The dashed line is the S1strong coupling theory predicted binding energy.

in fact the average of the two S1 bipolarons properties. We definitely encountered thissituation in the S1 regime for the large α = 3.125 case shown. At smaller couplingsthere is a larger difference between the d-wave S1 bipolaron and the s-wave S1 bipolaronenergies, and it is easier to project out the ground state and thus to separate the twostates. We show this in Fig. 4.11, where a comparison of the correlation function C(r)measured at two imaginary times, τ = 35 and respectively τ = 80 is made. At timeτ = 35 we see a smaller probability for the electrons to stay along the diagonal directions.This is an evidence that in our measurements we capture both the d-wave and the s-wavestates. When the measurements are taken at a larger time, the value of the correlationfunction at sites which correspond to the diagonal directions increases. For the presentedcase, C(r) does not change sensible if the measurement time is increased above τ = 50,thus we can conclude that the asymptotic regime is reached above this time. The fact


0 2 4 6 8U

0

0.2

0.4

0.6

0.8

0 5 10 15

0

0.2

0.4

0.6

0.8

1

C(0)4 C(1)4 C(2)

0 1 2 3 4 5 6 7U

0 5 10 15

α=3.125a) b)

c) d)

α=2.42

α=2 α=1.62

Figure 4.9: The electrons position correlation function C(0) , 4×C(1), and 4×C(2) (seeEq. (4.32)) versus U for different values of the electron-phonon coupling.

that at τ = 35 we see a decrease of C(5), i.e. a decrease of the correlation function atlarge distance, shows that the d-wave bipolaron in the intermediate coupling region is alarge state spread over many sites, like the s-wave ground state.

To conclude, in the intermediate coupling regime, with increasing U , the system evolvescontinuously from a S0 bipolaron to a weakly bound state spread over many sites with anenhanced probability that the electrons occupy nearest-neighbor sites, which we call largeS1 bipolaron. The symmetry of this states is s-wave. In the same region of parametersanother stable state with d-wave symmetry exists, with a smaller binding energy. Thespatial extent of this state is also large. When α is increased both s-wave S1 and d-waveS1 bipolaron wave functions become more nearest-neighbor localized evolving to the statespredicted by the perturbation theory. The energy difference between the s-wave and the d-wave states becomes exponentially small at large electron-phonon coupling. The existenceof S1 bipolaron is a consequence of the reduction of the polaron effective hopping, so thatthe exchange attraction, which is very weakly normalized by the phonons, can win overthe effective kinetic energy, resulting in the formation of two S1 bipolaron states.

In Fig. 4.12-a we plot the number of phonons as a function of Coulomb repulsion U ,for different values of the electron-phonon coupling. Before the S0-S1 transition to take

4.4. Results 77

0 10 20 30r

0

0.01

0.02

0.03

0.04

0.05

0.06

C(r

)

0 10 20 30 40r

a)

α=2.42

b)

α=1.62U=11 U=4.5

Figure 4.10: The correlation function C(r) in the intermediate electron-phonon couplingregime. The relative distance between electrons is given in circular coordinates.

-1 0 1 2 3 4 5 6 7 8 9 10r

0

0.05

0.1

0.15

0.2

0.25

C(r

)

τ = 80τ = 35

U=9.68α=2.42

Figure 4.11: The correlation function C(r) in the intermediate electron-phonon couplingregime for a value of U which corresponds to a large S1 bipolaron. The sites at r = 2,r = 5 and r = 9 are along the diagonal directions. The statistics are collected at twodifferent imaginary times. For the smaller τ = 35 time the results represent the averageof the s-wave symmetry ground state and the d-wave symmetry first excited state (noticethe small occupation probability of the sites along the diagonal directions). At the largertime, τ = 80, only the s-wave ground state remains.


0 5 10 15 20U

0

5

10

15

20

25

30nu

mbe

r of

pho

nons

α=3.125α=2.42α=2α=1.62α=1.125

0 5 10 15 20U

0

0.2

0.4

0.6

0.8

2m /

m*

a) b)

e

Figure 4.12: a) Average number of phonons in the bipolaronic cloud for different valuesof electron-phonon coupling versus U . b) Inverse of the bipolaron effective mass versusU . me is the electron mass. The horizontal lines correspond to the inverse of two freepolarons effective mass.

place, the number of phonons decreases slowly with U , and it is well approximated by thestrong coupling perturbation theory. The S0-S1 transition is sharp, and the bipolaronchanges from a very heavy state (S0) to a light one (S1), as can be seen in Fig. 4.12-bwhere the inverse of the bipolaron effective mass is plotted as a function of U . The S0-S1transition is sharp even in the intermediate coupling region. We found (not shown) thatfor smaller ω0 the transition is sharper.

The results presented up to here are calculated for K = 0 and in the singlet channel.For the strongly bound S0 bipolaron, in agreement with the exponentially small effectivehopping predicted by the perturbation theory, we found a flat dispersion resulting in anarrow band, of the order of the calculation error bars. We were not able to compute themomentum dependent properties of the weakly bound bipolarons for reasons described inSec. 4.3.3. In the triplet channel we found no bound states at any value of the parameters.

4.4.3 Phase Diagram

The phase diagram is shown in Fig. 4.13. We want to remind the reader that withour technique, the calculation becomes more difficult when the binding energy is small,simulation up to a large imaginary time being required in this case. The most difficultcomputations are at both large electron-phonon coupling and large U . The small bindingenergy implies large imaginary time simulations which are difficult because the number of

4.5. Conclusions 79

0 1 2 3 4α

0

10

20

30

U

S0

S1

large

polar

ons

trans

ition

pol

aron

s

small polarons

Figure 4.13: Phase diagram. The solid line is the bipolaron -two free polarons boundary.The dashed line separates the S0 and S1 bipolarons. The doted lines are the strongcoupling theory results.

phonons and the effective mass is always large (the ground state consists by two weaklyinteracting small polarons, and a small polaron has itself an exponentially large mass andcontains a large number of phonons). Therefore the largest errors we have are in thedetermination of the bipolaron-two free polarons boundary at large values of α. In thestrong coupling theory this boundary is determined by the critical value U = 8αt ≡ 4Ep.In the intermediate coupling regime we found that the bipolaron state disappears muchbefore that value. The value U = 8αt ≡ 4Ep should be taken as an upper limit, reachedasymptotically when α is increased, for the existence of the bipolaron state. The S1 regioncontains weakly bound states where the probability to have the electrons on nearest-neighbor sites is larger than to have them on the same site. Depending on the value ofelectron-phonon coupling the S1 bipolaron can be a large state with the wave functionspread over may sites, or a small state where the electrons are residing on nearest-neighborsites. The large S1 bipolaron breaks into two large or intermediate (transition) polaronsand the small S1 bipolaron evolves into two small polarons when the Coulomb repulsionU is increased. It is worthy to stress that the S1 bipolaron exists only for intermediateand strong α, where the polaron effective hopping (mass) is strongly reduced (enhanced),stabilizing the exchange singlet state.

4.5 Conclusions

In this chapter we studied the two-dimensional Hubbard-Holstein bipolaron propertiesusing a Diagrammatic Quantum Monte Carlo algorithm which calculates the zero tem-


perature Matsubara Green’s functions. The bipolaron properties are extracted from theGreen’s functions behavior at large imaginary time where the ground state is projectedout. Unlike the other Diagrammatic Quantum Monte Carlo simulations used for studyingdifferent polaron and exciton models, in order to avoid the sign problem, our algorithmproduces and sums real space (Wannier orbitals basis) diagrams. The code can be rela-tively easily modified for other bipolaron models with longer range electron-phonon and/orelectron-electron interaction. The dimensionality and the lattice symmetry can also bemodified.

We calculated the phase diagram in the parameter space defined by U and α. De-pending on the parameters value, different kinds of bound states are formed. We studiedboth their properties and the transition from one bipolaron type to another.

At small electron-phonon coupling two electrons form a weakly bound state evenfor U = 0. In this regime the binding energy increases with increasing the phononfrequency. For larger coupling the phonon induced attraction results in the formation ofstable bipolaron states even for large Coulomb repulsion U . Starting from two free polaronstates and decreasing U , two polarons first form a weakly bound S1 bipolaron which willtransform to a strongly bound S0 state at small U . The unrenormalized exchange energytogether which wins over the reduced polaron effective hopping is responsible for bindingthe S1 bipolaron state. The S1 bipolaron is a large state spread over many lattice sites forα corresponding to the polaron transition region, and becomes nearest-neighbor localizedwhen α is increased. The binding energy of the S1 bipolaron and the critical Coulombrepulsion U where the bipolaron state disappears are smaller than the values obtained inthe strong coupling perturbation theory. The S0 bipolaron forms at small values of U ,for couplings larger than αc ≈ 1. Its properties are well described by the strong couplingperturbation theory.

We found that the ground state always has s-wave symmetry. The transition betweendifferent kind of bipolaron states is continuous. In the intermediate and strong electron-phonon coupling regime, for values of U which correspond to a S1 ground state, anexcited d-wave stable state also exists. This state is spatially large for intermediate α andnearest-neighbor localized for large α, similar to the corresponding s-wave ground state.The excitation energy is larger at intermediate coupling and goes exponentially to zerowhen α is increased.

4.6 Appendix

4.6.1 Diagrams Updates

The rules which determine the diagrams weight can be easily found by inspecting Eq. 4.27.It will result in the following: (i) every electron hopping corresponds to a term t dτ , (ii)every electron-phonon vertex corresponds to a term g dτ , (iii) every phonon propagatorof length τ corresponds to a term e−ω0τ , and (iv) every interval where the electrons are

on the same site during a time τ corresponds to a term e−Uτ . An example is given inFig. 4.4.

In case that longer range electron-phonon interaction (Eq. 4.30) is considered, the

4.6. Appendix 81

a)

b)

Figure 4.14: Changing the distance between electrons at the diagram’s ends

process to create or destroy a “j” phonon by an “i” electron will be associated with theterm gij dτ . If long range electron-electron interaction is present (Eq. 4.31), then to an

interval of time τ with the electrons on “i” and “j” sites, a term e−Vijτ will correspond.

We define an interval as the part of a diagram between two successive hoppings ortwo successive electron-phonon vertices or one hopping (electron-phonon vertex) and oneelectron-phonon vertex (one hopping). Every interval contains two electron lines, onecorresponding to the spin up and the other to the spin down electron.

In order to generate with a finite probability all the possible diagrams, we propose thefollowing minimal set of updates.

Changing the diagram length This update is similar to the one presented for Holsteinpolaron. A time τ ′ is chosen with the probability

P(τ ′) = e−(nω0 − µ)(τ ′ − τ)dτ ′ (4.33)

between the last interval left end time and τmax. The transformation of the last intervalfinal (right end) time from τ to τ ′ has the acceptance ratio 1. The time of both electronpropagators is changed.

Changing the distance between electrons at the diagram ends This is donein two ways and both of them must be included in the code. The first, presented inFig. 4.14-a, implies the creation of two electron hoppings, one at each of the diagramends. Its counterpart subroutine removes two hopping, one at every diagram end. Thesecond, shown in Fig. 4.14-b, implies an electron hopping creation at one end and aremoval of an electron hoping at the other end. Writing the detailed balance condition isstraightforward and implies the choice of the hoppings time, the hoppings direction andthe electron lines (up or down) which suffer the hopping process.


Figure 4.15: Interchanging the electrons at the diagram’s ends

Interchanging the electrons at the diagram ends In order to separate the singletand the triplet channels, we have to generate diagrams where the electrons position atthe diagram ends are interchanged. One way to do this is presented in Fig. 4.15. Thisupdate starts form a configuration where the electrons at both ends are on the same site.Everything is identical with the previous introduced update procedures (which changes thedistance between electrons at the diagram ends) besides the fact that now, the electronsat the ends hop in different relative directions to each other.

One problem with the above proposed procedure is that it becomes very inefficient atlarge U , where the probability to have the electrons on the same site will be exponen-tially small. We created a different procedure for interchanging the electrons (not shown)starting form configurations where the electrons at the ends are on nearest-neighbor sites.This update moves the electrons one around the other until their relative position changethe sign.

Changing the distance between electrons on a interior interval We propose twoprocedures for this purpose. The first, presented is Fig. 4.16-a chooses an electron line froman interval and creates on it two electron hoppings is the same direction. The counterpart

yx

yx

a)

b)

Figure 4.16: Changing the distance between electrons on a interior interval.

4.6. Appendix 83

Figure 4.17: Translating the phononic configuration at one end

subroutine chooses an interval which finishes at both ends on the same electron line withhoppings in the same direction and then removes the hoppings.

The second is specific and obligatory to the two-dimensional lattice. It is presented isFig. 4.16-b. It applies on intervals which finish on the same electron line with hoppingsin perpendicular directions. It interchanges the direction of the respective hoppings.

Adding/Removing internal phonons These updates are similar to the polaron modelcorresponding ones.

Adding/Removing external phonons These updates are similar to the polaronmodel corresponding ones. Care should be taken for obtaining only phononic configu-rations identical up to a translation at both ends.

Translating the phononic configuration at one end Not only the electronic con-figurations at the ends should be identical up to a translation, but also the phononicconfigurations. However up to here we presented only updates which change (translate)the electronic configurations. Applying them we get into diagrams where the translationvector from the initial end to the final end of the electronic configurations is differentfrom the one corresponding to the phononic configurations. One possibility to solve theproblem is to define updates which translate both the electronic and the phononic config-urations in the same time. However the acceptance ratio of these procedures would be toosmall for an efficient sampling. Other possibility is working with two kind of updates. Onewhich translates only the electronic configurations and the other which translates only thephononic configurations. We generate all the possible diagrams but make measurementsonly on the diagrams where the electronic and the phononic configurations translationvectors coincide. We have already introduced the first kind of updates which translatesthe electrons. An update procedure which translates the phononic configuration at oneend is presented in Fig. 4.17. This update changes all the phonons position with one sitetowards the same direction. A phonon is moved by creating or deleting two neighboringelectron hoppings towards the same direction. The update moves all the phonons attachedat one chosen end in the same time.

The above introduced subroutines constitute a minimal set for ensuring the ergodic-ity condition. In order to increase the algorithm efficiency, we also used other updateprocedures not discussed here.


4.6.2 Estimators

Analogues to Holstein polaron model (Section 3.6.2), the value of the desired quantitiesare obtained by calculating the corresponding estimators average over the Monte Carlogenerated ensemble of diagrams.

Green’s function estimator The Greens’ function estimator is the same as the polaronmodel corresponding one

pn0lj (τ0; i) =

1

2a

P n0lj (τ0)

P nilj (τi)

Θ(a− |τi − τ0|)δni,n0 (4.34)

and implies collecting statistics in an arbitrary large time window, (τi − a, τi + a).

Energy estimator The energy estimator is determined in the same fashion as thecorresponding polaron one (see Eq. 3.57). It results in

e(i) =1

τi(ω0

∑

lph

τlph+ U

∑jU

τjU−Nv −Nh) (4.35)

where τi is the length of the diagram, lph counts the phonon propagators with τlphlength,

jU counts the intervals with double occupied sites of length τjU, Nv is the number of

electron-phonon vertices and Nh is the number of electron hoppings.

Inverse of the effective mass estimator At large imaginary time (see Eq. 4.28) wehave

Ps(λ, τ)

Ps(0, τ)−→ e−(E(λ)− E(0))τ = e−λ2τ/2m∗

(4.36)

Considering Eq. 4.25 the estimator for the left hand side of Eq. 4.36 results in

Q(i, λ) = e−iλxi = 1− iλxi − λ2x2i (4.37)

Therefore the estimator for the inverse of the bipolaron effective mass is

2me

m∗ (i) =x2

i

τi(4.38)

where xi is the translation vector between the time 0 end and the time τi end. me is thefree electron effective mass.

n-phonon probability estimator The probability to have n phonons is calculatedwith the estimator

zn(i) = δni,n (4.39)

where ni is the diagram “i” number of phonons.

4.6. Appendix 85

Total number of phonon estimator is

nph(i) = nδni,n (4.40)

Electrons relative position correlation function estimator The correlation func-tion (Eq. 4.32) has the estimator

c(r; i) = δr,Ri(4.41)

where Ri is the relative distance between electrons in the diagram“i” ends .

The measurements should be taken at long imaginary time for all the quantities definedabove, except for the Green’s functions which we measure at all the time values between0 and τmax.


References

[1] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).

[2] C. Schlenker, in Physics of Disordered Materials, edited by D. Adler, H. Fritzche,and S. Ovshinski (New York, Plenum, 1985).

[3] O. F. Schirmer and E. Salje, J. Phys. C 13, L1067 (1980).

[4] W. Bao, et al., Solid State Commun. 98, 55 (1996).

[5] G. Zhao, et al., Phys. Rev. B 62, 11949 (2000).

[6] S. Tojima, et al., Phys. Rev. B 35, 696 (1987).

[7] J. Scott, et al., Mol. Cryst. Liq. Cryst. 163, 696 (1983).

[8] A. S. Alexandrov and N. F. Mott, Rep. Prog. Phys. 57, 1197 (1994).

[9] J. Bonca, T. Kartrasnik, and S. Trugman, Phys. Rev. Lett. 84, 3153 (2000).

[10] A. La Magna and R. Pucci, Phys. Rev. B 55, 14886 (1997).

[11] G. Wellein, H. Roder, and H. Fehske, Phys. Rev. B 53, 9666 (1996).

[12] A. Weisse, H. Fehske, G. Wellein, and A. Bishop, Phys. Rev. B 62, 747 (2000).

[13] L. Proville and S. Aubry, Physica D 133, 307 (1998).

[14] L. Proville and S. Aubry, Eur. Phys. J B 11, 41 (1999).

[15] G. De Filippis, et al., Phys. Rev. B 64, 155105 (2001).

[16] E. Burovski, A. S. Mishchenko, N. Prokof’ev, and B. Svistunov, Phys. Rev. Lett.87, 168402 (2001).

[17] J. Bonca and S. A. Trugman, Phys. Rev. B 64, 09457 (2001).

[18] A. B. Migdal, Sov. Phys. JETP 7, 996 (1958).

[19] I. G. Lang and Y. A. Firsov, Sov. Phys. JETP 16, 1301 (1963).

[20] P. Anderson, Phys. Rev. Lett. 34, 953 (1975).

[21] R. Street and N. F. Mott, Phys. Rev. Lett. 35, 1293 (1975).

[22] D. Alder and E. J. Yoffa, Phys. Rev. Lett. 36, 1197 (1976).

[23] S. Robaszkiewicz, R. Micnas, and K. Chao, Phys. Rev. B 23, 1447 (1981).

Chapter 5

Dynamical Cluster Approximation

5.1 Introduction

Strongly correlated materials remain a puzzle for physicists, despite the enormous theoret-ical and experimental effort invested during the last decades in the study of their peculiarproperties. So far, there is no good theory for describing systems where the interactionand the kinetic part of the Hamiltonian are comparable as order of importance. Moreover,when this occurs in lower dimensional systems, strong quantum fluctuations causes meanfiled methods to fail.

From the theoretical point of view, aside from inventing new analytical approximationschemes, a lot of effort was put in developing numerical algorithms able to deal withstrongly correlated systems. However, the “approximations free” algorithms (like exactdiagonalization and finite size Monte Carlo) can tackle only small size systems and/or workonly for some particular regions of the parameter space. There are other algorithms whichcan deal with larger systems, but they imply approximations often difficult be to justified.One of them is the Dynamical Mean Field Theory (DMFT) [1, 2], which proved to besuccessful in describing systems where only the local correlations are relevant. It maps thelattice to an impurity embedded self-consistently in an effective medium, thus consideringexplicitly the local correlations at the impurity site. However, it neglects any nonlocalfluctuations and consequently fails to describe systems where these are important. Forexample the DMFT cannot be applied to high Tc superconductors which are characterizedby a nonlocal d-wave superconducting order parameter.

In the last years, the DMFT approach was extended to include, aside from local cor-relations also finite ranged nonlocal ones. The idea is to map the lattice into a finite-sizecluster embedded self-consistently in a host. In this way nonlocal correlations up to thecluster size are treated explicitly, while the physics on longer length scales is treated atthe mean field level. Therefore, good results should be expected for systems where onlythe short-range correlations are relevant, as in cuprates superconductors. Two practicalrealizations of this idea are in used nowadays. One is the Dynamical Cluster Approx-imations [3, 4], which considers a cluster with periodic boundary conditions embeddedself-consistently in a host. The other approach considers an open boundary conditionscluster, and it is called Cellular Dynamical Mean Filed Theory [5]. In the former ap-proach, discussed in this chapter and applied to Hubbard type models in the next two

87

88 Chapter 5. Dynamical Cluster Approximation

chapters, all the sites are equivalent and hybridizing with the host. In the later approachonly the sites at the cluster edge are hybridizing with the host. This results in a fasterconvergence of the first approach with increasing the cluster size. However, we noticedthat the periodic boundary conditions imposed on very small clusters (e.g. 2 × 2) intro-duce some artificial effects. Presumably this is due to the peculiar topology of the smallclusters with periodic boundary conditions, as we will discuss in Chapter 7. On the otherhand, the open boundary conditions break the translational symmetry inside the cluster,introducing artifacts too. However it is not our aim to make a comparison between thetwo approaches here. An interested reader can find this in [6–8].

Both algorithms are fully causal and reduce to DMFT when the cluster number ofsites is taken one.

5.2 Dynamical Mean Filed Theory

In DMFT the problem is reduced to an impurity embedded in a host. The host propertiesare determined self-consistently, assuming that the local lattice Green’s function (i.e.G(r = 0)) is equivalent to the impurity Green’s function. A detailed description ofDMFT can be found in the reviews by Pruschke et al. [9] and Georges et al. [10].

Formally DMFT can be derived by disregarding the momentum conservation at theinternal vertices of the self-energy. This is equivalent [11] to approximate the Laue func-tion

∆(k1, k2, k3, k4) =∑

r

ei(k1 + k2 − k3 − k4)r = Nδk1+k2,k3+k4 (5.1)

by

∆DMFT (k1, k2, k3, k4) =∑

r

ei(k1 + k2 − k3 − k4)r = 1 (5.2)

This assumption results in a free summation over the internal momentum labels of everyGreen’s function in the calculation of the self-energy, or equivalently to replacing everyGreen’s function propagator with the average over the entire Brillouin Zone of the Green’sfunctions.

It was shown that the DMFT becomes exact in the limit of infinite dimension [11, 12].The reason is that for D −→∞ the hopping t scales as t ∼ D−1/2, resulting in a Green’sfunction G(r) ∼ tr ∼ D−r/2. If the diagrams are represented in real-space, it can beshown easily that all the diagrams which connect two sites via more than two independentpaths are collapsing to a single site. Therefore the self-energy, the four legs irreducibleinteraction vertex functions and in general everything which can be derived from compactdiagrams ∗ becomes local.

We want to remind that even though the self-energy is local, it is fully frequencydependent, thus taking good care of the local dynamic correlations.

∗By definition a compact diagram is a close diagram which cannot be cut into two separated self-energy pieces.

5.3. Dynamical Cluster Approximation 89

5.3 Dynamical Cluster Approximation

The Dynamical Cluster Approximation is a generalization of DMFT to include short-range spatial correlations. It maps the lattice to a cluster embedded self consistently ina host. There are a variety of methods for solving the cluster hybridizing with a hostproblem. The most popular are the fluctuation exchange approximation (FLEX) [13], thenon-crossing approximation (NCA) [14] and the Quantum Monte Carlo (QMC) algorithmdeveloped by Hirsch and Fye [15]. The first two solvers give approximative results. TheQMC solver is numerically exact, but computational resources very consuming, implyingmassive parallel computing. For an extensive treatment of DCA the reader is asked to seeHettler et al. [3] and Jarrell et al. [16]. The purpose of this section is to describe briefly thebasics of the DCA + QMC algorithm we are going to use in the last two chapters of thisthesis. In the following we assume Hamiltonians with local interaction (like Hubbard orPeriodic Anderson models), even though the DCA can be generalized to include non-localinteraction terms.

5.3.1 Formalism

In DMFT, at any internal vertex of a compact diagram the momentum conservationis neglected (see Eq. 5.2). This implies that we can replace in the calculation of anycompact diagram the Green’s function with the coarse-grained over the entire BrillouinZone Green’s function, i.e.

G(ω) =1

N

∑

k

G(k, ω) (5.3)

Because the self-energy, Σ, and the irreducible interaction vertices, Γ’s, can be deriveddirectly from the compact part of the free energy ∗ [17, 18] we can use the same approxi-mation for their calculation. The result is a local self-energy and consequently the DMFTapproximation is expected to be valid for systems where only the local correlations areimportant.

DCA restores partially the momentum conservation at compact diagrams vertices.Traditionally, DCA is derived in momentum space. The Brillouin Zone is divided intoNc = LD cells of size ∆kD = (2π/aL)D. a is the lattice constant. Each cell contains N/Nc

momentum points, and is centered on a specific momentum K (see for example Fig. 6.13and Fig. 7.6 for the two-dimensional case with Nc = 4 and respectively Nc = 8). In DCAthe Laue function which dictates the momentum conservation at the compact diagramsvertices is approximated by

∆DCA(K1 + k1, K2 + k2, K3 + k3, K4 + k4) = NcδK1+K2,K3+K4 (5.4)

Here K + k specifies a momentum inside the cell centered on the momentum K. Thus,the DCA restores partially the momentum conservation. The momentum transfered be-

∗The compact part of the free energy, Φ[G], is a functional of G and is obtained by summing all theclose compact Feynman diagrams. The self-energy and the vertex functions can be written as Σ = δΦ

δG

and respectively Γ = δ2ΦδGδG

.


tween the cells, i.e the momentum larger than ∆k, is conserved. On the other hand, theconservation of momentum within the cell, i.e the momentum less than ∆k, is neglected.

In the spirit of DMFT, Eq. 5.4 implies that in the calculation of compact diagrams(and implicitly in the calculation of the self-energy), the Green’s function can be replacedby the coarse-grained Green’s function over the Brillouin Zone cells,

G(K,ω) =Nc

N

∑

k

G(K + k, ω) (5.5)

Here the summation is taken over all k belonging to the cell centered on K. The DCAassumption results in a self-energy, Σ(K,ω), which is a function of frequency and cellsmomentum K. Considering the real (site) space, it is natural to expect this approximationto be valid for systems where the non-local correlation range is shorter than π/∆k = La/2.Thus, the DCA is valid only for systems where the nonlocal correlations are short ranged.

The diagrammatic expansion of the DCA self-energy is identical to the diagrammaticexpansion of a Nc site cluster with periodic boundary conditions ∗, provided that thecluster Green’s function Gc(K,ω) is the same as the coarse-grained Green’s functionG(K,ω), i.e.

Gc(K,ω) = G(K,ω) (5.6)

The bare Green’s function of the cluster, G, is obtained from the equation

G−1(K,ω) = G−1c (K,ω) + Σ(K,ω) (5.7)

It can be written asG−1(K,ω) = iω + µ− εK − Γ(K,ω) (5.8)

and it corresponds to a cluster without interaction hybridizing with a host, via the hy-bridization function Γ(K,ω).

Therefore, the DCA maps the lattice into a Nc site cluster with periodic boundaryconditions embedded self-consistently in a host. The self-consistent condition is obtainedimposing Eq. 5.6, with G(K,ω) given by Eq. 5.5, where

G(K + k, ω) =1

iω + µ− εK+k − Σ(K,ω)(5.9)

We want to stress again that in DCA only the irreducible quantities are approximatedby the corresponding cluster ones. The lattice self-energy is identified to the clusterself-energy, but the lattice and the cluster Green’s functions are different, as can beseen from Eq. 5.9 and Eq. 5.6. Analogues, the lattice irreducible four legs interactionvertices, Γ’s, are approximated by the corresponding cluster ones, but the lattice andthe cluster susceptibility functions are not equivalent. In general it can be shown thatthe coarse-graining procedure introduces an error of order O(∆k6) in the calculation ofcompact diagrams and an error of order O(∆k2) (so much larger!) in the calculationof non-compact diagrams [19]. Therefore, only quantities like self-energy and irreducibleinteraction vertices which can be derived directly form the compact part of the free energyare well approximated by the corresponding cluster quantities.

The general prescription for the DCA algorithm is the following:

∗The non-local interactions, V (q), if present, should also be coarse-grained in the DCA.

5.3. Dynamical Cluster Approximation 91

1. Make a guess for the self-energy (e.g. take its perturbation theory value or take itzero).

2. Calculate the coarse-grained Green’s function (Eq. 5.5) using the lattice Green’sfunction (Eq. 5.9).

3. Calculate the bare cluster (cluster excluded) Green’s function (Eq. 5.7).

4. Calculate the full cluster Green’s function and the cluster self-energy. This step isthe most time consuming part of the algorithm, and different techniques can be usedfor this purpose. We use a Quantum Monte Carlo solver, which has the advantagethat it is exact. The QMC algorithm will be introduced briefly in the Section 5.3.2.

5. With the new self-energy go back to step 2. Repeat the cycle until the convergenceis reached.

6. After the convergence is reached, run one more cycle and calculate the cluster two-particle Green’s functions.

7. Identify the lattice self-energy with the cluster self-energy.

8. Having the cluster two-particle Green’s functions, invert the Bethe-Salpeter equa-tion, and extract the cluster irreducible Γc functions. Identify the lattice irreducibleΓ functions with the corresponding cluster ones. Use the Bethe-Salpeter equationto calculate the lattice susceptibilities. The calculation of susceptibility functions isdiscussed in Section 5.3.3.

5.3.2 Quantum Monte Carlo Algorithm

The Quantum Monte Carlo solver of the cluster embedded in a host problem is a gener-alization of Hirsch-Fye algorithm [15] used for impurity problems.

For the initial input we have the bare cluster Green’s function, G and the interactionof the cluster degrees of freedom which we assume to be local. In the path integralformulation, the cluster partition function is

Z =

∫Dγ∗Dγe−

∫ β

0dτdτ ′

∑i,j,σ γ

∗i,σ(τ)G−1(i, τ ; j, τ ′)γj,σ(τ ′)− ∫ β

0dτ

∑iHI(γ

∗i,σ(τ), γi,σ(τ))

(5.10)Here γ are the Grassmann variables which describe the cluster degrees of freedom. G(i, τ ; j, τ ′)is the bare cluster Green’s function written in site and imaginary time representation. HI

is the interaction part of the Hamiltonian, local in space and time. We consider it to haveHubbard like form

HI = U

Nc∑i=1

(ni↑ − 1/2)(ni↓ − 1/2) (5.11)

QMC algorithm considers the discrete version of Eq. 5.10. If the imaginary timeinterval is divided into Nl pieces of size ∆τ , so that β = Nl∆τ , up to an error of O(∆τ 2),


Eq. 5.10 can be written as

Z ∝∫ Nl∏

l,l′=1

Nc∏i,j=1

dγ∗il,σdγil,σe−∑

il,jl′,σ γ∗il,σG−1

il;jl′γjl′,σe

∑i,l−∆τHI(γ

∗il, γi(l−1)) (5.12)

The interacting part of the Hamiltonian, which is quadratic, can be made bilinear by intro-ducing the discrete, Ising type, Hirsch-Hubbard-Stratonovich auxiliary field (HHS) [20],s,

e−∆τHI = e−∆τU∑

i(ni↑ − 1/2)(ni↓ − 1/2) =1

2e−∆τU/4

∏i

∑si=±1

eαsi(ni↑ − ni↓)

(5.13)

where cosh(α) = e∆τU/2. At every site, the field si can take only two values, +1 andrespectively −1.

Considering Eq. 5.13, Eq. 5.12 can be written as

Z ∝ Trsil

∫ Nl∏

l,l′=1

Nc∏i,j=1

dγ∗il,σdγil,σe−∑

il,jl′,σ γ∗il,σG−1

il;jl′γjl′,σe−∑

i,l,σ αsilσγ∗il,σγi(l−1),σ

(5.14)Now the electronic degrees of freedom (Grassmann variables) can be integrated out, sothat

Z ∝ Trsil∏σ

det(G−1s,σ) (5.15)

where(Gs,σ)−1

il,jl′ = G−1il;jl′ + ασsilδi,jδl′,l−1 (5.16)

Thus, the calculation of the partition integral is reduced to a sum over the field con-figurations s of the product of two determinants. The ratio between two configurationsweight, corresponding to s′ and respectively s is

R ≡ P ′sPs

=det(G↑) det(G↑)det(G′↑) det(G′↑)

(5.17)

It is not difficult to show [16] that if two configurations differ by a HHS filed flip at sitem ∗, than

R =∏σ

[1 + (1−Gσ,m,m)(e−ασ(sm−s′m) − 1)]−1 (5.18)

and

G′σij = Gσij +(Gσim − δi,m)(e−ασ(sm−s′m) − 1)

1 + (1−Gσ,m,m)(e−ασ(sm−s′m))Gσmj (5.19)

The QMC procedure is initialized by setting G = G and choosing the HHS fileds = 0. Afterwards, using Eq. 5.19, we construct Green’s functions, Gσ, corresponding toa valid HHS filed, i.e. si = ±1. The Metropolis algorithm, introduced in Section 2.1,

∗m labels both the lattice site and the time index, i.e. m ≡ il

5.4. Conclusions 93

can be applied from now on. New HHS field configurations are proposed, and the tran-sition is accepted with the probability given by Eq. 5.18. If a transition to a new fieldconfiguration is accepted, it will be followed by a corresponding update of Gσ, accord-ing to Eq. 5.19. Aside from the cluster single-particle Green’s functions Gσ, the clustertwo-particle Green’s functions can also be measured. Because the introduction of HHSfield decouples the electronic interaction, for every particular HHS filed configuration thetwo-particle Green’s functions are in fact a sum of two single-particle Green’s functionsproducts. At the end of the QMC run, the cluster single-particle and two-particle Green’sfunctions are obtained by averaging the corresponding HHS field dependent quantitiesover the all generated field configurations.

5.3.3 Susceptibility Functions

The two-particle Green’s functions satisfy the Bethe-Salpeter equation

χq,σσ′(k, k′) = χ0

q,σσ′(k, k′) +

∑

k′′,k′′′χ0

q,σσ′′(k, k′′)Γq,σ′′σ′′′(k

′′, k′′′)χq,σ′′′σ′(k′′′, k′) (5.20)

The index k labels both the momentum and the frequency. χ0 is the noninteracting two-particle Green’s function obtained by convoluting two single-particle Green’s functions.

The susceptibility functions have the general form

χq =∑

k,k′vkk′,σσ′χq(k, k

′) (5.21)

The coefficients vkk′,σσ′ depend on the particular susceptibility function desired to becalculated.

In the last QMC run, the cluster two-particle Green’s functions are calculated. Theyobey the Bethe-Salpeter equation (Eq. 5.20) written for the cluster quantities

χc = χ0c + χ0

cΓcχc (5.22)

From this equation the cluster irreducible Γc functions can be extracted

Γc = χ0−1c − χ−1

c (5.23)

In DCA Γc is considered a good approximation for the lattice Γ. Therefore it can be usedin Eq. 5.20 to calculate the lattice two-particle Green’s functions. Subsequently Eq. 5.21will be used for computing the lattice susceptibility functions.

5.4 Conclusions

In this chapter we introduced the Dynamical Cluster Approximation technique. Weshowed that this is a generalization of DMFT to include short ranged non-local cor-relations and that DCA restores partially the momentum conservation at the internalvertices of the self-energy. Unlike the DMFT which maps the problem into an impurity


embedded self-consistently in a host, the DCA maps the problem into a cluster with pe-riodic boundary conditions embedded self-consistently in a host. Non-local correlationsup to the cluster size are treated explicitly, while the physics on longer length scales istreated at the mean field level. The properties of the embedded cluster can be calculatedexactly with a Quantum Monte Carlo (QMC) algorithm. The cluster self-energy is used tocalculate the properties of the host, and this procedure is repeated until a self-consistentconvergent solution is reached. The self-energy and vertex functions of the cluster arethen used to calculate lattice quantities. This technique is ideal for the problems whereshort-ranged correlations are predominant, as in the high-Tc materials.

5.4. Conclusions 95

References

[1] M. Jarrell, Phys. Rev. Lett. 69, 168 (1992).

[2] A. Georges and G. Kotliar, Phys. Rev. B 45, 4679 (1992).

[3] M. H. Hettler et al., Phys. Rev. B 61, 12739 (2000).

[4] M. H. Hettler et al., Phys. Rev. B 58, 7475 (1998).

[5] G. Kotliar et al., Phys. Rev. Lett 87, 186401 (2001).

[6] T. Maier and M. Jarrell, Phys. Rev. B 64, 041104 (2002).

[7] G. Biroli and G. Kotliar, Phys. Rev. B 65, 155112 (2002).

[8] K. Aryanpour et al., preprint, cond-matt/0301460.

[9] T. Pruschke et al., Adv. Phys. 42, 187 (1995).

[10] A. Georges et al., Rev Mod. Phys. 68, 13 (1996).

[11] E. Muller-Hartmann, Z. Phys. B: Condens. Matter 74, 507 (1989).

[12] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989).

[13] N. E. Bickers et al., Phys. Rev. Lett 62, 961 (1989).

[14] N. E. Bickers et al., Phys. Rev. B 36, 2036 (1987).

[15] J. E. Hirsch and R. M. Fye, Phys. Rev. Lett 56, 2521 (1986).

[16] M. Jarrell et al., Phys. Rev. B 64, 195130 (2001).

[17] A. A. Abrikosov, L. P. Gorkov, and I. Dzyaloshinski, Methods of Quantum FiledTheory in Statistical Physics (Dover, New York, 1975).

[18] G. Baym, Phys. Rev. 127, 1391 (1962).

[19] K. Aryanpour et al., Phys. Rev. B 65, 153120 (2002).

[20] J. E. Hirsch, Phys. Rev. B 28, 4059 (1983).

96

Chapter 6

Multi-Band Hubbard Model

6.1 Introduction

The high Tc cuprate superconductors are layered materials with relatively complex struc-tures and chemical composition. They are highly correlated, with an effective bandwidthroughly equal to the effective local Coulomb interaction. The short-range correlations areknown to play a paramount role in these materials. Therefore the DCA technique weintroduced in Chap. 5 we believe to be an ideal tool for the investigation of these systems.The undoped materials are antiferromagnetic insulators with a gap of approximatively2 eV . Upon doping the antiferromagnetism is destroyed and the system becomes super-conducting. At small doping, in the proximity of the antiferromagnetic phase, the normalstate physics cannot be described in terms of Fermi liquid theory and is characterized bythe presence of a pseudogap. An essential demand of every successful theory is to captureall these fundamental features at the same time.

A common characteristic all the cuprate materials share is the presence of quasi-two-dimensional CuO2 planes. It is widely believed that the relevant low-energy physics isdescribed by the degrees of freedom belonging to these planes. The full complexity of theorbital chemistry of just the CuO2 planes and the strong Coulomb repulsion on the Cuions would lead to models which are very difficult to study with conventional techniques.

Starting from the insulating parent materials, the photoemission experiments showthat the first electron-removal states have primarily oxygen p character, unlike the firstelectron-addition states which have Cu d character. This fact places these materials inthe charge-transfer region of Zaanen-Sawatzky-Allen scheme [1]. Early on, considering theligand field symmetry and band structure calculation [2–4], it was realized that the mostimportant degrees of freedom are the Cu dx2−y2 and the in-plane O pσ orbitals. Therefore,one of the first models proposed to describe the physics of high Tc materials was the socalled three-band Hubbard model introduced by Varma et al. [5] and Emery et al. [6, 7].This Hamiltonian considers only the oxygen pσ and the correlated cooper dx2−y2 orbitals.

One step further was taken by Zhang and Rice [8] who argued that the low-energyphysics of the hole-doped superconductors can be described by a one-band t-J model orby the closely related single-band Hubbard model. Starting from the three-band modeland neglecting the oxygen dispersion, Zhang and Rice showed , within second order per-turbation theory, that an extra hole added into the oxygen band binds strongly with a

97

98 Chapter 6. Multi-Band Hubbard Model

hole on the Cu, forming an on-site singlet. This singlet state, which has zero spin canbe thought as moving through the lattice like a hole in an antiferromagnetic background.Consequently, the physics can be described by a one-band t-J model and equivalentlythe low-energy electron-removal states can be mapped onto a lower Hubbard band of asingle-band Hubbard model.

Pertinent criticism to these simplified models were raised by various authors. Withrespect to the Cu degrees of freedom, H. Eskes et al. [9] stressed the possible importanceof the other d orbitals. In order to determine the relevance of different parameters andorbitals, they calculated the CuO4 cluster’s states considering all 12 O p and all 5 Cu dorbitals and compared the results with the photoemission and with the x-ray absorptionexperimental spectra. The d3z2−r2 , dxy, dxz and dyz orbitals enter in the formation of statesseparated only by ≈ 1.5 eV from the Zhang-Rice singlet states, this separation being muchsmaller than the Zhang-Rice singlet-triplet splitting (≈ 3.5 eV ). Especially the d3z2−r2

orbitals seem to be important since they are strongly stabilized by the presence of shortbond lengths with the apex oxygen atoms. Therefore, our model which will neglect allthe other Cu orbitals except the dx2−y2 ones, will not be suited to describe physics whichimplies excitations with energy larger than ≈ 1 eV . In fact it should be remarked thatall the simple model Hamiltonians suggested for the cuprates are not valid when tryingto describe the physics at energy scales larger than 1 eV . For example the optical spectraor the high energy photoemission lines shapes involve the full complexity of the manyorbital case at energy scales above 1 eV .

Investigating the relative importance of the various parameters describing the CuO2

planes [9–11] it was realized that in addition to the Cu on-site Coulomb repulsion (Udd ≈8 eV ) and Cu-O hopping integral (tpd ≈ 1.3 − 1.5 eV ) the O-O hopping integrals resultin a large O 2p bandwidth (W ≈ 5.2 eV ). Even though the Zhang-Rice theory is widelyaccepted nowadays, probably the most serious critique to it, is the neglect of the O 2pband structure. If one takes a different from Zhang and Rice starting point and considerstpd = 0, the first hole addition states will belong to the O band. When tpd is switchedon, these states will Kondo scatter with the Cu spins and some states will appear atthe bottom of the oxygen band. One important question we address here is whether thelow-energy electronic states are well separated from the non-bonding oxygen states, so theneglect of the remaining higher energy states and the reduction to a single-band modelis possible. This problem was previously considered by Eskes and Sawatzky [12] withinan impurity calculation approach, but there both the spatial correlation effects and thedispersion of the low-energy states were neglected.

Other objections to Zhang-Rice assumption were raised by Emery and Reiter[13–15],who showed that in a ferromagnetic background the low-energy states near the Γ point(M in their convention) are an admixture of the Zhang-Rice singlet and the correspondingtriplet. This implies a nonzero value for the oxygen spin, and destroys the equivalence ofthese states to holes. They claim that, even though, as a consequence of the strong Cu-Ohybridization low-energy states well separated from the non-bonding oxygen band statesappear, this does not necessarily mean that the physics can be reduced to a t-J or to aone-band Hubbard model. Using the DCA technique which provides a means of includingall the most important parameters and bands we present here a study of the validity ofthe single-band approach.

6.2. Reduction of Five-Band Hubbard Model to Two-Band Hubbard Model 99

Experimental data show that the phase diagram [16, 17] and other physical character-istics like the density of states (DOS) near the Fermi level of the electron [18] and hole [19]doped materials are profoundly different. There could be multiple reasons responsible forthis asymmetry. First, the electron and the hole doped materials are physically differentand aside from the CuO2 planes they contain other elements and have a complex chemi-cal structure. This structural and compositional differences can influence the low-energyphysics. Another reason, closely related with the validity of the Zhang-Rice approxima-tion, could be that, unlike the electron-doped materials, the hole-doped materials cannotbe described by a single-band Hubbard Hamiltonian. And if this reason does not stand upand Zhang and Rice are correct the motive for the electron-hole asymmetry should comefrom the different effective parameters which result after the reduction of the multi-bandmodel to a single-band one. Therefore with regard to the electron-hole asymmetry wepropose to answer to the following questions:

• Does the physics of a pure CuO2 plane contain the expected asymmetry or does theorigin of the asymmetry in real materials come from other influences?

• Is the Zhang and Rice approximation not valid so that the physics of the hole-dopedcuprates cannot be described by a single-band model?

• If Zhang and Rice is correct, what are the effective parameters of the one-bandmodel which are responsible for the electron-hole asymmetry?

6.2 Reduction of Five-Band Hubbard Model to Two-BandHubbard Model

6.2.1 Derivation of the Five-Band Hubbard Hamiltonian

Restricting ourselves to the energy scale below 1 eV , we can assume that the only relevantCu degrees of freedom are the dx2−y2 orbitals, as Eskes et al. [9] has shown. They couplewith the in-plane oxygen px and py orbitals. The situation is presented schematically inFig. 6.1. We work in the hole representation and consider the Cu(d10)O(p6) configurationas the vacuum state. The undoped materials have one hole per CuO2 unit. The param-eters were determined by comparing the Local Density Approximation (LDA) [11, 20],impurity [12] and cluster [9] calculations with experimental data. Their commonly ac-cepted values are around the ones given in the caption of Fig. 6.1. The dx2−y2 orbitalshybridize via the hopping integral tpd with the pσ orbitals (drawn with solid line). Asidefrom the hybridization with the dx2−y2 orbitals, the pσ orbitals hybridize via the hop-ping integral tp1 with the nearest-neighbor pσ and via the hopping integral tp2 with thenearest-neighbor pπ orbitals (dashed line). Therefore all degrees of freedom are coupled,and considering that there are two O and one Cu ions per unit cell, a five-band modelHamiltonian (with four O and one correlated Cu bands) is required.

Anticipating the major importance for the low-energy physics of the Cu-O hybridiza-tion, we choose the oxygen states basis in the following way. First, on every plaquette oftype 1 (see Fig. 6.2), made by four O around a Cu, we construct from the four pσ orbitals


−−

−

−

−

−

−

−− −

+

+

+

+ + +

+

+

+

+tpd

p1

t p2

t

Figure 6.1: A cooper dx2−y2 orbital and the four in-plane oxygen p orbitals which surroundit. The parameters in the hole representation are: tp1 = −0.65 eV , tp2 = 0.35 eV ,tpd = −1.3 eV , Udd = 8.8 eV , Upp = 6 eV , Upd = 1.3 eV , Ep − Ed = 3 eV .

1

Cu Cu

CuO

O

Cu O

OO

O O

O

2

x

Figure 6.2: The CuO2 plane. In constructing the oxygen bands, we consider two kindsof plaquettes. The plaquette of the type 1 (solid line) involves the four O ions around aCu ion and the plaquette of the type 2 (dashed line) includes the four O ions around thevacant site denoted with “×”in the figure.


a linear combination state with x2 − y2 (B1) symmetry, like in Fig. 6.3-a. These states(which we call B1σ) are the only states which hybridize directly with the dx2−y2 orbitals,and if the Zhang and Rice theory is correct they should be the only relevant states forthe low-energy physics. We use these plaquette states to construct translational invariant(band) states in the same fashion as in Ref. [8]. Afterwards we determine the remainingpσ and the other two pπ bands. The advantage of this particular choice of the oxygenstates basis is that the degrees of freedom considered in Zhang and Rice theory can beseparated explicitly from the remaining states.

At site“i” the B1σ state (Fig .6.3-a) is:

|iB1σs 〉 =

1

2

∑

δ

(−1)g(δ)|pσ,s〉i,δ (6.1)

with g(δ) = 0 if δ = (0, 12x) or δ = (−1

2y, 0) and g(δ) = 1 if δ = (1

2y, 0) or δ = (0,−1

2x). s

is the spin index. The anticommutation relation of these states is

cB1σis , cB1σ†

js′ = δss′(δij − 1

4δ<ij>) (6.2)

which implies that the nearest-neighbor states are not orthogonal. Here cB1σis (cB1σ†

is ) isthe annihilation (creation) operator of a B1σ oxygen hole with spin s and at site “i”.From these local states we construct orthogonal Bloch states,

|kB1σs 〉 =

1

β(k)

1

N

∑i

|iB1σs 〉 e−ikri (6.3)

were β(k) is a normalization factor consequence of the nearest-neighbor states nonorthog-onality (Eq. 6.2)

β(k) = [sin2(kx/2) + sin2(ky/2)]1/2 (6.4)

If we define

|pσ,s〉kx =1

N

∑i

|pσ,s〉ix e−ikri (6.5)

where |pσ,s〉ix is the pσ orbital along x direction at position “ri” and analogous

|pσ,s〉ky =1

N

∑i

|pσ,s〉iy e−ikri (6.6)

with |pσ,s〉iy being the pσ orbital along y direction at position “ri”, we obtain from Eq. 6.3and Eq. 6.1

|kB1σs 〉 =

i

β(k)[sin(kx/2) |pσ,s〉kx − sin(ky/2) |pσ,s〉ky] (6.7)

Now it is easy to see that the other pσ band is defined by the states

|kB2σs 〉 =

i

β(k)[sin(ky/2) |pσ,s〉kx + sin(kx/2) |pσ,s〉ky] (6.8)


−

−+ +

−

−+ +

−

−+ +

−

−+ +

−

−+ +

x

+

+

+

+− −

−

−

−

−

−

−

+

+

+

+

b)

a)

−

−+ +

−

−+ +

−

−+ +

−

−+ +

−

−+ +

+

+

+

+− −

−

−

−

−

−

−

+

+

+

+

c)

d)

x

B1σ πB1

σB2 B2π

Figure 6.3: a) B1σ state. Combination with x2 − y2 (B1) symmetry of the pσ orbitalson a plaquette of type 1. b) B2σ state. Combination with xy (B2) symmetry of the pσ

orbitals on a plaquette of type 2. c) B1π state. Combination with x2−y2 (B1) symmetryof the pπ orbitals on a plaquette of type 2. c) B2π state. Combination with xy (B2)symmetry of the pπ orbitals on a plaquette of type 1.

which are orthogonal on |kB1σs 〉 states.

Going back from momentum to site representation, we find that the |kB2σs 〉 states are

obtained by taking a linear combination with xy (B2) symmetry of the four O pσ orbitalsaround a plaquette of type 2 (Fig. 6.3-b), and afterwards constructing Bloch translationalinvariant states.

We construct the remaining two pπ bands in a similar way. First one can noticethat the situation is identical with the previous one (the one regarding the pσ bandsconstruction) if the Cu sites are interchanged with the vacant sites denoted by “×”inFig. 6.2 (or equivalently if the plaquette 1 is interchanged with the plaquette 2). TheB1π band is obtained from the states with x2 − y2 symmetry formed by the four O pπ

orbitals on a plaquette of type 2 (Fig. 6.3-c), and the B2π band from states with xysymmetry formed by the four O pπ orbitals on a plaquette of type 1 (Fig. 6.3-d).

|kB1πs 〉 =

i

β(k)[sin(kx/2) |pπ,s〉kx − sin(ky/2) |pπ,s〉ky] (6.9)

|kB2πs 〉 =

i

β(k)[sin(ky/2) |pπ,s〉kx + sin(kx/2) |pπ,s〉ky] (6.10)

with


|pπ,s〉kx =1

N

∑i

|pπ,s〉ix e−ikri (6.11)

|pπ,s〉ky =1

N

∑i

|pπ,s〉iy e−ikri (6.12)

The Hamiltonian written in the above constructed basis (which consists of the fivebands, namely d, B1σ, B2σ, B1π and B2π) is

H = H0 +HU (6.13)

H0 is the one-particle part of the Hamiltonian which describes the bands dispersion

H0 =∑

ks

Hks (6.14)

with

Hks =(d†ks cB1σ†

ks cB2σ†ks cB1π†

ks cB2π†ks

)×

×

Ed(k) Vpd(k) 0 0 0Vpd(k) E1(k) Vp1(k) Vp2(k) 0

0 Vp1(k) E2(k) 0 Vp2(k)0 Vp2(k) 0 E1(k) Vp1(k)0 0 Vp2(k) Vp1(k) E2(k)

dks

cB1σks

cB2σks

cB1πks

cB2πks

(6.15)

and

Ed(k) = Ed (6.16)

E1(k) = Ep + 8tp11

β(k)2sin2(kx/2) sin2(ky/2) (6.17)

E2(k) = Ep − 8tp11

β(k)2sin2(kx/2) sin2(ky/2) (6.18)

Vpd(k) = 2tpdβ(k) (6.19)

Vp1(k) = 4tp11

β(k)2[sin2(kx/2)− sin2(ky/2)] sin(kx/2) sin(ky/2) (6.20)

Vp2(k) = −4tp2 cos(kx/2) cos(ky/2) (6.21)

If tpd = 0 the oxygen bands with the same symmetry would be degenerate, as can beseen from Eq. 6.15. The bottom of the oxygen band has B1 symmetry and containsboth B1σ and B1π states. The mixing of B1σ and B1π given by Eq. 6.21 is strong,modifying significantly the DOS of the noninteracting B1 bands. When tpd is switched


on, the B1σ band mixes with the d one, and bonding low-energy states with mainly B1σand d character will appear. From the oxygen degrees of freedom in order of importance,with respect to the low-energy physics, the B1π are the next after the B1σ states, andthere are no priori reasons to neglect these states. However these pπ orbitals are moreimportant than the pσ ones which enter in the B2σ band.

HU contains the Coulomb repulsion terms

HU = Udd

∑i

ndi↑ndi↓ + Upp

∑

i,α,α′,s,s′npiαsnpiα′s′ + Upd

∑

i,s,s′,α

npiαsndis′ (6.22)

with α and s being the oxygen band and respectively the spin index. The first term ofEq. 6.22 is the Coulomb repulsion between two holes on a dx2−y2 orbital. The second termis the Coulomb repulsion between two holes on the same O and the third represents therepulsion between one O hole and one nearest-neighbor Cu hole. Even if theoretically it ispossible to treat within the DCA framework all these Coulomb terms, practically this isunfeasible at the moment because of the limited computer resources available. However,for the physical regime relevant for the high Tc superconductors, the occupation of theoxygen ions is rather small (below 30%), therefore we assume that treating the last twoterms of Eq. 6.22 in a mean-field way is pretty good. The mean-field approach results inan increase of the charge transfer parameter ∆ = Ep − Ed by Upp

np

2+ Updnd so

∆eff = Ep − Ed + Uppnp

2+ Updnd (6.23)

Starting with the parameters given in Fig. 6.1 and approximating the oxygen occupationwith np ≈ 0.3 we obtain a value of ∆eff = 4.8 eV . We did not find dramatic changes inthe physics with small modifications in ∆, therefore we are going to keep the value of ∆constant for all doping levels.

To conclude this section, our model Hamiltonian is given by the Eq. 6.13 with Upp = 0,Upd = 0 and ∆ = Ep − Ed = 4.8 eV . The other parameters value is given in Fig. 6.1’scaption.

6.2.2 DCA Applied to the Five-Band Hubbard Model

It is straightforward to apply the DCA technique to our five-band model. When U = 0,the Green’s function is defined by

G−10s (k, iω) = iω −Hks (6.24)

When the Coulomb interaction is switched on, the Green’s function is obtained fromEq. 6.24 by replacing Ed with Ed + Σ(K, iω). This is a 5× 5 matrix which can be easilyinverted analytically. During the coarse-graining process, G(k, iω) is needed in all k and ωpoints, therefore, for the computation speed reasons it is important to have an analyticalexpression of the Green’s function. We used Mathematica for the analytical inversion.The calculation of the cluster excluded Green’s function G implies only the d part of thecoarse-grained G

G−1(K, iω) = G−1dd (K, iω) + Σ(K, iω) (6.25)

and from this point the calculation of Σ(K, iω) and ΓQ,iν(K, iω;K ′, iω′) is identical withthe one presented in Chap. 5.


0

0.2

0.4

B1B1B2B2doping

0

0.2

0.4

-4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 00

0.2

0.4

σ

σπ

π

µ

a)

b)

c)

(eV)

Figure 6.4: Occupation of the four oxygen bands versus chemical potential at T = 1885 K.a) ∆ = 4.8eV, U = 8.8eV . b) ∆ = 3.5eV, U = 8.8eV . c) ∆ = 3.5eV, U = 11eV . Thesolid line represents the hole doping. Notice that up to 40% doping the occupation ofthe oxygen bands which do not hybridize directly with Cu is less than 1%. The secondoxygen band which starts to populate when doping increases above 40% is B1π.

6.2.3 Comparison of Five-Band and Two-Band Hubbard Models

In this section we study the importance of the different oxygen degrees of freedom bycalculating the five-band Hubbard model properties with the DCA technique. An indi-cation of the relevance to the low-energy physics of the different oxygen bands would begiven by their occupation number upon doping. Finally, we compare the five-band modelproperties with a reduced two-band model (which we suspect to approximate the low-energy physics) ones, and decide about the validity of this approximation. In the DCAalgorithm we take a cluster of 2× 2 Cu ions, which we believe is large enough to capturequalitatively the essential physics of cuprates.

Upon hole doping, the extra holes go primarily on the oxygen. We plotted in Fig. 6.4the oxygen bands occupation number as a function of the chemical potential. Notice thatincreasing the chemical potential is equivalent to increasing the hole doping. It can beseen that only one oxygen band, the B1σ, is populated up to 40% hole doping, the otherthree oxygen bands (B2σ, B1π, B2π) being very sparsely occupied (less than 1%) evenat high temperature (1885K). When the temperature is lowered their occupation numberis even less. The realistic parameters are the ones in Fig. 6.4-a, but our conclusion holdseven for different values of U and ∆, as long as the Cu-O hopping tpd is significant, ascan be seen form Fig. 6.4-b and Fig. 6.4-c. As we have discussed earlier, only the B1σband directly hybridizes with Cu. This hybridization is very strong and not only due tothe large tpd but mainly because of the symmetry and phase coherence between the dx2−y2


-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

10

0.2

0.4

0.6

0.8

1

U

∆

ω (ev)

DO

S

d8d

10

d L9

a)

b)

B1σ

µ

Figure 6.5: Density of states of the five-band Hubbard a) and the two-band Hubbard b)model. The doted-dashed line represents the total DOS before the Cu-O hopping tpd isswitched on. The solid line is the d part of the DOS calculated with MEM at T = 837Kin the undoped region, µ = −4.55 eV . Inset: Comparison of the low-energy partial dDOS between five-band (solid line) and two-band (dashed line) models.

and B1σ states (at the (π, π) point the hybridization matrix element is 2√

2tpd ≈ 3.7eV ).As a consequence, low-energy states with the energy well enough below the energy of thenon-bonding oxygen states and with only d and B1σ character will appear. The otherthree oxygen bands do not couple directly with Cu and they do not enter in the formationof the low-energy states.

The density of states (DOS) in the undoped case can be seen in Fig. 6.5-a. Whentpd = 0 the hole addition states have pure oxygen character. Their energy is ∆ −W/2.The DOS for tpd = 0 is shown with dash-dotted line. The B1σ part of the oxygen DOS isshown with dashed line. As long as tpd = 0 the B1σ density of states is degenerate withthe B1π one. The higher in energy part of the oxygen DOS is made by the degenerateB2σ and B2π bands. When tpd is switched on, the first hole addition states have a mixedcharacter (d and p) and are pushed to lower energies. The partial d DOS of these states isdrawn with the solid line, and was obtained using Maximum Entropy Method (MEM)[21]for the analytic continuation of the QMC data to real frequencies.

The B1σ density of states is strongly influenced by the hybridization of the B1σ stateswith the other oxygen states, especially with the ones belonging to B1π band. Therefore,even though the low-energy states have only d and B1σ character, the other oxygen bands,non-directly hybridizing with Cu, have an indirect influence. In order to estimate this


two-band Hubbard five-band Hubbard relative µdeviation

np 0.2650 0.2722 2.5%nd 0.7428 0.7376 0.7%χc

charge 0.2117 0.2206 4% -4.55 eV〈(ndi↑ − ndi↓)2〉 0.7328 0.7273 0.8%T ∗ χlocal 0.4482 0.4390 2%np 0.4448 0.4723 5%nd 0.8094 0.8074 0.2%χc

charge 0.2290 0.2288 0.3% -3.51 eV〈(ndi↑ − ndi↓)2〉 0.7433 0.7367 0.8%T ∗ χlocal 0.3377 0.3116 7%

Table 6.1: Comparison of different cluster quantities calculated for the two-band modeland the the five-band model at T = 1885K.

influence, we compare the properties of the five-band Hubbard model with the ones of atwo-band Hubbard model which considers only the Cu dx2−y2 and the oxygen B1σ states.The two-band Hubbard Hamiltonian can be obtained from the five-band Hubbard one(Eq. 6.13) taking Vp1 = 0 and Vp2 = 0 in Eq. 6.15. This leads to

H =∑

ks

Ep1(k)c†kscks + Edd

†ksdks + Vpd(k)(d

†kscks + h.c.) + U

∑i

ndi↑ndi↓ (6.26)

where now c† (c) represents the B1σ hole creation (annihilation) operator.The two-band Hubbard model density of states is shown in Fig. 6.5-b. The B1σ oxygenDOS looks very different from the five-band Hubbard model one. Nevertheless, we foundthat the low-energy DOS is very similar in both cases. The difference between them canbe seen in the inset of Fig. 6.5. In the five-band model the gap is a little smaller andthe first hole addition states are “pushed” a little further from the oxygen band. Thispresumably is due to the fact that the oxygen DOS, which mixes with Cu, is larger atthe bottom of the oxygen band (where the hybridization is the strongest). However, thedifference between the two cases is small and does not change the qualitative physics.Even the quantitative physics differs very little, as can be noticed from Table 6.1, wherea comparison between different cluster quantities calculated for the two-band and forthe five-band Hubbard models is presented. np and nd are the B1σ and respectivelythe Cu occupation number, χc

charge is the charge susceptibility measured on the cluster,〈(ndi↑ − ndi↓)2〉 is the unscreened moment on a Cu site and

T ∗ χlocal =T

N

∑i

∫ β

0

〈S−i (τ)S+i (0)〉dτ , (6.27)

with Si being the Cu spin operator at site “i’, is the screened Cu moment. The differencebetween the two cases is only a few percent.


So far we can conclude that for the low-energy physics the Cu-B1σ hopping integral isthe most important one, much more important than the hopping terms which determinethe oxygen band dispersion. As a consequence of this strong Cu-B1σ mixing, the low-energy states are well stabilized with respect to the initial (i.e. when tpd = 0) oxygenband and have only dx2−y2 and B1σ character. The other than B1σ oxygen bands can beneglected to a a very good approximation, and a two-band Hubbard model is well suitedfor describing the low-energy physics of the CuO2 planes.

At the end of this section we want to make a remark about the importance of theB1π band. As can be seen from Fig. 6.4, this is the first oxygen band after the B1σwhich starts to populate upon hole doping. Therefore, if for some reasons, like describingexperiments which explore high energy excitations (of order of 1 − 2 eV ), a three-bandmodel is necessary, we believe that it should contain all the bands with x2−y2 symmetry,i.e. dx2−y2 , B1σ and B1π, quite opposite to what is commonly proposed, namely a three-band model with the dx2−y2 and the pσ (B1σ and B2σ) bands.

6.3 Two-Band Hubbard Model

6.3.1 Phase Diagram and Other Properties

In this section we present the DCA calculation of the two-band Hubbard model. TheHamiltonian is given by Eq. 6.26 and according to the conclusions of Section 6.2 weassume it describes very well the low-energy physics of the CuO2 planes. Due to the largecomputational effort, our cluster is still the smallest one which allows an explicit treatmentof the short-range spatial correlation and permits d-wave pairing, and consists of four Cuions. As we are going to show, the calculation on even this small cluster, captures much of

-6 -5.5 -5 -4.5 -4 -3.5 -3µ

0.8

1

1.2

n

T= 420 KT=1885 K

Figure 6.6: The hole occupation number n, versus chemical potential µ.

6.3. Two-Band Hubbard Model 109

0 0.1 0.2T (eV)

0

0.05

0.1

χ

n=0.90, Nc=4n=0.93, Nc=4n=0.95, Nc=4n=0.98, Nc=4n=1.00, Nc=4n=1.02, Nc=4n=1.05, Nc=4n=1.00, Nc=1

-1 AFM

Figure 6.7: Inverse of the antiferromagnetic susceptibility χ−1AFM versus temperature cal-

culated for Nc = 4 at different hole densities and for Nc = 1 at n = 1.00.

0 0.2 0.4 0.6 0.8 1T (eV)

0

5

10

15

20

n=1.05n=0.95n=1.10n=0.90

χ-1 SC

Figure 6.8: Inverse of the d-wave pairing susceptibility χ−1SC versus temperature for differ-

ent hole densities.


the essential physics of cuprates, the phase diagram resembling the generic phase diagramof high Tc materials. We found regions characterized by antiferromagnetism, pseudogapand d-wave superconductivity for both electron and hole doped systems.

The undoped system is an insulator. This can be seen in Fig 6.5-b, where at T = 837 Kthe DOS exhibits a gap of 0.6 eV . When the temperature is lowered, the value of the gapincreases. Based on the results presented in Fig 6.6, where the hole occupation numberas a function of chemical potential is shown, we estimate a gap of ≈ 1.1 eV − 1.2eVat T = 0 K. This gap is smaller than the real gap found in the undoped cupratessuperconductors (≈ 1.5 eV − 2 eV ), but this is mainly due to our small cluster choice(Nc = 4). We expect an increase in the value of gap for larger clusters, similar to whatwas found in the one-band Hubbard model calculation [22].

The phase diagram was determined by calculating the relevant susceptibilities. Adivergence in the susceptibility function at a certain temperature indicates that the systembecomes instable, and a phase transition takes place. We calculated a large number ofsusceptibilities which are relevant for spin, charge and superconducting ordering, both atthe center and at the corner of the Brillouin Zone.

In the undoped regime and at small doping (both with holes and with electrons) wefound the antiferromagnetic susceptibility to diverge, meaning that a transition to anantiferromagnetic state takes place over there. In Fig. 6.7 we show the inverse of theantiferromagnetic susceptibility χ−1

AFM as a function of temperature, at different dopingconcentrations. The transition temperature (the Neel temperature) TN is the temperaturewhere χ−1

AFM reaches the value zero. In the same figure, for comparison, we also show theDMFT result (i.e. Nc = 1 case). The non-local fluctuations allowed in DCA reduce thevalue of the Neel temperature and make the transition critical exponent γ larger than 1(always γ = 1 in DMFT).

We computed different pairing susceptibilities, with different symmetries, as s-wave,p-wave and d-wave of both odd and even frequency and we looked for paring at boththe zone center and zone corner. From all of these only the d-wave pairing at the zonecenter diverges. In Fig. 6.8 we show the inverse of the d-wave pairing susceptibility χ−1

SC

versus temperature. The d-wave superconductivity starts at very low doping (≈ 1%) andpersists up to ≈ 24% electron doping and ≈ 30% hole doping. For the superconductingtransition the critical exponent γ is smaller than 1.

Using the Maximum Entropy Method we calculated the single particle density ofstates at different doping concentrations. The total d part of the low-energy DOS at5% doping is shown with thick solid line in Fig. 6.9, for both the electron and holedoped case. Below a characteristic temperature T ∗, larger than the superconductingcritical temperature Tc, we notice a suppression of states at the chemical potential, whichindicates the presence of a pseudogap. The pseudogap increases decreasing temperature.Since the MEM calculation of the DOS becomes computationally extremely expensive atlower temperatures, we plotted the DOS at the highest temperature where the pseudogapwas clearly seen. This temperature does not necessarily have to be the same for theelectron and hole doped case (see Section 6.3.2 for electron-hole asymmetry discussion).

The uniform spin susceptibility which is shown in the upper part in Fig. 6.10 alsodisplays evidence of the pseudogap. At the pseudogap temperature T ∗ the spin excitationsare suppressed and a downturn in the susceptibility function is seen. The suppression


-4 -3 -2 -1 0 1 2 3 4

0

0.2

0.4

0.6

0.8

Inte

nsity

(a.

u.)

-4 -3 -2 -1 0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

Inen

sity

(a.

u.)

total d DOS(0 , 0) d DOS(0 , ) d DOS( , ) d DOSπ

ππ

ω (eV)

a)

b) electron addition removal

pseudogap

electron pseudogap

T=900 K

T=730 K

Figure 6.9: a) Total d DOS and coarse-grained K dependent d DOS at 5% doping. a)hole doping case. b) electron doping case.

0 0.2 0.4 0.6 0.8 1T (eV)

0

0.5

1

1.5

2

2.5

3

n=1.05n=0.95n=1.10n=0.90

charge susceptibility

spin susceptibility

Figure 6.10: Uniform spin (upper part) and charge (lower part) susceptibilities versustemperature for different hole densities.


of the spin excitations below T ∗ was seen early on in the NMR experiments and wasassociated with the pseudogap [23–25]. An interesting feature is that the uniform chargesusceptibility (shown in the lower part of Fig. 6.10) is strongly enhanced in the underdopedregion below the same T ∗, showing that in the pseudogap phase the spin and the chargedegrees of freedom behave profoundly differently, i.e. the former are suppressed and thelater are enhanced.

The phase diagram calculated for Nc = 4, is shown in Fig. 6.11. At small doping, theoverlap of the antiferromagnetic phase and the superconducting one does not necessarilymean that the two phases coexists. Because our algorithm forces the system to be alwaysin a paramagnetic phase, it merely states that if the antiferromagnetic phase is suppressedfor some reasons (e.g. due to impurities or long-range interactions), a transition to asuperconducting state at a smaller temperature is possible. The phase diagram resemblesthe generic phase diagram of cuprates, except the fact that superconductivity phase existsat very small doping concentrations and the maximum Tc is around 5% doping (and notaround 15%, as in real materials). We believe that this is a consequence of our smallcluster choice, and a more realistic phase diagram would correspond to larger clusters, assingle-band Hubbard model calculation shows (see Chap. 7). Unfortunately, due to thelimited computational resources available we could not do calculations on larger clustersfor the two-band model.

Because of the large Cu−O hybridization the system is strongly covalent. For examplein the undoped regime the Cu occupation number is only ≈ 73%. This is in agreementwith cluster [9] and impurity [12] calculations and also with NMR measurements [26, 27].The Cu−O hybridization (Eq. 6.19) is strongly k dependent, its value taking values from2√

2tpd at (π, π) point to zero at (0, 0) point in the Brillouin Zone (BZ). Consequently thesystem exhibits a slightly doping dependent covalency. This is shown in Fig. 6.12-a wherethe Cu occupation number versus hole density is plotted. A constant covalency, equalto the one in the undoped regime (i.e. 0.73 Cu holes and 0.27 O holes per site), wouldcorrespond to the dashed line. It can be noticed that, for the electron doped regime,the Cu hole occupation number is decreasing faster than the hole concentration, whichindicates an increasing covalency with increasing the electron doping. This is because atlarge electron doping, i.e. when the filling with holes of CuO2 plane is small, the effectivehybridization is a result of a large Vpd(k) in the BZ. Increasing the number of holes, theBZ starts to fill up and a smaller Vpd(k) will be responsible for the hybridization, thereforethe covalency is decreasing. For the hole doped regime, the extra holes go primarily on theoxygen band, and the system becomes more complicated. In Fig. 6.12-a the unscreenedmoment on Cu orbitals is also shown. It is defined as in the Figure 6.12’s caption andcan be written also as

µ2 = nd − 2〈ndi↑ndi↓〉 (6.28)

The difference between nd and µ2 is a measure of the double occupancy with holes of Cusites. In the electron doped regime the double occupancy is very small, but it increasessubstantially in the hole doped regime, which indicates that the low-energy hole additionstates contain double occupied Cu configurations in a significant measure. An importanteffect of the extra holes is to screen the spins on the Cu sites, as can be seen in Fig. 6.12-b.

In the Zhang- Rice scenario an extra hole perfectly screens one spin on Cu forminga strongly bound on-site singlet which would contain a significant amount of the double


0.7 0.8 0.9 1 1.1 1.2 1.3n

0

200

400

600

800

1000

1200

1400

1600

T(K

)T_NT_cT*

d-wave d-wave SC

pseudogap

AFM

electron doped hole doped

SC

Figure 6.11: Two-band Hubbard model phase diagram.

0.6

0.7

0.8

n

n * n

0.7 0.8 0.9 1 1.1 1.2 1.3n (filling)

0.1

0.15

0.2

0.25

2

0

µ

Τ = 538 Κ

d

d

χlocalΤ = 538 Κ

screened moment on Cu, T *

a)

b)

Figure 6.12: a) The Cu occupation number, nd, the unscreened Cu moment, µ2 = 〈(ndi↑−ndi↓)2〉 versus hole filling. b) The screened Cu moment, T ∗ χlocal (Eq. 6.27) versus holefilling.


occupied Cu configuration. So, our results do not contradict the Zhang and Rice theory,but also do not exclude other scenarios where the extra holes form more complicatedbound states which involve more than one Cu spin. Quantitative analysis based on theamount of screening as function of hole doping cannot give an answer to the validity ofZhang and Rice assumption, because, aside from the screening due to the oxygen holes,there are also non-local processes which contribute to the screening of Cu moments (forexample the possible formation of inter-site spin singlets associated with the ResonanceValence Bond scenario). Therefore, in order to decide about the validity of Zhang and Riceapproximation, in Sec. 6.4 we are going to take another approach, making a comparisonof the single-particle and the two-particle properties of the two-band and single-bandHubbard models.

6.3.2 Electron-Hole Asymmetry

The DCA results show that the two-band Hubbard model is not symmetric with respect toelectron and hole doping. The asymmetry is seen in the phase diagram, in the one-particlespectral functions and in the susceptibility functions.

Looking at the phase diagram (Fig. 6.11) it can be seen that, comparative to thehole doped regime, the antiferromagnetism in the electron doped regime persists up to alarger critical doping. With respect to the d-wave superconductivity phase the situationis quite opposite, in the electron doped case the superconductivity being destroyed atsmaller critical doping. These features are in qualitative agreement with experimentalfindings (see Fig. 1.3). The pseudogap temperature measured as the temperature wherethe spin excitations are suppressed also shows strong asymmetry, but one has to becareful here because a larger T ∗ in the electron-doped case does not necessarily mean thatthe pseudogap is stronger (i.e. more developed), as we are going to discuss in the lastparagraph of this section.

Recent photoemission experiments on electron-doped materials [18] show that theangle resolved photoemission spectra (ARPES) are profoundly different from the onescorresponding to hole-doped materials. The 2× 2 cluster limits us to coarse the Brillouinzone in four cells aroundK=(0, 0), (0, π), (π, 0) and (π, π) (see Fig. 6.13), and approximatethe lattice self-energy with Σ(K,ω). Due to this rough coarse-graining the comparison toARPES is not possible aside from gross features, like pseudogap, occurring over a widerange of k space. With respect to the pseudogap, an important difference between the un-derdoped electron and the underdoped hole doped regimes can be seen in Fig. 6.9. There,besides the total d states DOS we also show the d coarse-grained momentum dependentDOS, which would correspond to the average over all k belonging to a coarse-grainedcell of single particle spectra A(k, ω). Unlike the total DOS which looks qualitativelysimilar and displays a pseudogap at the chemical potential, the K dependent DOS arevery different. In the hole doped case the pseudogap appears around (0, π) as can beseen form the thin-solid line plot in Fig. 6.9-a. For the electron doped case we did notdetect any suppression of states around (0, π) (see the thin-solid line plot in Fig. 6.9-b)even though the pseudogap is clearly present in the total DOS. These features are inagreement with experimental data. At small doping, the ARPES of the normal statehole-doped systems exhibits well defined quasiparticle peaks around the (π/2, π/2) point


kx

ky

Figure 6.13: Coarse-graining of the Brillouin Zone in four cells (Nc = 4) aroundK=(0, 0), (0, π), (π, 0) and (π, π).

in Brillouin Zone, and broad gapped states around the (0, π) and (π, 0) points [19]. In theelectron-doped samples, the ARPES measurements show Fermi surface pockets around(0, π) point (and around the symmetry related (π, 0) point), and gapped states near the(π/2, π/2) point [18]. With DCA the presence of the pseudogap at (π/2, π/2) for the elec-tron doped system can by only checked by increasing the cluster size, and some resultsregarding this matter are presented in the Chapter 7.

As can be seen from Fig. 6.10 and Fig. 6.8 the electron and the hole-doped spin,charge and pairing susceptibility functions are also different both for the divergence tem-peratures and the temperature and doping dependence. In the hole doped case, the spinsusceptibility at the pseudogap temperature T ∗ strongly increases with doping unlike inthe electron doped case where it decreases upon doping. At the same doping the holedoped spin susceptibility is much larger than the electron doped one. For the hole dopedcase the downturn in the susceptibility (which defines T ∗) is sharp and moves to lowertemperatures when the doping increases. For the electron doped case the downturn ismuch flatter and moves very slowly to smaller temperatures with doping. The main effectof doping is to make the downturn flatter. We found an almost flat low-temperature spinsusceptibility at 15% electron doping, which is an indication of Fermi-liquid behavior. Asharp downturn results in a rapid transition to the pseudogap physics, and we observedthat at the same temperature the pseudogap in the hole doped case is more developedthan in the electron doped case, even though the electron regime corresponding T ∗ islarger. Another interesting feature is the very strong increase of the uniform charge sus-ceptibility for the electron-doped case in the underdoped region (5% doping), suggesting


a tendency for phase separation (see Chap. 7). With respect to the asymmetry of thed-wave pairing it can be noticed that above Tc the pairing correlations increases with thedoping in the electron doped case and remains more or less the same (actually it slightlydecreases) in the hole doped case. The superconductivity transition critical exponent issmaller in the hole doped case, indicating stronger fluctuations.

6.4 Reduction of Two-Band Hubbard Model to Single-Band Hubbard Model

An important conclusion of the previous section is that the electron-hole asymmetry is anintrinsic property of the CuO2 plane. In this section we are going to address two closelyrelated problems, one concerning the reason for the asymmetry and the other concerningthe reduction of the two-band Hubbard model to a single-band Hubbard one.

In order to decide whether the low-energy physics of cuprate superconductors can bedescribed by a single-band Hubbard Hamiltonian we are going to compare the one-particlespectral functions (density of states) and the two-particle properties (susceptibility func-tions) of the two-band and single-band Hubbard models. Without claiming that thesimilarity of the two models DOS and susceptibility functions would give a completeproof of the fact that the one-band Hubbard model is the effective Hamiltonian whichdescribes the two-band Hubbard model low-energy physics ∗, we believe this is a com-pelling approach for establishing the validity of the Zhang and Rice assumption. Theexperimental information is obtained measuring the response functions behavior (and inalmost all cases the two-particle operators or the one-particle ones, as in photoemission,are involved). It is also true that a comparison of the dynamic susceptibilities would berequired, but with our Quantum Monte Carlo based algorithm the calculation of thesequantities for the two-band model is extremely computational resources consuming andhas not been done yet. However partial information about the relevant excited states iscontained in the temperature behavior of the static susceptibilities.

If Zhang and Rice theory is right, the single-band Hubbard model suited for describingthe physics of CuO2 planes should include at least the next-nearest-neighbor hoppingintegral, therefore we start by comparing the two-band Hubbard model with a t− t′ − Usingle-band Hubbard model

H = −t∑

<i,j>,σ

d†iσdjσ − t′∑

¿i,jÀ,σ

d†iσdjσ + U∑

i

ndi↑ndi↓ (6.29)

In Eq. 6.29 < > and ¿ À means summation over the nearest-neighbor sites and respec-tively next-nearest-neighbor sites. The inclusion of higher than nearest-neighbor orderhopping terms would be a necessary condition for explaining the electron-hole asymme-try, being known that a simple t−U Hubbard model is symmetric with respect to electronand hole doping. Even if different parameters are used for the electron and respectively

∗To prove this we should show that the matrix elements of the single-band Hubbard model resolventoperator are the same with the two-band Hubbard model ones corresponding to the low-energy states [28].This would also imply a comparison of more-than-two-particle operators correlation functions.

6.4. Reduction of Two-Band Hubbard Model to Single-Band Hubbard Model 117

0.7 0.8 0.9 1 1.1 1.2 1.3n

0

200

400

600

800

1000

1200

1400

1600

1800

T(K

)T_N, t’=0T_N, t’/t=-0.30T_c, t’=0T_c, t’/t=-0.30T*, t’=0T*, t’/t=-0.30

electron dopedhole doped

t-t’-U phase diagram

Figure 6.14: t-t’-U Hubbard model (solid line) and t-U Hubbard model (dashed line)phase diagrams for t = −0.45 eV, t′/t = −0.3, U = 0.3.6 eV .

-4 -3 -2 -1 0 1 2 3 4

0

0.3

0.6

0.9

1.2

1.5

Inte

nsity

(a.

u.)

-4 -3 -2 -1 0 1 2 3 4

0

0.3

0.6

0.9

1.2

1.5

Inen

sity

(a.

u.)

total DOS(0 , 0) DOS(0 , ) DOS( , ) DOSπ

ππ

ω (eV)

a)

b) electron addition removal

pseudogap

electron pseudogap

T=560 K

T=560 K

Figure 6.15: a) t− t′−U total DOS and coarse-grained K dependent DOS at 5% dopingfor t = −0.45 eV, t′/t = −0.3, U = 0.3.6 eV . a) hole doping case. b) electron dopingcase.


0 0.2 0.4 0.6 0.8 1T (eV)

0

5

10

15

20

n=1.05n=0.95n=1.10n=0.90

χ-1 SC

t-t’-U Hubbard modelt=0.45 eVU=3.6 eVt’/t=-0.3

Figure 6.16: t-t’-U Hubbard model. Inverse of the d-wave pairing susceptibility χ−1SC versus

temperature for different hole densities.

0 0.2 0.4 0.6 0.8 1T (eV)

0

0.5

1

1.5

2

2.5

3

n=1.05 n=0.95n=1.10n=0.90

charge susceptibility

spin susceptibility

t-t’-U Hubbard model t=0.45 eVU=0.36 eVt’/t=-0.3

Figure 6.17: t-t’-U Hubbard model. Uniform spin (upper part) and charge (lower part)susceptibilities versus temperature for different hole densities.

6.4. Reduction of Two-Band Hubbard Model to Single-Band Hubbard Model 119

for the hole doped regimes, some features like the susceptibilities behavior upon doping orthe momentum dependent one-electron spectral functions could not be explained with asimple nearest-neighbor only hopping Hubbard Hamiltonian. The most significant higherorder hopping parameter is the next-nearest-neighbor one, t′, as explicitly cluster calcu-lations show [29], which has different effective sign for the electron and respectively forthe hole doped case. In a single-band t− t′−U model the effective sign of t and t′ is alsodifferent in the electron and in the hole representation. In the hole representation t < 0and t′ > 0. In order to avoid confusions we must specify that we define the hole (elec-tron) representation as the one where the filling 1+ δ corresponds to the δ hole (electron)doping. The sign of t is only a matter of convention, the change of it would correspondto a translation with a (π, π) vector in the Brillouin Zone, but the t′ hopping sign turnsout to have a major influence on the Hubbard model physics.

We have found very similar physics for the two models. In Fig. 6.14 the phase diagramof the t− t′−U model is shown with solid line ∗. In order to see the effect of t′, the phasediagram of t−U Hubbard model (i.e. t′ = 0 case) is also shown with dashed line. At halffilling t′ introduces an effective antiferromagnetic exchange J ′ = 4t′2/U between the samesublattice spins and subsequently frustrates the antiferromagnetism. However at finiteelectron doping, t′ favors the antiferromagnetism, making it persist up to a larger doping.On the other hand, in the hole doped case, the antiferromagnetism is always suppressedby t′. With respect to superconductivity, the presence of t′ results in a smaller (larger)critical electron (hole) doping at which the superconductivity disappears. t′ has no majorinfluence on the maximum superconductivity critical temperature Tmax

c .In Fig 6.15 we present the t − t′ − U Hubbard model DOS at 5% doping. The one-

particle spectral functions resemble the corresponding two-band Hubbard model ones.The presence of the t′ parameter with the right sign is responsible for the location of thepseudogap in the Brillouin Zone (see also Fig. 7.7 where the DOS calculated for Nc = 8cluster is shown).

The d-wave paring susceptibilities shown in Fig. 6.16 exhibit similar asymmetric fea-tures as the corresponding two-band model ones. In the hole doped regime the t′ hoppingintegral decreases the d-wave paring before the superconductivity transition and increasesthe pairing fluctuations (the critical exponent is smaller, the deviation from the mean fieldvalue 1 being larger).

The uniform spin and charge susceptibilities are shown in Fig. 6.17. The similarity tothe two-band Hubbard model case is evident. The spin susceptibility at the pseudogaptemperature T ∗ is strongly increasing with doping for the hole doped case and an oppositeeffect is seen for the electron doped case. The downturn at T ∗ in the spin susceptibilityis much sharper for the hole doped regime indicating a fast transition to the pseudogapphysics. The electron-doped charge susceptibility is strongly increased below T ∗ in theunderdoped region, similar to what happens for the two-band Hubbard model.

The necessity of the t′ hopping (and even of the third neighbor hopping t′′) in explain-ing the measured ARPES line shape and the electron-hole asymmetry [30–36] was realizedearly on. Representing hopping terms on the same sublattice, these parameters are notseverely renormalized in an antiferromagnetic background and therefore their influence

∗Here we kept the sign of t′ always positive and modified the density of holes, thus that the δ hole(electron) doping would correspond to 1 + (−)δ hole filling per unit cell.


0 0.2 0.4 0.6 0.8T (eV)

-0.06

-0.04

-0.02

0

0.02

0.04

ener

gy (

eV)

0.2 0.4 0.6 0.8 1T (eV)

total energy gainkinetic energy gain

5% doping2.5 % dopinga) b)

t’=0.3 (hole doping)

t’=-0.3 (electron doping)

t’=0.3 (hole doping)

t’=-0.3 (electron doping)

Figure 6.18: The total energy gain (i.e. Etotal(t′ 6= 0) − Etotal(t

′ = 0)) and the kineticenergy gain (i.e. Ekinetic(t

′ 6= 0) − Ekinetic(t′ = 0)) when the t′ parameter is switched on.

The upper part corresponds to t′ > 0 (i.e. hole doping) and the lower part to t′ < 0 (i.e.electron doping). a) At 1.025 unit cell filling. b) At 1.05 unit cell filling.

turns out to be important. The exact diagonalization results [34, 36] of the t − t′ − Jmodel, which are in agreement with ours, are very helpful for understanding the t′’s in-fluence on the ARPES spectra and magnetic properties. The one hole in a t − J model(t′ = 0) calculation predicts a dispersion of the form

E0(k) = −0.55J(cos(kx) + cos(ky))2 (6.30)

which is flat along the (0, π) − (π, 0) direction, in disagreement with experiments. Theintroduction of t′ would add the term

E1(k) = −4t′eff cos(kx) cos(ky) (6.31)

with the t′eff effective value roughly proportional with t′, which will lift up the degeneracy.For positive t′ (hole doped case) the lowest energetically addition state is at (π/2, π/2)in the Brillouin Zone. For negative t′ (electron doped case) the closest to the Fermi levelparticle addition states are situated at (0, π) and (π, 0). Exact diagonalization calculationshows that the hopping process with t′ < 0 (electron doped case) is energetically favorableunlike the hopping with t′ > 0 (hole doped case). The DCA results lead us to the sameconclusion. In Fig. 6.18 the gain in the the total and in the kinetic energy when the t′

hopping is switched on is shown at δ = +0.025 and at δ = +0.05 doping. For t′ < 0(electron doped) the system is lowering the total energy due to a large gain in the kineticenergy and a small loss in the potential energy. For t′ > 0 (hole doped) the systemhas a larger energy due to the increase of kinetic energy. The Neel-type configurationsare favorable to the next-nearest-neighbor hopping process, therefore for the electron

6.5. Conclusions 121

doped case these configurations are stabilized (which will result is a small increase in thepotential energy and a large decrease in the kinetic energy) and the antiferromagnetismis enhanced. For the hole doped case the t′ hopping process is discouraged and this leadsto the suppression of antiferromagnetism.

We believe that the inclusion of t′′ in a one-band Hubbard model will improve theagreement with the two-band model results. However for a quantitative comparison onehas to be careful. The amplitude of the different one-band model parameters does nothave necessarily to coincide in the electron and in the hole doped regimes, as clustercalculation shows explicitly [29]. The slightly doping dependent Cu-O covalency whichwe found for the two-band Hubbard model (Fig. 6.12-a) presumably implies modestlydoping dependent parameters for the corresponding reduced one-band Hubbard model.

6.5 Conclusions

As a result of the large oxygen bandwidth in cuprates, as a starting point, any realisticmodel should take fully into account all the oxygen hopping integrals. Starting from afive-band Hubbard Hamiltonian which considers all the oxygen degrees of freedom whichcouple with the Cu dx2−y2 orbitals we calculate the properties of the CuO2 plane usingthe DCA technique on a 2× 2 cluster.

We found that up to 40% hole doping only the oxygen band which couples directly withthe Cu band is significantly occupied (the occupation of the other being less than 1%).This, together with a comparison of different properties of the five-band Hubbard Hamil-tonian to the reduced two-band Hubbard Hamiltonian which shows very small differencebetween the two models, lead us to the following conclusions:

• The reduction of the five-band model to a two-band model which considers only theCu dx2−y2 and the oxygen B1σ band is a good approximation for the low-energyphysics of cuprates.

• The low-energy physics of the hole doped cuprates is determined by the mixeddx2−y2 −B1σ states which are pushed out from the non-bonding oxygen bands as aconsequence of the strong Cu-O hybridization. The non-bonding oxygen states donot contribute to the low-energy physics.

Calculating the two-band Hubbard model properties we found:

• The phase diagram resembles the generic phase diagram of the cuprates containingregions characterized by antiferromagnetism, pseudogap and d-wave superconduc-tivity.

• The system is strongly covalent. In the electron doped regime the Cu-O mixing isslightly doping dependent, increasing with electron doping.

• The phase diagram, one-particle spectral functions and charge, spin and d-wavesusceptibility functions show asymmetric features with respect to the electron andthe hole doping regimes in agreement with experimental data, proving that thisasymmetry is an intrinsic property of CuO2 plane.


We compared the two-band Hubbard model properties with a single-band t − t′ − UHubbard ones. If the Zhang and Rice theory is correct and the cuprates physics can bedescribed by a one-band model, in order to explain the electron-hole asymmetric featuresat least the next-nearest-neighbor hopping term t′ should be included. The t′ has adifferent sign in the electron (t′ < 0) and respectively in the hole (t′ > 0) representation.We conclude the following:

• The two models exhibits very similar low-energy physics provided that the next-nearest-neighbor hopping term t′ is set appropriately.

• The next-nearest-neighbor hopping term t′’s sign is the principal culprit for theobserved electron-hole asymmetry.

• The mapping of the two-band Hubbard model into a single-band one is a goodapproximation, as Zhang and Rice theory claims, provided we include the t′ hopping.

6.5. Conclusions 123

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Chapter 7

Single-Band Hubbard Model

7.1 Introduction

The single-band Hubbard model is believed to describe well the low-energy physics ofhigh Tc superconductors. Our calculations presented in Chapter 6 bring further evidencein favor of this statement. We started with a more realistic model which includes, asidefrom the correlated Cu dx2−y2 orbitals, also the in-plane px and py oxygen orbitals andshowed that the physics can be well approximated by a single band t − t′ − U Hubbardmodel. The next-nearest-neighbor hopping integral t′ has an essential role in explainingthe electron-hole asymmetry observed in the phase diagram, susceptibility functions andone-electron spectral functions.

Since the discovery of high Tc superconductors the Hubbard model and the closelyrelated t− J model were under intense investigation. Many approximation schemes bothanalytical and numerical were used, but reliable results are difficult to obtain because theapproximations involved are uncontrollable. The Dynamical Cluster Approximation tech-nique, which considers a periodic cluster of size Lc × Lc embedded self-consistently in aneffective-medium and takes (ξ/Lc)

2 as the small parameter which controls the calculation,we believe to be very promising in this respect. Here ξ represents the antiferromagneticcorrelation length which, based on neutron scattering experiments data [1], it is believedto be short-ranged in cuprates. Even the small 2× 2 cluster seems to be large enough tocapture the basic physics of the cuprates. The phase diagram calculated by M. Jarrell etal. [2] and shown in Fig. 6.14 with the dashed line exhibits antiferromagnetic, pseudogapand d-wave superconductivity regions, resembling the general phase diagram of cuprates(see Fig. 1.3). The DCA provides another important result, also in agreement with therecent experimental data, which shows that the superconductivity mechanism in cupratesis driven by the reduction of the kinetic energy below the critical temperature. Early on,analyzing the photoemission spectra, Norman et al. [3] pointed out that the supercon-ductivity in cuprates could be a result of a kinetic energy gain rather than a potentialone characteristic for normal BCS superconductors. This was confirmed by recent opticalexperiments [4] which show a transfer of the conductivity spectral weight from large tosmall frequency below Tc. Using DCA, Th. Maier et al. [5] calculated the kinetic and thepotential energies of the normal and superconducting states below Tc. For the supercon-ducting state the DCA code is modified to allow electron pairing, i.e. a non-zero value

125

126 Chapter 7. Single-Band Hubbard Model

-0.278

-0.276

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-0.270

kine

tic e

nerg

y

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pote

ntia

l ene

rgy

T

δ=0.05 Tc

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δ=0.20 Tc

NSSC

-0.252

-0.250

-0.248

-0.246

-0.244

0 0.01 0.02 0.03 0.04

T

δ=0.20 Tc

NSSC

Figure 7.1: Kinetic (top) and potential (bottom) energies of the normal (NS) and super-conducting state (SC) as a function of temperature for low doping (δ = 0.05, left) andhigh doping (δ = 0.20, right). The vertical dotted lines represent the value of Tc. Pairingis mediated by a reduction of the kinetic energy. Calculated by Th. Maier et al. [5] witht = 0.25 eV , U = 2 eV .

for the off-diagonal elements of the Green’s function, Eq. 1.16, is permitted. The normalstate is calculated imposing all the group symmetries on the Green’s function. The re-sults are shown in Fig. 7.1. In the superconducting state, unlike the potential energy, thekinetic energy strongly decreases with decreasing the temperature. The kinetic energy isalso lower than the normal state one, showing that that the superconductivity is a kineticenergy driven mechanism.

Considering a larger cluster is the next natural step in DCA calculation. The com-putational time required for the calculation scales as N3

c , but the rapid increase in theperformance of supercomputers around the world makes us very optimistic that in thevery next future we will be able to perform systematic calculations on 16 site and evenlarger clusters. In this chapter we are going to present the results obtain for t − U andt− t′−U Hubbard models when a Nc = 8 cluster is considered. Compared to the Nc = 4cluster case, most of the qualitative physics remains the same but also some importantdifferences are noticed, and also new physics, as phase separation in the electron-dopedt− t′ − U Hubbard model, appears too.

Looking carefully at the data available so far it can be noticed that the Nc = 4 caseis rather special. M. Jarrell et al. [6] calculated the Neel temperature at zero doping

7.1. Introduction 127

ing transition, we will add a hopping t' ~Ref. 37! into thethird dimension between an infinite set of weakly coupledHubbard planes

t'~kx ,ky ,kz!522t'~cos kx2cos ky!2 cos kz . ~72!

We take t'!t , and treat the additional coupling at the DMFAlevel, so the self-energy is independent of kz . This is accom-plished by modifying the coarse-graining cells into rectangu-lar solids of dimensions Dk , Dk , and 2p in the kx , ky , andkz directions, respectively. After coarse graining, the problemis reduced to a two-dimensional cluster. Information relevantto the mean-field coupling between the planes is containedwithin G.

A. Results at half filling

The physics of the half filled model is a severe test of theDCA as well as finite-sized simulations ~FSS! due to thequantum critical point at zero doping. As this point is ap-proached, both the dynamical and spatial correlation lengthsdiverge, and both the DCA and FSS are expected to fail.

1. Antiferromagnetism

Earlier finite size simulations38,39 employing the QMCmethod have led to the conclusion that the ground state is anantiferromagnetic insulator at half filling. Since the model istwo dimensional, we know from the Mermin-Wagner theo-rem that the transition temperature is necessarily zero. But asfound in infinite dimensions,32 the DMFA predicts a finitetemperature transition even in two dimensions. This spuriousbehavior may be attributed to the lack of nonlocal correla-tions in the DMFA. These correlations are known to inducestrong fluctuations particularly in reduced dimensions andare responsible for the suppression of the finite temperaturetransition. The DCA which includes these nonlocal correla-tions is thus expected to progressively drive the spuriousfinite temperature transition found in the DMFA towards zerotemperature as the cluster size increases.

This behavior is illustrated in Fig. 8, where the inverseantiferromagnetic susceptibility is plotted versus temperaturefor d50 and various values of Nc which preserve the latticesymmetries as discussed in Sec. III C. At high temperatures,the susceptibility is independent of Nc , due to the lack ofnonlocal correlations. In contrast to FSS calculations, thelow temperature susceptibility diverges at T5TN , indicatingan instability to an antiferromagnetic phase. As Nc increasesNc.1, the nonlocal dynamical fluctuations included in theDCA suppress the antiferromagnetism. For example, whenNc51, the susceptibility diverges with an exponent g'1, asexpected for a mean-field theory; whereas the susceptibilitiesfor larger Nc values diverge at lower temperatures withlarger exponents indicative of fluctuation effects.41 At first,these effects are pronounced; however, as Nc increases, TNfalls and g rises more slowly with increasing Nc . This canbe understood from the singular nature of the spin correlationlength, which at least in the large U limit is expected to varyas jeA/T, where A is a constant of the order of the exchangecoupling. For this quantum critical transition, we expect the

DCA to indicate a finite temperature transition once j ex-ceeds the linear cluster size. Since correlations build expo-nentially, large increases in the cluster size will only reducethe DCA transition temperature logarithmically.

Note that the data for TN(Nc) falls on a smooth curve,except for TN(Nc54). This behavior was seen previously inthe transition temperature of the Falicov-Kimball model, cal-culated with DCA.7,8 The Nc54 data falls well off the curveproduced by the other data, and has a much larger exponentindicating that fluctuation effects are more pronounced. Pres-ently this behavior is not completely understood but may berelated to the fact that the maximum coordination number forNc54 is two, whereas it is greater than two for cluster sizeslarger than Nc54.

An interplanar coupling can significantly alter the phasediagram. However, since the superexchange coupling variesroughly like the square of the hopping, it is necessary tomake t' a significant fraction of the intraplanar coupling t inorder to see an effect. For example, if t' /t50.4, the ratio ofthe interplanar to intraplanar exchanges is roughly J' /J'0.16. In Fig. 9 the antiferromagnetic transition temperatureis plotted versus Nc when t' /t50,0.4,1.0 when U5W52and d50. For both t' /t50.4 and t' /t51.0, the transitiontemperatures for Nc516 and 32 are the same to within thenumerical error. Thus, the finite-temperature transitionsfound for small clusters, can be preserved as Nc→` by in-troducing the interplanar coupling.

2. Mott transition at half filling

In the strong coupling limit, a Mott Hubbard gap is ex-pected to open in the charge excitation spectra. In the weakcoupling limit, the situation is less clear. Since the groundstate of the half filled model is always an antiferromagnet,the system remains insulating, but the nature of the insulat-ing state in weak coupling is less clear, and depends upon thedimensionality. In one dimension, Lieb and Wu40 showedlong ago that a charge gap opens as soon as U.0. There is

FIG. 8. The inverse antiferromagnetic susceptibility versus tem-perature for various cluster sizes. The lines are fits to the function(T2TN)g. In the inset, the corresponding Neel temperatures, deter-mined by the divergence of the susceptibility, are plotted. The lineis a polynomial fit to the data, excluding Nc54.

M. JARRELL, TH. MAIER, C. HUSCROFT, AND S. MOUKOURI PHYSICAL REVIEW B 64 195130

195130-16

Figure 7.2: Undoped t−U Hubbard model. Inverse of the antiferromagnetic susceptibilityversus temperature for various cluster sizes. In the inset, the Neel temperature versus thecluster size. Notice that TN for the Nc = 4 case does not fit on the general curve. Dataobtained by M. Jarrell et al. [6]. t = 0.25 eV .

for different cluster sizes. We show their results in Fig. 7.2. From the inset it can beseen that, except for the Nc = 4 case, the antiferromagnetic temperature is decreasingslowly with the number of sites in the cluster. Because the mean-field character of theDCA calculation results in a finite temperature transition when the correlation lengthexceeds the cluster size, this indicates that the spin-correlation length increases rapidlywith decreasing the temperature ∗. However the Neel temperature for Nc = 4 cluster does

∗For example, if the the spin-correlation length is assumed to vary as ξ = eA/T where A is a constantof the order of the exchange coupling, we should expect a logarithmic decrease of TN with increasingNc [6].


N = 4c

1

2 3 4

5 6 7

8

8

6 752 3 4

1

8

7432

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8

5 6

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43

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3

1 2

3 4

3

1 2 1 2

3 43 4 4

a) b) cN = 8

Figure 7.3: a) Nc = 4 cluster. b) Nc = 8 cluster.

not fit the general behavior, being much smaller. The critical exponent γ is also verylarge (γ = 1.29 compared to γ = 1.07 corresponding to the Nc = 8 case) indicating strongfluctuations. Presently, this anomalous behavior of the 2 × 2 cluster is not completelyunderstood. We believe it is related with the fact that the periodic boundary conditionsimposed by DCA on the cluster introduce some artificial effects. It was shown that theseerrors become quickly negligible when the cluster size is increased, the DCA convergingas O(1/L2

c) [7]. The convergence is much faster than it would be if a cluster with openboundary conditions embedded in a host is considered, the later approach being calledCellular Dynamical Mean Field Theory [8, 9] and converging asO(1/Lc). However the 2×2cluster with periodic boundary conditions has a fundamentally different topology fromlarger clusters. The coordination number for the Nc = 4 cluster is two while for the largerclusters it is four. This can be seen from Fig 7.3-a where the Nc = 4 cluster with periodicboundary conditions is shown. It is known that a small coordination number (smaller thanfour) favors the “singlet liquid” Resonance Valence Bond (RVB) [10] kind of solution tothe antiferromagnetic Neel one. Therefore this could be the main reason for such a smallTN obtained in the four site cluster calculation. In fact the exact diagonalization of theHeisenberg model on a 2 × 2 cluster with periodic boundary conditions shows that forthis cluster the ground state is exactly the nearest-neighbor RVB state and this is a verystable solution, the excitation gap being much larger than the one corresponding to otherclusters (the gap is equal to 2J which is 3.5 times larger than the 4× 4 cluster excitationgap) [11]. Therefore it is reasonable to assume that even when the cluster is coupled to theeffective host the special topology introduced by the periodic boundary conditions on the2 × 2 cluster will stabilize the RVB configurations in detriment to the antiferromagneticNeel ones, resulting in a much smaller TN .

Based on the fact that TN obtained for the Nc = 8 cluster fits well the curve shownin Fig. 7.2 which describes the general behavior of the antiferromagnetic temperatureversus the cluster size, we believe that the eight site cluster calculations is more realisticthan the corresponding four site cluster calculations. However one still has to be careful,because we expect the next-nearest-neighbor correlations to be significantly influenced

7.2. Phase Separation in the Electron Doped Regime 129

0.05 0.1 (eV)

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sity

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8

12

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arge

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b)

T = 0.0375 eV

Figure 7.4: t− t′ − U Hubbard model with t = 0.45 eV , t′/t = −0.3 and U = 3.6 eV . a)Charge susceptibility versus temperature at 5% electron doping. b) The hole density ver-sus chemical potential. In the shaded region, for a particular µ there are many solutions,each of them corresponding to a different number of holes.

by the cluster topology now. As a result of imposing the periodic boundary conditions,the next-nearest-neighbor coordination number is two, as can be seen from Fig 7.3-b. Inorder to estimate the errors due to the cluster topology, comparison to larger clustersshould be done. Some preliminary Nc = 16 results show that the Nc = 8 and Nc = 16physics is pretty close (even though some quantitative differences which have to be dueto the topology of the eight site cluster, like the too small pseudogap temperature, areobserved). Further investigation in this direction is required.

Nevertheless most of the physics is common for Nc = 4, Nc = 8 and (as much as weknow so far) Nc = 16 clusters. Therefore we believe that all the common features aregenuine characteristics of the single-band Hubbard models.

7.2 Phase Separation in the Electron Doped Regime

The electron doped cuprates are described by a t− t′−U Hubbard model, with a positivenearest-neighbor hopping integral, t′, in the hole representation. The Nc = 4 chargesusceptibility in the underdoped region (see Fig. 6.17) increases strongly with decreasingtemperature below T ∗. When the eight site cluster is considered this effect becomesdramatic developing into a real divergence of the charge susceptibility. This can be seen inFig. 7.4-a where the charge susceptibility versus temperature is shown for the 5% electrondoped case. The divergence occurs very close to T ∗, i.e. close to the same temperature


where the downturn in the spin susceptibility is observed. However, this is an accidentspecific only to the eight site cluster which, for reasons related to the cluster topology,provides a too small T ∗. For the Nc = 16 cluster, the pseudogap temperature is largerand therefore larger than the temperature where the charge susceptibility diverges (whichwe find is not significantly influenced by the cluster size).

The charge susceptibility reflects the modification in the hole number when the chem-ical potential is slightly changed

χcharge =δn

δµ(7.1)

Therefore the divergence of χcharge indicates an instability in the hole density and suggeststhat the system suffers a first order transition to an inhomogeneous phase where regionswith different hole densities are spatially separated.

The phase transition is characterized by the dependence of the hole number on thechemical potential. The instability in the number of holes makes the calculations whichpreserve the doping and adjust the chemical potential at every iteration very difficult.Usually this is the way in which our computations are done. But we can also run calcula-tions which keep the chemical potential fixed. We noticed that, depending on the initial“guess” of self-energy, for a fixed chemical potential the DCA can converge to differentsolutions characterized by different hole density. This is shown in Fig. 7.4-b. The solu-tions shown with circles (solid line) are obtained starting from a chemical potential whichcorresponds to zero doing, where the divergence in the charge susceptibility is not presentand where only one solution exists. The next calculation is done for a slightly smaller µand the present calculated self-energy is taken as the initial “guess”. The same procedureis repeated in the subsequent calculations. When the chemical potential corresponds toabout 8% doping the divergence in the susceptibility disappears and, again, there is onlyone solution. Let’s call this solution the small doped solution. On the other hand if westart with a µ corresponding to 8% doping and apply the same procedure, but with thedifference that now at every subsequent calculation we increase the chemical potential,the result will be the one shown with squares (dashed line). We call this solution the largedoped one. All the calculations described above are done at a temperature smaller thanthe susceptibility divergence temperature corresponding to 5% electron doping. The (n,µ) characteristic has a hysteresis shape showing explicitly that the transition is first order.For the temperature shown in the figure, the doping range where the system is instabletowards phase separation is between a very small doping (0.5%) and about 8% doping.Presumably the hysteresis range increases at lower temperatures. We can conclude thatthe system likes to separate in two regions, a very lightly doped one and a about 8%doped one. The situation is similar to the familiar gas-liquid first order phase transition,the pressure and the volume being replaced by the chemical potential and respectively bythe number of holes.

The exact diagonalization of the t− t′−J model with a positive t′ (which correspondsto electron doping) shows that two holes on a small cluster like to stay in the vicinity ofeach other, suggesting the possibility of phase separation [12, 13]. This is not true for anegative or zero next-nearest-neighbor hopping integral when the holes prefer to stay ata large distance from each other. However the cluster size is too small to prove a realphase separation, but at least the exact diagonalization shows that our results are not

7.3. Phase Diagram 131

0.8 0.9 1 1.1 1.2n

0

300

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900

1200

1500

1800

2100

T(K

)

T_c, t’=0T_c, t’/t=-0.3T_N,t’=0T_N, t’/t=-0.3T*,t’=0T*,t’/t=-0.3

AFM

phase-seppseudogap

&psdgap

SC SC

electron doped

hole doped

Figure 7.5: t-t’-U Hubbard model (solid line) and t-U Hubbard model (dashed line) phasediagrams calculated for a Nc = 8 cluster. t = −0.45 eV, t′/t = −0.3, U = 3.6V

quite unexpected.

7.3 Phase Diagram

In this section we discuss the t−U and t−t′−U Hubbard model phase diagrams calculatedwith DCA on a Nc = 8 cluster. They are shown in Fig. 7.5. The electron-hole asymmetrict − t′ − U model is plotted with solid line and the symmetric t − U model with dashedline.

Compared to the Nc = 4 phase diagram (see Fig. 6.14), the Nc = 8 one shows someimportant differences. The most significant one is the large Neel temperature and thelarge critical doping where the antiferromagnetism disappears (the antiferromagnetismpersists up to 20% doping for the t− U Hubbard model).

Normally, in order to account for correlations larger than the cluster size antiferro-magnetic, below TN we should work with a broken symmetry solution, i.e. to couple thecluster with an antiferromagnetic host whose properties follows to be determined self-consistently. This work requires some modification to the present DCA code and has notbeen done yet. Instead, even below TN we keep working with a paramagnetic solutionwhich implies that the cluster is coupled to a medium characterized by spin rotationalinvariant properties. In this way we suppress (artificially) the antiferromagnetic corre-lations longer than the cluster size. Therefore we should consider the physics obtainedbelow TN as a characteristic of the Hubbard model with suppressed long-range antiferro-magnetic correlations. It is possible that the same features remain when the longer-range


correlations are allowed, but this has to be checked.

The next-nearest-neighbor hopping integral has the same influence on the antiferro-magnetic properties as we determined for the Nc = 4 calculations. In the undoped regimeTN is decreased by t′. The critical doping where the antiferromagnetism disappears isreduced (increased) in the hole (electron) doped regime.

Bellow TN we find pseudogap and d-wave superconductivity, similar to experimentaldata. The pseudogap temperature T ∗, which is determined from the downturn in thespin susceptibility is much lower than the antiferromagnetic temperature. Preliminarycalculations show that it increases again for the Nc = 16 cluster (but it still remains lowerthan the corresponding TN). This suggests that the next-nearest-neighbor correlations(which presumably are the most affected by the periodic boundary conditions imposedon the Nc = 8 cluster) have an important contribution in this matter and are suppressingthe pseudogap in the Nc = 8 case. The effect of t′ on the spin susceptibility is the same asfor the Nc = 4 case. The t′ corresponding to hole doped regime makes the susceptibilityincrease and the downturn at T ∗ to sharpen with increasing the doping. The oppositeeffect is seen for the electron doped case.

For the Nc = 4 cluster we found that the superconductivity persists up to a very smalldoping (less than 1%) with a maximum Tc around 5% doping. The d-wave superconduc-tivity region looks more realistic for the Nc = 8 cluster, for example the t − U Hubbardmodel doping range is between 5% and 20% with a maximum Tc around 15% doping. Pre-liminary calculations with a Nc = 16 cluster indicates that these features remain whenthe cluster size is increasing. Switching on t′ the superconductivity moves towards largerdopings in the hole doped regime. In the electron doped case the critical doping where thesuperconductivity disappears is reduced and the superconducting region shrinks. Similarto what we observed for the Nc = 4 case, t′ has no significant influence on the maximumvalue of the critical temperature, Tmax

c .

As we discussed in Section 7.2, in the electron-doped regime t′ gives rise to phaseseparation when the doping is smaller than 8% − 9%. For the Nc = 8 cluster we findthat the phase separation temperature Tps more or less coincides with the pseudogaptemperature T ∗. For the Nc = 16 cluster Tps remains the same but T ∗ increases, thus thephase separation takes place below the pseudogap temperature.

7.4 Density of States and Pseudogap

One of the great advantages of the DCA technique is the possibility to calculate thedensity of states and the momentum dependent spectral functions, thus making possiblea direct comparison between the calculated spectra and the experimental photoemissionones. The Nc = 4 cluster calculation of t − t′ − U Hubbard model shows that the holedoped and the electron doped spectral functions are different. In the hole doped casethe pseudogap is located around the (0, π) and (π, 0) points in the Brillouin Zone, inagreement with experimental data. On the other hand, in the electron doped case we didnot detect any pseudogap features around these points. This seems to be in agreementwith the experiment too, the photoemission showing in this case a pseudogap at (π/2, π/2)and Fermi pockets at (0, π) and (π, 0) [14]. However, the small four site cluster implies a

7.4. Density of States and Pseudogap 133

k y

k x

Figure 7.6: Coarse-graining of Brillouin Zone in eight cells (Nc = 8) around K =(0, 0), (±π/2,±π/2), (0, π), (π, 0) and (π, π).

-4 -3 -2 -1 0 1 2 3 40

0.1

0.2

0.3

0.4

Inte

nsity

total DOS DOS DOS(π/2,π/2)

(0,π)

Nc=8t’/t=-0.35% electron doping

ω (eV)

T= 320 K (0.028 eV)

Figure 7.7: t-t’-U Hubbard model DOS at 5% electron doping. t = −0.45 eV, t′/t =−0.3, U = 3.6V


too rough coarse-graining of the Brillouin Zone (see Fig. 6.13) and does not allow us tostudy the spectral function around (π/2, π/2). But the Nc = 8 cluster is large enough forthis purpose, the coarse-grained Brillouin Zone containing a cell around (π/2, π/2) (seeFig. 7.6).

The total density of states (thick solid line) and the momentum dependent spectralfunctions around (π/2, π/2) (thin solid line) and around (0, π) (dashed line) at 5% electrondoping is shown in Fig. 7.7. The temperature is below T ∗ and the pseudogap is clearlyseen in the total DOS and in the momentum dependent spectra around (π/2, π/2), thusin very good agreement with experiment.

However some important remarks should be made about the calculation of DOS inthe electron underdoped regime. We found the pseudogap temperature, T ∗, to be equalto the phase separation temperature, Tps. In order to see a well developed pseudogap,spectra at temperatures well below T ∗ should be calculated. But this implies that we gointo the instable region where the phase separation takes place and where for the same µa smaller doped and larger doped solutions exist (and of course any linear combination ofthem). The DOS presented in Fig. 7.7 corresponds to the large doped solution (analogueto the dashed line one shown in Fig. 7.4-b, the difference being that we are at a lowertemperature). It would be interesting to see if the other solution (the small doped one)exhibits similar DOS features.

The eight site cluster DOS calculation for the hole doped regime has not been doneyet.

7.5 Conclusions and Discussions

In this chapter we studied the properties of the single-band t − U Hubbard and t − t′ −U Hubbard models with Dynamical Cluster Approximation using an eight site cluster.Increasing the number of cluster sites, the Nc = 8 cluster is the first after the Nc = 4 onewhich preserves the lattice symmetry.

Most of the Nc = 8 physics is similar to the Nc = 4 one, but also some majordifferences are present. The most important one is the large Neel temperature and thelarge critical doping where the antiferromagnetism disappears. This causes almost all theinteresting physics (pseudogap, superconductivity) to take place below TN . The largeTN is not a peculiarity of the Nc = 8 cluster because similar large TN are also observedwhen larger clusters are considered, but rather the Nc = 4 cluster is special. We arguethat this anomalous behavior of the 2× 2 cluster comes form the special topology whichresults when the periodic boundary conditions are imposed. As exact diagonalizationshows the periodic boundary conditions on a four site cluster stabilize strongly the singletliquid (nearest-neighbor RVB) configurations, and we believe this is the reason for thesuppression of the antiferromagnetism.

Normally, the calculation below TN should be done by coupling the cluster to anantiferromagnetic host. This work has not been done yet. Instead we keep working belowTN with the same paramagnetic code (rigorously suitable only above TN) which constrainsthe host to be symmetric. This will destroy artificially the longer than the cluster sizeantiferromagnetic correlations. Therefore the validity of our conclusions is restricted to

7.5. Conclusions and Discussions 135

systems with short-range antiferromagnetic correlations. The conclusions might still betrue when longer than the cluster size correlations are allowed, but this has to be checked.

Below TN , in the underdoped region, looking at the spin susceptibility and at thedensity of sates we find the characteristics of the pseudogap physics. The d-wave super-conductivity present at very small doping in the Nc = 4 cluster calculation is suppressednow. The doping range of superconductivity is more realistic, being between 5% and 20%for the t−U model. The next-nearest-neighbor hopping integral, t′, influences the physicsin the same way we found for the Nc = 4 case. In the hole doped regime it increases thecritical doping where the superconductivity disappears and it has an opposite effect forthe electron doped regime. Opposite to that, t′ enhances the antiferromagnetism in theelectron doped case and suppresses it in the hole doped case.

For the Nc = 4 cluster, in the small electron doped region, the next-nearest-neighborhopping results in a strong increase of the charge susceptibility below T ∗. For the Nc = 8cluster, t′ results in more dramatic effect, the charge susceptibility diverging at a temper-ature close to T ∗. The same effect is also noticed for the Nc = 16 cluster, but here T ∗

is larger than the phase separation temperature, Tps. The divergence of charge suscep-tibility indicates a first order transition to a phase separated state. We find that belowTps, for a fixed chemical potential, depending on the initial guess of the self-energy, theDCA converges to two solutions, a small doped and a large doped one. Thus the particlenumber-chemical potential characteristic has a hysteresis shape, and studying it we con-clude that the system is instable towards separation in two regions with different doping,a very lightly doped one, (≈ 0.5%) and another one with a larger doping (≈ 8%).

Form the Nc = 4 calculation we concluded that t′ is responsible for the electron-holeasymmetry seen in the ARPES spectra. However, because of the coarse-graining of theBrillouin Zone in only four cells, we could not study the spectral function near (π/2, π/2)where the pseudogap was experimentally found in electron-doped cuprates. The Nc = 8cluster makes this possible, and we find a pseudogap at (π/2, π/2) indeed. However weshould mention that the spectra were calculated below Tps, thus in the region instabletowards phase separation, and our DOS corresponds to the larger doped solution. If thesame features are present for the other solution remains to be checked.

The Nc = 8 Dynamical Cluster Approximation study of the Hubbard model bringsnew results but also reveals some problems with the technique when it is applied to verysmall clusters where the imposed periodic boundary conditions have a significant effect.However the exact diagonalization results shows that for clusters larger than or equal to4× 4 these effects become negligible, and the present computing resources already allowsus to study the 4 × 4 cluster. But even the small Nc = 4 and Nc = 8 clusters resultsare valuable, but one has to be cautious with the physical interpretation. Comparingthe Nc = 4, Nc = 8 and some preliminary Nc = 16 calculations we noticed a similarphysics. They all exhibits antiferromagnetism, pseudogap and d-wave superconductivity.The difference consists only in magnitude of these phenomena. We believe that throughan understanding of the periodic boundary conditions influence on the DCA results, wecan find a way to control these properties. For example, the Nc = 4 cluster enhancesthe next-nearest-neighbor RVB configurations and as a result the antiferromagnetismis reduced and the pseudogap and the superconductivity are enhanced. Allowing long-range antiferromagnetic correlations we can determine if the superconductivity and the


pseudogap are enhanced, suppressed or whether they disappear completely. Based onthe Nc = 4 results we believe that one of the last two options is true. If so, in orderto explain the experimentally determined properties of the cuprates we can argue thatsome other interaction which stabilize the singlet formation in detriment to long-rangeantiferromagnetism should be considered. A possible and reasonable candidate for thispurpose could be the electron-phonon interaction, which we found in Chapter 4 to favorthe nearest-neighbor singlet formation when the Coulomb repulsion is large. Besides wehave the advantage that phonons can be introduced easily in DCA, so that the study ofthe above proposed scenario is perfectly possible. That would be a continuation of thepresent work, making also the natural connection between the first and the last part ofthis thesis.

7.5. Conclusions and Discussions 137

References

[1] T. R. Thurston, et al., Phys. Rev. B 40, 4585 (1989).

[2] M. Jarrell, T. Maier, M. H. Hettler, and A. N. Tavildarzadeh, Europhys. Lett. 56,563 (2001).

[3] M. R. Norman, M. Randeria, B. Janko, and J. C. Campuzano, Phys. Rev. B 61,14742 (2000).

[4] H. J. A. Molegraaf, C. Presura, and D. van der Marel, Science 295, 2239 (2002).

[5] Th. A. Maier, M. Jarrell, A. Macridin, and C. Slezak, cond-mat/0211298, submitedto PRL.

[6] M. Jarrell, Th. A. Maier, C. Huscroft, and S. Moukouri, Phys. Rev. B 64, 195130(2001).

[7] Th. A. Maier and M. Jarrell, Phys. Rev. B 65, 041104 (2002).

[8] G. Biroli and G. Kotliar, Phys. Rev. B 65, 155112 (2002).

[9] G. Kotliar et al., Phys. Rev. Lett 87, 186401 (2001).

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[13] T. Tohyama and S. Maekawa, Supercond. Sci. Technol. 13, R17 (2000).


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Samenvattig

Kort na het ontstaan van de bandentheorie werd opgemerkt dat overgangsmetaalverbindin-gen niet in metalen en isolatoren kunnen worden verdeeld met behulp van het algemeneschema dat ze classificeert aan de hand van hun elektronband bezetting. Men realiseerdezich snel dat de reden voor het falen van de bandentheorie lag in de verwaarlozing vande sterke Coulomb afstoting tussen elektronen die zich op hetzelfde ion bevonden. Dezeeffecten kunnen niet goed beschreven worden in de conventionele gemiddeldeveld be-naderingen. Sindsdien zijn veel andere materialen ontdekt met “vreemde” eigenschappen,die slecht door de conventionele theorie worden beschreven. De belangrijkste ontdekkingwas de hoge Tc supergeleiding in cupraat gebaseerde materialen, die onmiddellijk gevolgdwerd door een buitengewone ontwikkeling van zowel experimentele als theoretische tech-nieken, ontworpen om met gecorreleerde systemen om te kunnen gaan. Helaas is menzelfs nu nog ver van een tevreden stemmende oplossing van het probleem en is er geengoede theorie voor sterk wisselwerkende elektron systemen.

Behalve de sterke wisselwerking tussen de elektronen, worden veel gecorreleerde syste-men gekarakteriseerd door een sterke koppeling tussen de ladingsdragers en de vrijheids-graden van het kristalrooster. Deze grote koppelingssterkte valt buiten de toepasbaarheidvan conventionele theorieen die systemen met elektron-fonon wisselwerking behandelen,en die bijvoorbeeld worden toegepast op gewone supergeleiders.

De laatste jaren is een groot aantal numerieke algoritmes ontwikkeld om wisselw-erkende systemen te kunnen beschrijven. Hiervan bleek de Diagrammatische QuantumMonte Carlo (DQMC) techniek zeer krachtig te zijn voor het oplossen van polaron- enbipolaronachtige problemen. De oplossing van deze problemen is uitermate belangrijkvoor het begrip van de eigenschappen van quasideeltjes en voor het paringsmechanismein supergeleiders. Een andere krachtige techniek die met multi-elektron problemen kanomgaan is de Dynamische Cluster Benadering (DCA). We hebben deze methode gebruiktom modellen van het Hubbard type op te lossen, waarvan wordt gelooft dat ze de lage-energie fysica van cupraat supergeleiders beschrijven.

Het eerste hoofdstuk is ter introductie. We beginnen met een korte discussie van debasis problemen die men tegen komt bij de bestudering van gecorreleerde systemen. Wevervolgen door een aantal belangrijke eigenschappen van hoge Tc supergeleiders te to-nen en met een opsomming van enkele van de belangrijkste theoretische scenario’s diein dit vakgebied zijn voorgesteld. Een belangrijk deel wordt toegewijd aan de elektron-fonon wisselwerking. We introduceren de conventionele theorieen, die zijn gebaseerd opMigdal’s benadering, die zowel de normale als de supergeleidende toestand van metallis-che systemen beschrijven en slechts geldig zijn in de benadering van kleine elektron-fononkoppeling en grote Fermi energie. Echter het grootste deel van de gecorreleerde materialen

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wordt gekenmerkt door een grote elektron-fonon koppeling en een kleine Fermi energie,wat dus beide geldige redenen zijn voor het falen van Migdal’s theorema. We besprekenkort twee alternatieve theorieen voor supergeleiding. De eerste, die ontwikkeld werd doorPietroniero et al., is een veralgemenisering van de gebruikelijke Migdal-Eliashbergtheoriewaarbij de eerste vertex verbeteringen meegenomen worden in de Green’s functie bereken-ing. De tweede theorie, ontwikkeld door Alexandrov et al., is gebaseerd op de ontwikkelingrond de sterke koppelingslimiet en leidt tot bipolaron vorming gevolgd door Bose-Einsteincondensatie van het bipolaron gas.

In het tweede hoofdstuk laten we de algemene ideeen van de Diagrammatische Quan-tum Monte Carlo (DQMC) techniek zien. Het doel is om de Green’s functies van hetsysteem te berekenen en dit probleem kan in het algemeen gereduceerd worden tot eensommatie van een oneindige set van diagrammen (Feynman’s diagrammen). Dit komtmathematisch overeen met een berekening van een som van integralen met een steedstoenemende integratieveelvoud. Het bijzondere van deze techniek wordt bepaald doorde manier waarop de configuraties met verschillende veelvoud worden gegenereerd. Dealgemene principes van de Diagrammatische Quantum Monte Carlo techniek worden ge-bruikt in hoofdstukken drie en vier om codes op te bouwen die bruikbaar zijn voor hetbestuderen van polaron en bipolaron problemen.

Het derde hoofdstuk behandelt het Holstein polaron met behulp van DQMC. HetHolstein model beschrijft een elektron dat lokaal gekoppeld wordt aan scherpe disper-sieloze fononen. Zowel de grondtoestand als de impuls afhankelijke eigenschappen wor-den bestudeerd. Het polaron ontwikkelt zich continu van de grote polaron toestand, diekarakteristiek is voor het zwakke koppelingsregime, naar de kleine polaron toestand diehoort bij het sterke koppelings regime. We berekenen de grondtoestandeigenschappenzoals energie, gewicht van het quasideeltje, gemiddeld aantal fononen, fonon verdeling eneffectieve massa van een- en tweedimensionale polaronen voor een groot aantal waardesvan de koppelingsconstante. In het overgangsgebied suggereert de fonon verdeling vande polaronische wolk dat de grondtoestand een mengsel is tussen polaron toestanden diegroot en klein lijken te zijn. Uit de spectraalanalyse van de Green’s functie blijkt dater meerdere stabiele aangeslagen toestanden bestaan in dit in dit overgangsgebied. Deeigenschappen van polaronen in het zwakke koppelingsregime bij een impuls nul zijn fun-damenteel verschillend van de eigenschappen bij grote impuls. Als het polaron bij lagek een zwak aangekleed elektron is, dan is het bij grote impuls hoogstens een nul-impulspolaron plus een fonon dat al de impuls draagt. Dit betekent dat het een quasideeltjes-gewicht van nul heeft en een vlakke (fononachtige) dispersie.

In het vierde hoofdstuk bestuderen we de bipolaron vorming. We beschouwen tweeelektronen op een vierkant rooster, het model wordt gekarakteriseerd door ”on-site” (lokaalper roosterpunt) elektron-rooster koppeling, on-site elektron-elektron Coulomb afstotingen een nauwe-binding (tight-binding) dispersie. We hebben het fasediagram berekend inde parameterruimte die gedefinieerd is door de Coulomb afstoting en de elektron-fononkoppeling. Afhankelijk van de parameter waardes worden verschillende gebonden toes-tanden gevormd. We bestuderen zowel hun eigenschappen als de overgang van het enebipolaron type naar het andere. Bij kleine elektron-fonon koppeling vormen twee elek-tronen zelfs voor U = 0 een zwak gebonden toestand. Bij grotere koppeling en kleine Uis het bipolaron een sterk gebonden toestand waarbij de elektronen zich voornamelijk op

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dezelfde plaats bevinden (S0 bipolaron). Echter, bij grote en middelmatige koppeling,maakt de door fononen geınduceerde aantrekking de vorming van bipolaron toestandenzelfs mogelijk bij grote Coulomb afstoting U . Hiervoor is de exchange energie (uitwissel-ingsenergie) verantwoordelijk, die niet door fononen wordt gereduceerd en die het wintvan de sterk door fononen gerenormalizeerde kinetische energie. Daarom is de bipolarontoestand in dit gebied van de parameterruimte vooral een dichtsbijzijnde-buur singlet(S1 bipolaron). We ontdekten dat de grondtoestand altijd s-golf symmetrie heeft. Deovergang tussen de verschillende soorten bipolaron toestanden is continu. In het middel-sterke en sterke elektron-fonon koppeling regime, bij waardes van U die overeenkomenmet een S1 grondtoestand, bestaat ook een stabiele aangeslagen d-golf toestand. We be-handelen in hoofdstuk vier ook hoe de bipolaron DQMC code die we ontwikkeld hebbenveranderd kan worden voor de beschrijving van bipolaron modellen met langere afstandelektron-fonon koppeling en/of elektron-elektron wisselwerking.

Het vijfde hoofdstuk introduceert kort het Dynamische Cluster Benadering algoritme(DCA). Het DCA is een uitbreiding van de Dynamische Gemiddeld Veld Theorie (DMFT).De DMFT projecteert het rooster probleem op een onzuiverheid die zelfconsistent is in-gebed in een gastheeromgeving en verwaarloost daarom ruimtelijke correlaties. De DCAprojecteert het rooster op een cluster van eindige grootte die ingebed is in een gas-theeromgeving. Niet-lokale correlaties op lengteschalen kleiner dan de cluster grootteworden expliciet behandeld, terwijl de natuurkunde op langere lengteschalen behandeldwordt op het gemiddeld veld niveau. Hier berekenen we de eigenschappen van een in-gebed cluster met een Quantum Monte Carlo (QMC) algoritme. De zelfenergie van hetcluster wordt gebruikt om de eigenschappen van de gastheer te berekenen and deze pro-cedure wordt herhaald tot een zelfconsistente convergerende oplossing wordt bereikt. Dezelfenergie en vertex functies van het cluster worden dan gebruikt om de rooster groothe-den te berekenen. Deze techniek is ideaal voor problemen waar korte afstand correlatiesdomineren, zoals in de hoge-Tc materialen.

Op het gebied van de natuurkunde van hoge Tc supergeleiders, bestuderen we inhoofdstuk zes en zeven de Hamiltonianen van het Hubbard type. Als gevolg van de grotezuurstof bandbreedte in cupraten, moet elk realistisch model tenminste alle zuurstof hop-ping integralen (hup-integralen) meenemen. Beginnend bij de vijf-band Hubbard Hamil-toniaan, die alle zuurstof vrijheidsgraden die met de Cu dx2−y2 banen koppelen beschouwt,berekenen we de eigenschappen van het CuO2 vlak met behulp van de DCA techniek opeen 2 × 2 cluster. We vonden dat alleen de zuurstof band die direct met de Cu bandkoppelt relevant is voor de lage energie fysica. Als gevolg hiervan is een reductie naar eentwee-band Hubbard model mogelijk. Verder berekenen we een twee-band Hubbard modeldat de CuO2 vlak eigenschappen bepaalt. Het verkregen fasediagram lijkt op het generiekefasediagram van de cupraten en laat antiferromagnetisme, d-golf supergeleiding, pseudo-kloof en Fermi-vloeistof gebieden zien. Het fase diagram toont ook een sterke asymmetrieten aanzien van de elektron en gaten gedoopte regimes, in overeenkomst met de experi-menten. Asymmetrische kenmerken worden ook gezien in de een-deeltje spectrale functiesen in de lading, spin en d-golf parings susceptibiliteit functies. In de vergelijking tussende eigenschappen van het twee-band Hubbard model met die van het een-band Hubbardmodel, hebben we erg gelijksoortige eigenschappen gevonden, vooropgezet dat de een-na-dichtsbijzijnde-buur hopping term t′ naar behoren is afgesteld. De t′ is de hoofdschuldige

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voor de elektron-gat asymmetrie die in de cupraten wordt gezien. We concluderen dat deprojectie van het twee-band Hubbard model op een een-band model een goede benader-ing is, vooropgezet dat we t′ meenemen en een bescheiden doping afhankelijkheid van deparameters nemen.

In hoofdstuk zeven hebben we de eigenschappen van het een-band t− t′−U Hubbardmodel met DCA bestudeerd, gebruik makend van een cluster met acht posities. In tegen-stelling tot een cluster met vier posities, staat een cluster met acht posities wel de studievan de toestandsdichtheid rond het (π/2, π/2) punt in de Brillouin zone toe. We vondenvoor het elektron-gedoopt systeem een pseudo-kloof rond (π/2, π/2), in overeenkomst metexperimentele data en in tegenstelling tot de gaten gedoopte situatie waarbij de pseudo-kloof bij (π, 0) ligt. De ladingssusceptibiliteit divergeert in het elektron gedoopte regime,en toont zo een eerste orde overgang naar een gescheiden fase toestand. De Nc = 8fysica lijkt op die voor Nc = 4, maar er zijn ook een paar belangrijke kwantitatieveverschillen. We geloven dat dit het resultaat is van de periodieke randvoorwaarden dieworden opgelegd door DCA, en die sterk de topologie van kleine clusters beınvloeden.Voor het Nc = 4 cluster stabiliseren de periodieke randvoorwaarden de resonante valen-tie binding (RVB) configuraties, onderdrukken ze antiferromagnetisme en bevorderen zesupergeleiding. De Nc = 8 cluster topologie onderdrukt de pseudo-kloof temperatuur.Toch ontdekten we gelijksoortige fysica bij de vergelijking van Nc = 4, Nc = 8 en enkelevoorlopige Nc = 16 berekeningen. Ze laten allen antiferromagnetisme, pseudo-kloof end-golf supergeleiding zien. Het verschil bestaat alleen in de grootte van deze fenomenen.We geloven dat we door een begrip van de invloed van de periodieke randvoorwaardenop de DCA resultaten, een manier kunnen vinden om deze eigenschappen te controleren.Op basis van de conclusies van hoofdstuk vier, geloven we bijvoorbeeld dat het mogelijkis dat fononen de RVB configuratie stabiliseren en supergeleiding bevorderen, en dat zeeen gelijksoortig effect hebben als dat van de periodieke randvoorwaarden op een clustermet vier posities. Het is mogelijk om fononen mee te nemen in de DCA berekening en ditzal het onderwerp van een volgend onderzoeksproject worden.

Publications

• Quantum Decoherence in the Spectral Functions of Undoped LaMnO3J. Bala, G. A. Sawatzky, A. M. Oles, A. MacridinPhys. Rev. Lett. 87, 67204 (2001)

• Kinetic Energy Driven Pairing in the CupratesTh. A. Maier, M. Jarrell, A. Macridin, and C. Slezaksubmitted to PRL, cond-mat/0211298

• Electron-hole asymmetry in cuprates and the validity of one-band Hubbard modelA. Macridin, G. A. Sawatzky, Th.A. Maier, and M. Jarrellsubmitted to PRL

• On the origin of the pseudogap in underdoped cupratesTh. A. Maier, M. Jarrell, A. Macridin, F.-C. Zhangpreprint cond-mat/0208419

• Two-Dimensional Hubbard-Holstein BipolaronA. Macridin and G. A. Sawatzkyto be submitted

• Physics of cuprates: From five-band to one-band model HamiltonianA. Macridin, G. A. Sawatzky, Th. A. Maier and M. Jarrellto be submitted

• Simulations Employing the Dynamical Cluster ApproximationA. Macridin, Th. Maier, M. Jarrell and J. P. Hagueproposal for computation time atOak Ridge National Laboratory Supercomputer Center

• Phase separation in the electron-doped cupratesA. Macridin, G. A. Sawatzky, Th. A. Maier and M. Jarrellin preparation

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Acknowledgements

This thesis is a result of more than four years of research, most of the time spent at theUniversity of Groningen. During this period of time, many people contributed to thisproject, and herein I would like to thank them all.

I would like to start thanking my promotor, George Sawatzky, for his guidance. I havebeen always fascinated by his great physical intuition and it is his merit that now I seeso clear that theoretical physics and applied mathematics are two very different things.I would also like to thank him for giving me the opportunity to discuss with so manyfamous physicists and not at least for the very pleasant time I have had visiting him inVancouver.

Mark Jarrell has a paramount contribution to this work. I want to thank him forlearning me so much about physics, computers and numerical algorithms. I have greatlyenjoyed the time spent in Cincinnati mainly due to Mark’s hospitality and to the relaxedatmosphere in his group.

I am grateful to Nikolai Prokof’ev for learning me the Diagrammatic Quantum MonteCarlo technique and, as well, for the great time I have had in Amherst, Massachusetts.

Discussing with Maxim Mostovoy has been always very useful and nice and I wouldlike to thank him for that.

I want to thank Andrei Mishchenko for the nice collaboration we had regarding thepolaron problem and to Robert Eder for providing and explaining me the Lanczos exactdiagonalization code.

I would like to show my appreciation to Dirk van der Marel and Daniel Khomskiifor their valuable comments regarding the manuscript and, as well, for the many usefuldiscussions about physics that we have had.

I have enjoyed talking with Hao Tjeng and Fu-Chun Zhang. I have also appreciatedthe collaboration with Seiji Yunoki and Thomas Maier.

Special thanks to Peter Steeneken, my roommate, for translating the “samenvattig” inDutch and for being such a good colleague and friend. I wish him a happy life with Nicole.I have greatly enjoyed my discussions with Ilya Elfimov. I am also thankful to all the otherpeople from my group: Oana, Ronald, Andri, Mirwais, Karina, Adrian Lupascu, CristianSarpe, Veronica, Salvatore, Ruth, Michel, Peter, Sarker, Mark, Mizokawa, Yutaka, Anita,Renate and Harry. I have great memories about the people from another groups, too.I’ve had a lot of fun talking with Michele Zaffalon and Gwilherm Nenert. I want alsoto thank Hajo, Alexey, Andrea, Markus, Katarzyna, Artem, Patricio, Diana, Serghei,Luben, Wilma, Sjoerd, Efremov and Amir.

I would like to thank to the nice people I’ve met in Cincinnati, Karan, Orlando, Cyrill,Sumith and Nimal from the physics department, and the Romanians Sofia, Mircea, George

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and Irina.In Groningen, since I started my Ph. D the number of Romanian people has been

increasing rapidly, especially in Zernike complex. I’ve had so much fun talking with themevery day, so that I hope they don’t mind if I blame them for my still poor English andfor my practically inexistent skills in the Dutch language. However, I want to thank themall for being great friends, and especially to Gabi and Eftimie for accepting to be myparanymphs. Of course, it is my duty to mention Andrei “the crying baby” Filip and toassure him that, contrary to what he might believe, it is highly unlikely for someone tocontact Ebola or other similar hemorrhagic disease in the Netherlands and however thisis not much related to the amount of hair he possesses. So, he shouldn’t be afraid andshouldn’t bother his aunt for hair removing lotions.

I want to show my gratitude to my friend Inaltimea Sa Calu Catina, who has beenvisiting me from time to time only to check the state of the plum brandy that I had beennot able to drink alone.

And finally, special thanks to my wife Corina, for supporting me and heroically toler-ating all my moods during these years.

Documents

Phonons, charge and spin in correlated systems Macridin