Delocalization effects and charge reorganizations induced by repulsive interactions in strongly disordered chains

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Delocalization effects and charge reorganizations induced byrepulsive interactions in strongly disordered chains

Dietmar Weinmann1,2, Peter Schmitteckert1,3, Rodolfo A. Jalabert1, and Jean-Louis Pichard4

1 Institut de Physique et Chimie des Materiaux de Strasbourg, UMR 7504, CNRS-ULP23 rue du Loess, 67037 Strasbourg Cedex, France

2 Institut fur Physik, Universitat Augsburg, 86135 Augsburg, Germany3 Fridolinstr. 19, 68753 Waghausel, Germany4 CEA, Service de Physique de l’Etat Condense, Centre d’Etudes de Saclay, 91191 Gif-sur-Yvette Cedex, France

November 6, 2000

Abstract. We study the delocalization effect of a short-range repulsive interaction on the ground state ofa finite density of spinless fermions in strongly disordered one dimensional lattices. The density matrixrenormalization group method is used to explore the charge density and the sensitivity of the ground stateenergy with respect to the boundary condition (the persistent current) for a wide range of parameters(carrier density, interaction and disorder). Analytical approaches are developed and allow to understandsome mechanisms and limiting conditions. For weak interaction strength, one has a Fermi glass of Ander-son localized states, while in the opposite limit of strong interaction, one has a correlated array of charges(Mott insulator). In the two cases, the system is strongly insulating and the ground state energy is essen-tially invariant under a twist of the boundary conditions. Reducing the interaction strength from large tointermediate values, the quantum melting of the solid array gives rise to a more homogeneous distributionof charges, and the ground state energy changes when the boundary conditions are twisted. In individualchains, this melting occurs by abrupt steps located at sample-dependent values of the interaction wherean (avoided) level crossing between the ground state and the first excitation can be observed. Importantcharge reorganizations take place at the avoided crossings and the persistent currents are strongly enhancedaround the corresponding interaction value. These large delocalization effects become smeared and reducedafter ensemble averaging. They mainly characterize half filling and strong disorder, but they persist awayof this optimal condition.

PACS. 72.15.-v Electronic conduction in metals and alloy – 73.20.Dx Electron states in low-dimensionalstructures – 72.10.Bg General formulation of transport theory – 05.60.Gg Quantum transport

1 Introduction

One of the beauties of Condensed Matter Physics is thepossibility of understanding a variety of phenomena withina (weakly interacting) quasi-particle approach, despite thealways present strong Coulomb interaction between elec-trons. As it has been understood from the early days [1],the applicability of one-particle approaches can be tracedto the Pauli exclusion principle, and in first approxima-tion the interactions simply account for a renormalizationof single-particle quantities (like the effective mass or themean field potential felt by individual electrons).

This traditional view is challenged when studying ar-tificially confined mesoscopic systems or very dilute low-dimensional electron gases. Mesoscopic phenomena havetheir origin in the coherence of electronic wave functionsacross a small sample. The reduced dimensions are ex-pected to render electronic correlations more importantthan in the bulk. Going to lower dimensions and/or very

dilute limits results in a poorer screening of the electron-electron interaction, enhancing the role of Coulomb re-pulsions. When the disorder is strong, Anderson localiza-tion becomes also detrimental to screening, thereby fur-ther magnifying the role of interactions.

The above considerations are closely related to threeexperimental findings which have recently dominated theattention in Mesoscopic Physics. Firstly, the large val-ues of the persistent current measured in metallic meso-scopic rings [2,3,4] cannot be accounted by the theoret-ical predictions based on the single electron picture [5],nor by perturbative approaches taking into account (toinfinite order) the effect of electron-electron interactions[6,7]. Secondly, the discovery of a metallic behavior intwo-dimensional gases of electrons (Si-MOSFET [8]) orholes (GaAs heterostructures [9] and SiGe quantum wells[10]) is at odds with the conclusions of non-interactingtheories that predict an insulating behavior for any valueof the disorder strength [11]. For a recent review on the

http://arXiv.org/abs/cond-mat/0011335v1

2 Dietmar Weinmann et al.: Delocalization due to repulsive interactions in strongly disordered chains

metal-insulator transition in 2D electron and hole gasessee Ref. [12]. The metallic behavior is observed in a param-eter region of the system which is believed to be limitedby its crystallization threshold [13,14]. Finally, the con-ventional view of an infinite zero-temperature quasiparti-cle lifetime at the Fermi energy has been questioned bythe interpretation of measurements yielding a saturationof the electronic decoherence rate with decreasing tem-perature [15]. While a complete understanding of thesefindings is still missing, it is widely accepted that they areinfluenced in a non-trivial way by large interaction effects.Recent theoretical attempts suggest a possible relation be-tween these different phenomena [16,17,18].

In this work we study the persistent current and thelocalization within a model of spinless electrons on a dis-ordered one-dimensional (1D) ring with nearest-neighborinteractions. Clearly, we do not aim to describe the metal-lic quasi-one dimensional rings of Refs. [2,3,4] nor the two-dimensional electron gas where the metal-insulator tran-sition has been observed. However, the interplay betweenlocalization and interaction can be readily studied in thissimple model. The interest on disordered one-dimensionalinteracting models of fermions (with and without spin)can also be assessed from the large variety of analytical[19,20,21,22,23,24] and numerical [25,26,27,28,29,30,31,32] techniques that have been applied to them.

In the absence of interactions, a 1D disordered sys-tem is an Anderson insulator, with the electron wave-functions localized on the scale of the one-particle local-ization length ξ1. The dependence of the conductance onthe size M of the system is given by an exponential de-crease, the characteristic length scale being determined byξ1. If the system is closed into a ring threaded by a mag-netic flux, its orbital response (the persistent current) alsoscales exponentially with M/ξ1 [33].

In rotational invariant continuum systems the persis-tent current does not depend on any kind of interactions[22,24]. This is directly connected to the fact that theDrude weight (to be defined in the sequel) is not affectedby electron-electron interactions in Galilean invariant sys-tems [34]. When the rotational invariance is broken bya lattice or by the presence of disorder, interactions canmodify the value of the persistent current [30].

The problem of spinless fermions in a disorder-freechain with nearest-neighbor interactions is exactly solv-able [19]. In particular, at half filling we have a Mott in-sulator with a finite gap and a charge density wave concen-trated on alternating sites of the lattice. Attractive inter-actions favor an inhomogeneous density (clustering). Re-pulsive interactions favor a homogeneous density (chargedensity wave or Mott insulator). Disorder tends to distortthose arrangements by favoring the occupancy of the lowpotential sites. In the intermediate regime between theAnderson and Mott insulators, disorder, interaction andkinetic energy are relevant. The competition between dis-order and interactions that we study throughout our workexhibits then a non-trivial character.

The localized character of an electron system deter-mines the behavior of the conductance, which is a trans-

port property, as well as the persistent current, which is athermodynamic property. As first shown by Kohn [35], inthe zero temperature limit, both properties can be related.We recall the basic ingredients of such a relationship [22,35,36] for the particular case of the insulating regime.

The linear response of the electronic system to aspatially uniform, time-dependent electric field is thefrequency-dependent conductivity

σ(ω) = σ1(ω) + iσ2(ω) . (1)

A 1D ring containing M sites, threaded by a magneticflux Φ has a flux-dependent many-particle ground stateenergy E(Φ). We choose units such that ~ = e2 = c = a =1 (a is the lattice constant), and Φ = 2π corresponds toone flux quantum threading the ring. Kohn showed thatthe second derivative of E(Φ) (the charge stiffness or Kohncurvature)

Dc = M

(

d2E(Φ)

dΦ2

)∣

∣

∣

∣

Φ=0

(2)

is related to the imaginary part σ2(ω) of the conductivitythrough

Dc = limω→0

ω σ2(ω) . (3)

Using the Kramers-Kronig relations between the realand the imaginary part of the conductivity (see, e.g. [22]),it is found that Dc also gives the weight of the zero-frequency peak in the real part of the conductivity

σ1(ω) = πDcδ(ω) + σreg1 (ω) . (4)

Therefore, Dc is sometimes called the Drude weight. Thisestablishes a link between transport properties and per-sistent currents.

In the insulating regime the amplitude of the flux-dependent oscillation of the ground state energy is typi-cally much smaller than the energy gap between the many-body ground state and the first excited state. Therefore, itis easy to see (i.e. Sec. 4) that a perturbation theory in thehopping matrix elements across the boundary is enough todescribe the flux dependence of the ground state energy,which yields

E(Φ) = E(0) − ∆E

2(1 − cosΦ) . (5)

Here ∆E = E(0)−E(π) can also be interpreted as thedifference of ground state energy between periodic (Φ=0)and anti-periodic (Φ = π) boundary conditions, since amagnetic flux through the ring is equivalent to introduc-ing a change of the boundary conditions. The sign of ∆Edepends only on the parity of the number of particles N(∆E < 0 for odd N and ∆E > 0 for even N) [37,38].

The simple Φ-dependence of the ground state energyin Eq. (5) allows to relate ∆E to the persistent current

J(Φ) = −dE(Φ)

dΦ=∆E

2sinΦ , (6)

Dietmar Weinmann et al.: Delocalization due to repulsive interactions in strongly disordered chains 3

and the Drude weight

Dc = −M2∆E . (7)

In this paper we will work extensively with the phasesensitivity

D = (−1)N M

2∆E , (8)

that for a strongly disordered one-dimensional ring is sim-ply given by the absolute value of the Drude weight and,at the same time, determines the magnitude of the persis-tent current.

The possibility of a negative charge stiffness (or Dc)arising for spinless fermions and Hubbard rings [39,40] in-dicates that the orbital response may be paramagnetic.This is a peculiar behavior since Dc is believed to deter-mine the zero-frequency behavior of σ1.

The previous numerical work on interacting disorderedrings has necessarily been restricted to finite samples [26,27] or it has relied on strong approximations (like Hartree-Fock [28,31]). Direct diagonalization of small systems (lat-tices with M=6 and 10 sites at half filling) with Coulombinteraction [26] yielded an (impurity) average persistentcurrent that is suppressed by effects of the interactionsexcept for strong disorder and weak interaction strength,where it is weakly enhanced. Direct diagonalization in one-dimensional rings of spinless fermions with short-range in-teractions in lattices of up to 20 sites [27] found that both,disorder and interactions, always decrease the persistentcurrent by localizing the electrons. The above simulationdeals with values of the disorder that are not strong incomparison with the typical kinetic energy of the elec-trons. Similar results have been obtained in this regimeusing the density matrix renormalization group (DMRG)method [30]. Hartree-Fock calculations also yielded a sup-pression of the persistent current as the strength of theinteraction increases [28].

The conclusions that we extract from our numericalcomputations of the phase sensitivity are somehow differ-ent than that of the previous numerical studies. Work-ing at large disorder, we find that when increasing thestrength of the interactions, abrupt charge reorganizationsof the many-body ground state take place (at sample-dependent values of the interaction strength) and are asso-ciated with anomalously large persistent currents [41,42,43]. In this work we extend our previous numerical cal-culations showing such an effect and we also present an-alytical work aiding towards its understanding. We pointout to the importance of considering the physics of indi-vidual samples and we show that the delocalization effectof interactions persists in the thermodynamic limit.

A sizeable increase of the persistent current withthe strength of the interaction had been obtained formoderately disordered 1D electronic systems with spin(Anderson-Hubbard model) from renormalization groupapproaches [23] or perturbation and numerical calcula-tions [32]. Our results show that the spin does not seemto be a necessary ingredient to obtain an enhancement of

persistent currents due to interactions. Even without spin,repulsive interactions can increase the persistent current,provided the disorder is important enough. This lets usexpect an even more dramatic increase at strong disorderin models with spin.

It is interesting to remark that the enhancement oftransport properties by the effect of a repulsive interactionhas been proposed in other contexts than the one of thiswork. A system with strong binary disorder, where the twopossible values of the disorder give rise to two separatedbands, presents in the absence of interactions a gap if thefilling is such that the lower band is filled completely [44].Then, the broadening of the bands due to the interactioncan lead to an overlap allowing for metallic behavior. Also,in a system of two interacting particles on a disorderedchain it has been shown that the localization length isenhanced by the effect of interactions [45,46,47,48].

The remainder of the paper is organized as follows.In Section 2, we present the model and the numericalmethod. In Section 3, we perform numerical studies forthe ground state structure and the phase sensitivity forthe case of half filling. A scaling with the system size al-lows to demonstrate the delocalization effect of repulsiveinteractions in the presence of strong disorder. In Section4, analytical work is presented, which aims to describethe basic physical mechanisms leading to the effects whichwere found numerically. In Section 5, we extend the nu-merical studies away from half filling and establish thecriterium for observing a delocalization effect due to theinteractions. Finally, we present in Section 6 our conclu-sions and outline some open problems.

2 Model and Method

2.1 Model Hamiltonian

We consider spinless fermions on a chain with nearest-neighbor interaction

H = −tM∑

i=1

(c†i ci−1 + c†i−1ci) +

M∑

i=1

vini + U

M∑

i=1

nini−1

(9)

and twisted boundary conditions, c0 = exp(iΦ)cM . The

operators ci (c†i ) destroy (create) a particle on site i and

ni = c†ici is the occupation operator. The on-site randomenergies vi are drawn from a box distribution of width W .The strength of the disorder W and the interaction U aremeasured in units of the kinetic energy scale ( t=1). Theuse of twisted boundary conditions allows to represent aring of M sites pierced by a flux Φ.

2.2 Numerical Method

The numerical results are obtained with the density ma-trix renormalization group (DMRG) algorithm [49,50]. In


this method, successive iterations are obtained by build-ing larger (real space) blocks from smaller components.The Hamiltonian of each block can be diagonalized sincein each iteration we truncate the states determined in theprevious step. In the truncation process, states in the smallblocks are selected as a function of their projection on theground state of the larger block, and not as a function oftheir energies. The projection is computed from the re-duced density matrix of the smaller blocks. The iterationof this procedure allows to calculate ground state proper-ties in disordered 1D systems with an accuracy compara-ble to exact diagonalization, but for much larger systems[30].

It is important to recall that the eigenenergies of themany-body states of the disordered ring need to be ob-tained with large precision since the phase sensitivity isgiven as the difference of ground-state energies. We cantypically achieve system sizes corresponding to 50 parti-cles on 100 lattice sites by keeping up to 2000 states perblock, and then the largest matrices that we have to diag-onalize have a dimension of the order of 10 millions.

3 Half filling

We will center our numerical studies on the charge den-sity and the Φ-dependence of the ground state energy. Westart in this section by treating the case of half filling (thenumber of electrons is N =M/2), leaving the case of anarbitrary filling for section 5.

3.1 Charge reorganization in individual samples

3.1.1 Charge density

We start our analysis with the reorganization of theground state induced by the nearest-neighbor (NN) repul-sion (Fig. 1), plotting the charge density ρ (expectationvalue of ni) as a function of U and site index i for a typicalsample (M =20 andN=10). In order to favor the inhomo-geneous configuration, the disorder is taken large (W =9)such that the localization length (a rough guess is givenby the perturbative result for weak disorder ξ1≈100/W 2)is of the order of the mean spacing k−1

F = 2 between thecharges. For U≈0, one can see a strongly inhomogeneousdensity, while for large U a periodic array of charges setsin. These two limits are separated by a sample-dependentcrossover regime.

The charge reorganization can be clearly observed inthe Fourier transform of ρ with respect to the space vari-able i (Fig. 2). For weak interactions the Fourier transformdoes not show significant structure. On the other hand, asingle peak appears above a certain interaction strength,reflecting the establishment of a regular periodic array ofcharges. For intermediate interaction strengths, there is atendency towards a periodic array, but defects persist andthere are still several Fourier components present.

iU

Fig. 1. Charge configuration for a typical sample (d of Fig. 3)for N =10 particles on M =20 sites at W =9 as a function ofthe lattice site i and the interaction strength U .

kM=2U~

Fig. 2. Fourier transform of the charge configuration of Fig.1.

3.1.2 Density-Density correlation function

In order to quantitatively describe the sample-dependentreorganization of the charge density, we calculate [42] thedensity–density correlation function

C(r) =1

N

M∑

i=1

ρiρi+r (10)

for values 0 ≤ r ≤ M/2. The parameter γ =maxrC(r) −minrC(r) is used to distinguish betweenthe electron liquid with constant density (γ=0), and theregular crystalline array of charges (γ = 1). Since we in-clude the translation r = 0 in the definition of γ, we getγ 6= 0 for the electron glass. Thus, γ measures charge crys-tallization from an electron liquid as well as the meltingof the glassy state towards a more liquid ground state.

Fig. 3 shows the dependence of γ on the interactionstrength for four individual samples representing differentbehaviors. For certain impurity configurations, like in sam-ple a, the periodic array is obtained at a weak repulsive


ab

cd

U

Fig. 3. Density–density correlation parameter γ for four sam-ples with N =10, M =20 and W =9. Thick dots: Average overup to 166 samples.

interaction, while one needs a strong interaction for othersamples like b and d. Typically, γ assumes a minimum fora small repulsive interaction at a sample-dependent valueUc of the order of the kinetic energy scale t. This meansthat the charge distribution is closest to a liquid aroundUc and suggests a maximum of the mobility of the chargecarriers. This is an indication for a delocalization of theground state by repulsive interactions. In addition, mostof the samples show small steps in the interval 0 ≤ U ≤ 2t,caused by instabilities between different configurations ofsimilar structure. The formation of the regular array ofcharges imposed by strong repulsive interactions occurs ata sample-dependent interaction strength Um. The step-likeincrease of γ shows that the regular array is establishedabruptly at Um.

Typically the first and last dips of γ for repulsiveU , signaling the above described charge reorganizations,are separated by a transition region. In order to describeour problem as a transition between phases we have toconsider the average behavior of γ and extract sample-independent values of the critical interaction strengths.The small dispersion of Uc yields an average γ presentinga minimum at an interaction strength UF of the order of t.This demonstrates the delocalization effect of a small re-pulsive interaction accompanied by a “more liquid” chargedensity. The increase of the persistent current in 1D mod-els with spin was traced back to the effect of repulsive in-teractions making the charge density more homogeneous[23]. The present study shows that this mechanism alsoapplies for spinless fermions in strongly disordered 1Dchains.

Unlike the case of small U , for large interactions thejumps of γ are widely spread and therefore smeared outin the ensemble average. Thus, starting from the sample-dependent values Uc and Um, it is possible to identifya sample-independent interaction strength UF associatedwith the first charge reorganization by the minimum of〈γ〉, but the last charge reorganization does not give aclear signature on 〈γ〉. In the next subsection we give the

estimation of the typical interaction strength needed toestablish the Mott phase. Recent works [13,51,52] on 2Ddisordered clusters with Coulomb interaction also showsthat one goes from the Fermi glass towards the pinnedWigner crystal through an intermediate regime (locatedbetween two different Coulomb-to-Fermi energy-ratios rfsand rws ). The difference with our problem is that, in the2D case, a topological change can be observed in the pat-tern of the driven currents, which cannot exist in the 1Dcase. However, in the two problems, we are addressing thedifficult question of the quantum melting of a solid arrayof charges (Mott insulator in our case, Wigner crystal for2D Coulomb repulsion) in the presence of a random sub-strate. This melting can occur through a crossover regime(or an intermediate quantum phase in 2D) where a liq-uid of “defectons” may co-exist with an underlying solidbackground, as suggested by Andreev and Lifschitz [53].

3.1.3 Size dependence of the correlation parameter

The typical interaction strength needed to establish a per-fectly regular array of charges can be estimated from thecompetition between interaction and potential energiesin some special configurations. Starting from the perfectMott configuration with alternating charges, we see thatwe can gain potential energy by going to configurationswhich are perfect only on parts of the lattice. The mostfavorable of this kind should typically be the case wherethe odd sites are occupied on half of the ring and the evensites on the other half of the ring, with two domain wallsbetween these regions. The cost in interaction energy isU , while the gain in potential energy is of the order of(W/2)

√

N/2.The periodic array over the whole system is favored

when the cost of the defect is larger than the gain in po-tential energy. This typically happens for

U > UW =W

2

√

N/2 . (11)

At half filling (N = M/2), the critical interaction strengthUW increases as the square root of the system size. In thedependence of the ensemble averaged correlation parame-ter on U for different system sizes (Fig. 4) this tendencyis clearly visible. The interaction scale on which 〈γ〉 in-creases to its maximum value 1, characterizing the Mottinsulator, is shifted to higher interaction values when thesystem size is increased. In the thermodynamic limit, oneexpects that no finite interaction strength (provided it isshort ranged) can be sufficient to impose a perfectly or-dered Mott insulator in a disordered system, consistentwith the findings of Refs. [21,54].

3.2 Phase sensitivity and localization

In this section, we present calculations of the phase sensi-tivity of the ground state which is defined, in Eq. (8), as


U

h iM = 40M = 30M = 20M = 10

Fig. 4. Ensemble average of the density–density correlationparameter γ as a function of the interaction strength for dif-ferent system sizes M at half filling and W = 9.

the energy difference between periodic (Φ= 0) and anti-periodic (Φ=π) boundary conditions. This phase sensitiv-ity is a measure of the localization of the electronic wave-functions. The more insulating the system is, the weakerwill be the effect of boundary conditions. As discussed inthe introduction, the phase sensitivity conveys similar in-formation, in the localized regime, as other measures of theresponse of the ground state to a flux threading the ring:the Kohn curvature [35] (charge stiffness) ∝ E

′′

(Φ = 0)

and the persistent current J ∝ −E′

(Φ).

3.2.1 Phase sensitivity in individual samples

In Fig. 5 we show the phase sensitivity D(U) for the foursamples presented in Fig. 3, which were at half filling andwith large disorder (W = 9). Both for U ≈ 0 and U ≫ 1,D(U) is very small, but sharp peaks appear at sample-dependent values Uc, where the phase sensitivity can be4 orders of magnitude larger than for free fermions. Re-markably, the curves for each sample do not present anysingularity at U=0 which could have allowed to locate thefree fermion case (that appears simply as an intermediatecase). Peaks can be seen at positive and negative valuesof U .

It is important to notice that each peak of D(U) in anindividual sample corresponds to a charge reorganization.This can be seen, directly, by following the evolution of thecharge density as a function of U , or more systematically,by considering the one-to-one correspondence between thepeaks of D(U) and the dips of the density-density corre-lation parameter γ (Fig. 3). The information of D(U) andγ(U) is complementary, but not equivalent: strong peaksofD (happening for small values of U) correspond to smalldips in γ, while the last charge rearrangement (leading tothe Mott phase) is associated with a large dip in γ and asmall peak (or a shoulder) of D.

The free fermion case U=0 corresponds to an Ander-son insulator. A small repulsive interaction typically tends

a bcdU

logD

Fig. 5. Phase sensitivity D(U) for four different samples withN = 10, M = 20, and W = 9 in (decimal) logarithmic scale.Thick solid line and dots: average of log(D).

to delocalize the system and increases D(U) until a chargereorganization takes place. In some cases (like sample a),the first charge reorganization for positive U immediatelydrives the system to the homogeneous array of charges. Insome other cases we go from the inhomogeneous densityto the periodic array in a few steps signaled by additionalpeaks of the phase sensitivity. Examining the U depen-dence of the density of those samples, one can note insome cases local defects in the periodic array subsistingup to large values of U . As discussed in Sec. 3.1.3, for agiven interaction strength U , the appearance of defects be-comes more and more likely as we increase the system sizeM or the disorder W . Once the regular array of chargesis established, the system becomes more and more rigid(pinned by the random lattice), and D(U) decreases as afunction of U . In Sec. 4.3 we calculate, by perturbationtheory, the phase sensitivity of the Mott insulator in adisordered potential.

3.2.2 Mean values and statistics of the phase sensitivity

The critical values Uc of the interaction strength are sam-ple dependent. Therefore, for a given value of U we have avery different behavior for the samples where U is close toa critical value than for those where it is not. Mixing thetwo situations leads to a widely fluctuating distribution ofD(U) and if we average logarithms, as usually done in thelocalized regime, we obtain the smooth curve of Fig. 5. Wehave checked that the probability distribution of D(U) isin fact log-normal (see insets of Fig. 7).

In Fig. 6 we see that, for a given size and filling(M = 20, N = 10), the log-average 〈logD(U)〉 increaseswhile decreasing the disorder (W = 9, 7 and 5). For U=0we have the usual non-interacting behavior, while for largeU the phase sensitivity becomes only weakly dependenton disorder (we will get back to this point in Sec. 4.3).We also see that 〈logD(U)〉 decreases with the interac-tion strength U , except for small U (U . t) and strongdisorder (W = 9 and 7). The regions for U ≈ t, for W = 9


0 2 4 6 8 10 U

−8

−6

−4

−2

<lo

g D

(U)>

0 1 2−4.6

−4.4

0.0 0.5 1.0−3.2

−3.1

W =7

W =9

Fig. 6. Mean values of log D(U) for M = 20, N = 10 andthree values of the disorder strength: W =5 (triangles), W =7(squares) and W =9 (circles). Upper and lower insets: blow upof the small-U region for the last two cases showing a delocal-ization effect.

and 7, are blown up in the insets (lower left and upperright respectively), and show that the interaction scale forthe first charge reorganization is weakly increasing withthe disorder. The delocalization effect increases with thevalue of the disorder, but it is always very small. Our re-sults for the average phase sensitivity are consistent withthose of Ref. [26], finding a small increase of the averagepersistent current in small 1D systems (up to 5 electronson 10 sites) by direct diagonalization. We show in ourwork that such an effect persists in larger systems and itssmall magnitude is to be contrasted with the spectacularenhancement found in individual samples.

The small delocalization effect on the average phasesensitivity together with the large sample-to-sample fluc-tuations makes it necessary to consider many impurityrealizations (up to 5000) in order to confirm the enhance-ment beyond the statistical uncertainty. Comparing theinformation from Figs. 3 and 6 we see that the averagedelocalization effect is more easily seen on the less fluctu-ating correlation parameter γ than on D.

The widely fluctuating values of D(U) can be repre-sented to a very good approximation by a log-normal dis-tribution for values of U smaller than 8 (Fig. 7). Thevariances σ2(U) = 〈δ(logD(U))2〉 are of the order of|〈logD(U)〉|. They increase slowly with U in the regionof U . t (roughly the interval over which we see the delo-calization effect), and more rapidly for larger values of U(where the effect of the disorder becomes less important).Once the Mott regime is attained σ2 no longer increaseswith U (and becomes strongly dependent on W ).

For a given sample the ratio between the phase sensi-tivity at a value of U 6= 0 and at U=0 is also a widely fluc-tuating variable. Our numerical simulations (not shown)demonstrate that the variable η=log(D(U)/D(0)) is nor-

U<2 (U)> logD(0)

P W = 9

logD(2)P W = 9

Fig. 7. Variance of the phase sensitivity as a function of Ufor three values of the disorder strength: W = 5 (triangles),W = 7 (squares) and W = 9 (circles). Upper and lower insets:probability distribution (dots) of log D(0) and log D(2) respec-tively, calculated from 10000 samples (M =20, N =10, W =9).The mean values and variances are < log D(0) > ≈ −4.486,σ2(0) ≈ 1.49 and < log D(2) > ≈ −4.495, σ2(2) ≈ 1.66. Thesolid lines represent Gaussian distributions with these param-eters.

mally distributed [41]. Such a behavior does not simply fol-low from the log-normal distributions for D(U) and D(0),since these are not independent random variables. Thewidth of the η-distribution is increasing with U , and forU =2 variations of D over more than an order of magni-tude are typical [41]. The samples a and b shown in Figs. 3and 5 are characterized by extremely small and large val-ues of η, respectively, while c and d are typical sampleschosen around the center of the distribution.

3.2.3 Size dependence and thermodynamic limit

In the previous sections we described the increase of thephase sensitivity as a signature of the charge reorganiza-tions that eventually lead to a Mott insulator upon in-creasing the interaction strength. On the other hand, asdiscussed in Sec. 3.1.3, in the thermodynamic limit the dis-order will lead to defects in the Mott phase at any finitevalue of U . Two questions naturally arise at this point.Firstly, do charge reorganizations still exist and yield anenhancement of the phase sensitivity as we go to largerand larger systems? Secondly, will the delocalization ef-fect survive in the thermodynamic limit?

In order to address the first question, we consider inFig. 8 the typical case of a sample with a strong disorder(W = 7). The actual realization of the impurity potentialis sketched at the bottom panel (thick solid) for latticesites going from i= 1 to i= 40. If we consider a sampleof half such a size (M = 20) at half filling (N = 10) weobtain for the charge density (upper left panel) and forthe phase sensitivity (filled squares, upper right panel)the kind of behavior previously discussed. A first chargereorganization takes place around U = 1 where a peak in


0 2 4 6 8 10−10

−8

−6

−4

−2

5 150.0

0.5ρ i

5 150.0

0.5

1.0

ρ i

0 10 20 30 40 i

−22

vi

U

Fig. 8. Charge density as a function of the lattice site at halffilling for a sample with M = 20, N = 10 (upper left panel)and with M = 40, N = 20 (central panel) with the impurityconfiguration vi represented by a thick solid line in the lowerpanel (W =7). Stars and dotted lines correspond to ρi(U =0),while pluses and solid lines to ρi(U = 3). In the upper rightpanel the (decimal) logarithm of the charge sensitivity as afunction of U is shown for the two samples: M = 20 (filledsquares) and M =40 (empty squares).

D(U) is observed. If we now consider the whole sample(M = 40) at the same filling (N = 20), logD(0) as wellas the overall values of logD(U) are reduced by a factorof 2, but the charge reorganization involving the first halfof the sample still takes place (medium panel) giving riseto a sharper peak of D(U) at a slightly shifted criticalvalue (empty squares, upper right panel). This resonant-like behavior is generic. Upon increasing the sample sizethe peaks of D(U) become sharper (and slightly shifted),and new charge reorganizations may yield supplementarystructure in D(U) (like the shoulder observed around U=7).

Having answered the first question for the affirmativewe still need to settle the second one, since it is not obviousthat the sharper peaks yield a delocalization effect on theaverage. A detailed size-dependence study is needed, andwe will focus on the interaction dependence of the local-ization length. In the absence of interactions, a 1D disor-dered system is an Anderson insulator, with the electronwave-functions localized on the scale of the one-particlelocalization length ξ1. Then, the persistent current andthe phase sensitivity scale with the system size M pro-portional to exp (−M/ξ1) [33].

For our many-particle system, the (many-body) local-ization length ξ(U) depends on the interaction strengthand can be defined from the scaling

〈lnD(U,M)〉 = A(U) − M

ξ(U), (12)

0 1 2 3 4 5 U

1.5

2.0

2.5

ξ(U

) 0 1 2

1.76

1.80

Fig. 9. Localization length obtained from the size-dependenceof the phase sensitivity (Eq. 12) as a function of the interactionstrength U for half filling and two values of the disorder: W =7(squares) and W = 9 (circles). Inset: blow up of the small-Uregion for the case of W =9 showing a delocalization effect onξ.

which in the non-interacting case (U=0) yields the stan-dard one-particle localization length characterizing thespatial decay of the electron wave-functions. The previousscaling is well satisfied in the accessible space of parame-ters [43] and allows to extract the values ofA(U) and ξ(U).The former is weakly dependent on U for 0≤U ≤ 5 andmore strongly dependent for U >5. The localization lengthexhibits a similar behavior as 〈logD(U)〉. It decreases withthe interaction strength except for small U and strong dis-order, where a small delocalization is observed (Fig. 9).For W =9 (where the single-particle localization length issmaller than 2 lattice sites) a small repulsive interactionresults in a 2% effect on ξ, for W = 7 the delocalizationeffect is smaller, consistent with the behavior of D(U). Wetherefore conclude that the delocalization on the averageis not a finite-size effect.

4 Physical Mechanisms

We have seen in the previous chapter how upon increas-ing the strength of the electron-electron interaction we gofrom the Anderson to the Mott insulating regimes throughnon-trivial transitions. In particular, our numerical simu-lations show that the intermediate regime is signed bycharge reorganizations associated to an enhanced phasesensitivity. In this chapter we will try to understand themechanism underlying such behavior and we will developa perturbation theory yielding the phase sensitivity in theMott regime in the presence of disorder.


4.1 Simplified model of two particles on three sites

The relation between charge reorganization and enhancedpersistent current can be understood in a simple toy modelof two particles (spinless fermions) on three sites. We con-sider the Hamiltonian of Eq. (9) with M=3 and switch offthe interaction between the extreme sites 1 and 3. There-fore, we write

H(3) = −t3∑

i=1

(c†i ci−1 + c†i−1ci) +

3∑

i=1

vini + U

3∑

i=2

nini−1 .

(13)

We keep the twisted boundary conditions, c0 = exp(iΦ)c3that allow us to address the phase sensitivity. The site-dependent interaction mimics the fact that in the large-M case the sites 1 and 3 are joint through the rest of thechain (that we do not include in the description of thepresent model).

For two spinless fermions on three sitesthe Hilbert space has dimension 3 and the set|1, 1, 0〉, |1, 0, 1〉, |0, 1, 1〉 is a convenient basis spec-ifying the occupation of the sites. We can readilydiagonalize our 3 × 3 matrices for any value of ourparameters t, U and vi. However, in order to simulate thelocalized regime we will only consider the case where atleast one of the on-site potentials vi is much larger thanthe kinetic energy scale t. We then study the transitionfrom U=0 (where the potentials vi determine the chargedistribution) to values of U much larger than all the vi

(where the ground state is mainly directed along thevector |1, 0, 1〉 in order for the electrons to avoid eachother).

4.1.1 Three-site model with fixed impurity configuration

In a fist step we further simplify our problem by restrict-ing the disorder to v1 = v2 = −ǫ and v3 = ǫ ≫ t. Thisconfiguration clearly favors the state |1, 1, 0〉 at U =0. Inthe two extreme cases the ground state energies are givenby

E(U=0) ≃(

−2(ǫ

t

)

−(

t

ǫ

)

+1

2

(

t

ǫ

)2

cosΦ

)

t , (14a)

E(U≫ǫ) ≃(

−2

(

t

U

)

− 2

(

ǫt

U2

)

+ 2

(

t

U

)2

cosΦ

)

t ,

(14b)

The term −2ǫ in Eq. (14a) is the on-site energy of the state|1, 1, 0〉, the next order terms in t/ǫ take into account theenergy gain due to hopping. In the large-U case there isno term in ǫ since the ground state is close to |1, 0, 1〉 andwe have chosen v1 =−v3. From Eqs. (14) we can calculatethe phase sensitivity in the two limiting cases and obtain

D(0) ≃ 3

2

(

t

ǫ

)2

t≫ D(U≫ǫ) ≃ 6

(

t

U

)2

t , (15)

in agreement with the fact that the phase sensitivity inthe Mott phase is reduced with respect to that of theAnderson phase. However, the decrease of D with U isnot monotonous; for the critical value Uc = 2ǫ we have adegeneracy that leads to an increased phase sensitivity:

D(Uc) ≃3

2

(

t

ǫ

)

t≫ D(0) . (16)

The enhancement factor η = D(Uc)/D(0) is then given by

η = ǫ/t = Uc/2t , (17)

and even though both, D(Uc) and D(0) decrease when thedisorder strength ǫ/t is increased, their ratio is increasing.

Our toy model then captures the physics of a charge re-organization associated with an enhanced sensitivity withrespect to a perturbation at a point of degeneracy. The ef-fect is reminiscent of the Coulomb blockade phenomenon.When the occupation numbers are approximately goodquantum numbers (0 or 1) transport is blocked; in thetransition between such extreme configurations the occu-pation numbers are no longer good quantum numbers andtransport is favored. In the Coulomb blockade problem wehave a true transport situation and the degeneracy is be-tween the state having N electrons in the dot and 1 inthe leads with the state having N +1 electrons in the dot.Moreover, the constant charging energy model [55] allowsus to think in terms of a one-particle problem. In our casewe do not have a transport configuration but a delocal-ization of wave-functions and the degeneracy is betweenmany-body ground states of the ring.

4.1.2 Disorder average in the three-site model

The result above for a given potential realization can beused to determine the ensemble average of the critical in-teraction strength Uc within this toy model.

First of all, we relax the restriction to the disorderrealizations, allowing for arbitrary on-site energies vi ∈[−W/2,W/2], assuming box-distributions with probabil-ity density

P (v) =1

WΘ(W/2 − |v|). (18)

For symmetry reasons, the exchange of the values v1 ↔ v3leaves D(U) unchanged and we consider always v1 > v3.

No level crossings between the ground state at U = 0(adapted to the disorder configuration), and the groundstate at U ≫ W (close to |1, 0, 1〉), occurs when the dis-order realization favors the latter already at U = 0. Thisis the case if

v2 > v1, v3 , (19)

which is fulfilled in 1/3 of the whole parameter space.Then, no positive Uc exists within our toy-model, and noenhancement of D(U) with respect to D(0) can be ex-pected for U > 0.


dU

logDlog

Fig. 10. The energy spacing ∆ between the ground state andthe first excited state of sample d of Figs. 1,3 and 5 (upperline) together with the corresponding phase sensitivity (lowerline).

We now concentrate on the averaged enhancement fac-tor 〈η〉, where the average is taken over the parameterspace v2 < v1, in which an avoided level crossing occurs.In the absence of hopping (t = 0), the energies of thestates |0, 1, 1〉 and |1, 0, 1〉 are given by v2 + v3 + U andv1 + v3, respectively, such that a level crossing occurs atUc = v1 − v2. Following the argument of the previoussection, the hopping then leads to a D(U) which is en-hanced at Uc by the factor η = (v1 − v2)/2t. The averageover the part of the parameter space (v1, v2, v3) in whichv2, v3 < v1, using the box distributions yields

〈η〉 =3

8

W

t. (20)

The average enhancement of the phase sensitivity in-creases proportionally to the disorder strength, as the av-erage peak position, 〈Uc〉 = (3/4)W . This behavior is con-sistent with our numerical findings for larger systems.

4.2 Avoided level crossings

In the toy model presented in the previous section we sawhow an enhanced phase sensitivity is linked with a chargereorganization of the ground state. The charge reorgani-zation took there a very simple form: at a certain criticalvalue of the interaction strength an electron jumps fromone site to its neighbor. Or, more precisely, the groundstate that was, for small U , mainly given by the vector|1, 1, 0〉 became, after Uc, approximately aligned in the di-rection of |1, 0, 1〉. That can be seen as a crossing of twolevels as a function of the parameter U . In this sectionwe go back to our numerical simulations for large M anddemonstrate that the physics of level crossings is still validdespite the fact that the charge re-accommodation is notnecessarily local (i.e. the electron may jump many sitesacross).

The sharp changes in the ground state structure andthe peaks observed in the phase sensitivity are the conse-quences of avoided crossings between the ground state andthe first excitation, obtained upon increasing U . While theground state at weak interaction is well adapted to the dis-ordered potential, another state with a different structureand better adapted to repulsive interactions, becomes theground state at stronger interaction. In the case of largedisorder, with a one-particle localization length ξ1 whichis of the order of the mean distance between the particles,the overlap matrix elements between the different nonin-teracting eigenstates due to the interaction are very smalland the levels almost cross. There is only a very small in-teraction range where a significant mixing (hybridization)of two states is present. This is exactly where the peaksof the phase sensitivity appear.

This scenario is confirmed by Fig. 10, where the phasesensitivity D and the energy level spacing ∆ between theground state and the first excited state are shown as afunction of U for sample d. Here, we used the Lanczosalgorithm for direct diagonalization in order to obtain theenergies of a few excited states.

Minima of ∆ appear at the interaction values of thepeaks in the phase sensitivity. A gap of increasing sizeopens between the ground state and the first excitationafter the last avoided crossing at interactions U > Um,when the Mott insulator is established. A study of manyother samples leads to the same conclusions. The statisticsof the first excitation energy is therefore determining thebehavior of the phase sensitivity. In 2D with Coulombrepulsion, this has recently been addressed giving rise tointermediate statistics [56] at the opening of the quantumCoulomb gap.

The importance of level hybridization in determiningthe persistent current of many-particle systems has alsobeen demonstrated in Ref. [57] within a slightly differ-ent model: A ring enclosing a magnetic flux coupled to aside stub via a capacitive tunnel junction. In particular,it was shown that the passage through the hybridizationpoint is associated with the displacement of charge in realspace, and also that strong enough Coulomb interactionscan isolate the ring from the stub, thereby increasing thepersistent current.

4.3 Phase sensitivity in the Mott insulator

4.3.1 Perturbation theory in t/U

In the non-interacting limit, disorder leads to Andersonlocalization and the problem can be treated by an expan-sion in terms of t/W [58]. In the Mott insulator limit, theinteraction dominates, and it is possible to use an expan-sion in terms of t/U . In this second regime we decomposethe Hamiltonian of Eq. (9) as

H = H0 +H1 (21)


with an unperturbed part containing disorder and inter-action

H0 =M∑

i=1

vini + UM∑

i=1

nini−1 (22)

and the perturbation given by the hopping term

H1 = −tM∑

i=1

(c†ici−1 + c†i−1ci) . (23)

The solutions of the unperturbed part H0 are simple.The eigenstates are products of on-site localized Wannier-states for each particle

|ψα〉 =

(

N∏

k=1

c†ik(α)

)

|0〉 (24)

(|0〉 is the vacuum state) and the corresponding eigenen-ergies are given by

Eα =N∑

k=1

vik(α) + UNNN(α) . (25)

NNN(α) is the number of particle-pairs in the configura-tion α which occupy nearest-neighbor sites.

It is important to notice that, in contrast to H0, theperturbing part H1 depends on the boundary conditionc0 = exp(iΦ)cM . For periodic (p, Φ = 0) and anti-periodic(ap, Φ = π) boundary conditions one obtains

Hp1 = −t

M∑

i=2

(c†i ci−1 + c†i−1ci) − t(c†1cM + c†Mc1) (26)

and

Hap1 = −t

M∑

i=2

(c†i ci−1 + c†i−1ci) + t(c†1cM + c†Mc1) , (27)

respectively.The nth order correction to the phase sensitivity is

given by

Dn(U) = (−1)N M

2(Ep

n − Eapn ) , (28)

where Epn and Eap

n are the nth order terms in the per-turbation expansion of the ground state energies for theHamiltonians Hp

1 and Hap1 , respectively.

4.3.2 Phase sensitivity without disorder

The result for the problem without disorder has alreadybeen mentioned in the literature [29]. We present here itsderivation and generalize it to the case with disorder. Inthe limit t/U=0, the ground state of our Hamiltonian athalf filling is given by a regular array of charges without

nearest-neighbor sites simultaneously occupied. There aretwo possibilities to realize such an array, either all theparticles seat on the odd sites of the chain

|ψo0〉 =

(

N∏

k=1

c†2k−1

)

|0〉 , (29)

or the N particles are on the even sites

|ψe0〉 =

(

N∏

k=1

c†2k

)

|0〉 . (30)

These states both have zero energy and therefore, in or-der to study the effect of the boundary conditions, wehave to use degenerate perturbation theory in the sub-space spanned by |ψo

0〉 and |ψe0〉. For each boundary con-

dition, we can build the perturbation expansion in t fromthe matrix elements

Hs,s′

n =∑

α1,α2,...,αn−1

× (31)

〈ψs0|H1|ψα1

〉〈ψα1|H1|ψα2

〉 . . . 〈ψαn−1|H1|ψs′

0 〉(E0 − Eα1

)(E0 − Eα2) . . . (E0 − Eαn−1

),

with s, s′ = o, e and H1 given by Hp1 or Hap

1 dependingon the value of φ (0 or π, respectively). The sums run overall the eigenstates α of H0 except the two degenerate ba-sis states (29) and (30). There are two different types ofmatrix elements: the diagonal ones (He,e

n and Ho,on , start-

ing and finishing at |ψe0〉 and |ψo

0〉, respectively), and theoff-diagonal ones (Ho,e

n and He,on , starting and ending at

different states).The numerators of Eq. (31) contain matrix elements

〈ψαi|H1|ψαi+1

〉 of the perturbing Hamiltonian. Since H1

consists of one-particle hopping terms, non-zero matrix el-ements can arise only if the two states |ψαi

〉 and |ψαi+1〉

differ by nothing else than the position of one of the parti-cles. In addition, since the hopping terms allow only hop-ping to adjacent sites and we are dealing with spinlessfermions, the order of the particles on the chain is con-served in the subsequent hoppings.

In Fig. 11 we sketch a sequence of states α connect-ing |ψo

0〉 with itself for an n=6 diagonal contribution. Thefirst three steps of the sequence, connecting |ψo

0〉 and |ψe0〉,

represent an off-diagonal contribution to n= 3. We indi-cate with U the interaction energy associated with eachof the intermediate states of the sequence. Generally, thesequences in the states α can go “forward” and/or “back-wards”. For instance, a possible term contributing to n=2is that in which after the state α1 we go directly back to|ψo

0〉. However, since we are interested in the difference be-tween periodic and anti-periodic boundary conditions, itis only the φ-dependent terms that are relevant. That is,those involving a hopping between the sites M and 1.

In the case of periodic boundary conditions all the hop-ping terms have a negative sign. Thus, the numeratorsappearing in Eq. (31) are proportional to (−t)n. For anti-periodic boundary conditions the numerators are propor-tional to (−t)n(−1)hb , where hb is the number of hoppings


j o0iU 1U 23 (j e0i)U 4U 5j o0i

Fig. 11. A lowest order sequence contributing to the phasesensitivity for the example of N = 3 and half filling.

across the boundary between the sites M and 1. There-fore, the corrections to the ground state energy due to thepresence of finite hopping are the same for both boundaryconditions, except for the contributions to the sums withodd hb. These last contributions are the only ones relevantfor the finite phase sensitivity.

The sequences with n < N yield vanishing non-diagonal matrix elements, since we need at least N hop-pings to go between |ψo

0〉 and |ψe0〉. In the diagonal matrix

elements with n < M the sequences are such that eachparticle returns to its starting point, by doing forward andbackward hoppings, and therefore hb is necessarily even.Thus, these diagonal matrix elements are independent onthe boundary condition, and in addition Ho,o

n = He,en (the

denominators do not depend on the boundary condition,nor on the initial state).

The above described behavior of diagonal and non-diagonal matrix elements shows that the degeneracy isnot lifted for n < N . The lowest order in the perturbationexpansion which lifts the degeneracy, and at the same timeyields a contribution to the phase sensitivity, is n = N . Forthis order of the perturbation we have finite non-diagonalmatrix elements given by sequences where each of the par-ticles is moved by one site, the final state being the otherbasis state of the degenerate subspace. The connectionbetween the two states |ψo

0〉 and |ψe0〉 can be done by a

sequence where all the particles hop forward, and also bya sequence of backward hoppings. If one of these two se-quences crosses the boundary (yielding hb = 1, changingthe sign in the case of anti-periodic boundary conditions),the other does not (yielding hb = 0).

For the sequences that go between |ψo0〉 and |ψe

0〉 cross-ing the boundary, either with periodic or anti-periodicboundary conditions, we have to consider an additionalsign (−1)N−1 arising from the permutations needed to re-

cover the initial ordering of the fermionic operators in thefinal state. For an odd number of particles N and anti-periodic boundary conditions, the contributions of theforward and backward sequences cancel each other andHo,e

N = He,oN = 0, while with periodic boundary conditions

both sequences add and we have non-zero off-diagonal ma-trix elements that lift the degeneracy. For an even numberof particles, the opposite behavior occurs: The non-zerooff-diagonal matrix elements are those corresponding toanti-periodic boundary conditions. In this way there isa difference between periodic and anti-periodic boundaryconditions for all possible values of N . The corrections tothe ground state energies are given by the lowest eigen-value of the matrices HN . Since the diagonal matrix el-ements are the same, and independent of the boundaryconditions, for odd N we have Ep

N − EapN < 0, and for

even N we have EpN − Eap

N > 0. These signs ensure thatDN is positive, in agreement with a general theorem pro-posed by Leggett [37] and the result for a Luttinger liquid[38], fixing the sign of Ep −Eap according to the parity ofthe number of electrons.

The splitting of the energy levels (and therewith thephase sensitivity) is given by the size of the off-diagonalmatrix elements, which we estimate in the sequel. Athalf filling, the ground state configurations are the onlyones which do not contain any nearest-neighbor sites si-multaneously occupied, and their interaction energy van-ishes. The eigenenergies of the excited states of H0 areEα = U, 2U, 3U . . . , depending on the number of particlesplaced next to each other. Thus, the absolute values ofthe denominators in Eq. (31) are of the order Un−1. Wehave shown that the numerators are of order tn, there-fore, the parametric dependence of the lowest order (N th)correction to the phase sensitivity is given by

DN (U) ∝ U

(

t

U

)N

(32)

with an M -dependent prefactor, in agreement with theresult mentioned by Tsiper and Efros [29]. Since higherorder (n > N) terms contain higher powers of t/U , theresult (32) is the leading correction in the limit of stronginteraction t/U ≪ 1.

4.3.3 Phase sensitivity with disorder

The Mott insulator survives the introduction of disorderonly for finite-size samples and not too strong disordersuch that U > W

√N . We will place ourselves in this

regime in order to calculate perturbatively the phase sen-sitivity. The analysis presented for the clean case must bemodified, and leads to a qualitatively different result. Firstof all, the two different possibilities to realize the regulararray of charges will still be the energetically lowest con-figurations in the limit of strong interaction U ≫ t,W ,but the two states |ψo

0〉 and |ψe0〉 are no longer degenerate.


b d

U

logD D(U) / U 18

Fig. 12. The dependence of the phase sensitivity on the inter-action strength for the samples b,c, and d of Figs. 3 and 5, athalf filling (N =10, M =20, W =9), in double-logarithmic rep-resentation. Sample a exhibits the same asymptotic power law,but with a much smaller prefactor such that the correspondingcurve lies outside the scale of the figure. The dashed line showsthe perturbatively predicted slope D(U) ∝ 1/U2N−2.

Their energies at t=0 are given by

Eodd =N∑

k=1

v2k−1 and Eeven =N∑

k=1

v2k , (33)

respectively, and differ typically by W√N . If this differ-

ence is much larger than their coupling due to the hoppingterms (which, according to a perturbative argument likethe one presented above, is of the order of tN/UN−1), theground state is given by one of the two base states, butnot by a superposition of them. For N sufficiently large,or in the limit of large U where we are working, this isalways true.

As a consequence, one can use standard non-degenerate perturbation theory and only sequences of hop-ping terms with hb 6= 0 which connect the ground stateto itself can give rise to a phase sensitivity. The lowestorder contribution to the phase sensitivity must includenow M = 2N hopping terms to translate each of the par-ticles starting at odd (or even) sites by two sites such thatthe final configuration is equal to the initial one. Thissequence contributes in Eq. (31) to the diagonal matrixelement, connecting the ground state (|ψo

0〉 or |ψe0〉) with

itself, proportionally to (−t)M (−1)N−1 in the case of pe-riodic boundary conditions. In the case of anti-periodicboundary conditions we must add a sign (−1) associatedwith the one traversal of the boundary. In contrast to theclean case, forward and backward sequences always crossthe boundary and yield contributions of the same signto the diagonal matrix elements. Since the denominatorsdo not depend on the boundary conditions, the differenceEp

M −EapM has the correct sign [37] to yield a positive DM .

As illustrated in Fig. 11, one of the intermediate statescan be the regular array with the opposite parity than theinitial state. The energy difference associated to this state

is not U (like for all the other intermediate states withnearest-neighbor sites occupied) but |Eodd − Eeven|. Thedenominator containing the lowest power of U is then ofthe order WU2N−2 yielding

D2N(U) ∝ U

(

U

W

)(

t

U

)2N

, (34)

as the dominating behavior when U≫t,W . This is verifiedin our numerical simulations (Fig. 12) when consideringindividual samples after their last charge reorganization.The weak dependence with respect to W can be seen fromthe large-U behavior in Fig. 6.

4.3.4 Localization length for the Mott insulator

From the exponential size-dependence of Eqs. (32) and(34), one can extract the localization length in the Mottinsulator. Imposing the size scaling of Eq. (12) we obtainfor large M a localization length

ξclean = 2

(

ln

(

U

t

))−1

, (35)

in the clean case, while with disorder we have

ξdirty = 1

(

ln

(

U

t

))−1

. (36)

Both typical lengths shrink with increasing interactionstrength. Interestingly, the presence of disorder reducesthe localization length of the Mott insulator to one half ofthe clean value, independent of the disorder strengthW . Itis important to recall that the size-dependence that yieldsthe typical length ξdirty of Eq. (36) has to be restricted to

the condition U > W√N > tN/UN−1, allowing N to vary

over several orders of magnitude when U is large. ξdirty de-scribes the exponential decrease of D with the system sizewithin this range.

The logarithmic dependence of the localization lengthon the interaction parameter is also found for the Hubbardmodel in a clean one-dimensional system [59] as well as ina disordered 2D model with Coulomb repulsion [60].

5 Dependence on filling

In the previous sections we have been mainly concernedwith the interaction effects of spinless fermions at half fill-ing. At half filling, and with strong disorder, a short-rangeinteraction has a dramatic effect on the charge density. Wehave seen that reducing the disorder increases the localiza-tion length and results in less important charge reorgani-zations. Reducing the electron density increases the inter-particle distance, making a short-range interaction less ef-fective. In this section we analyze the effect of short-rangeinteractions as we move away from the optimal conditionsof strong disorder and half filling.


ab

dU

logD

Fig. 13. Phase sensitivity D(U) for four samples with theimpurity configurations of those of Fig. 5, but with only N =9particles (M = 20, W =9). The average over many samples isrepresented by the thick dots.

5.1 Fillings close to one half

In Fig. 13 we present the phase sensitivity for strong disor-der (W =9) and a filling close to one half (N=9, M=20).In analogy with the case of half filling presented in Fig. 5,we see strong peaks of D for individual samples and abroad maximum for the average values (thick dots). Wetherefore confirm that the physics of charge reorganiza-tions is robust, and it is not a simple commensurabilityeffect restricted to half filling.

5.2 Phase sensitivity in the large-U limit

The similarities between Figs. 5 and 13, that we havepointed out above, concern the charge reorganizations andthe corresponding peaks of D for relatively weak interac-tions (of the order of t). In the large-U limit there ap-pear some differences between the cases of half filling andless than half filling. For instance, we can see a tendencytowards saturation of D(U) for the samples a and c ofFig. 13. In fact, in a larger U -range we observe that allthe curves for densities less than one half saturate for suf-ficiently large U .

This saturation is easy to understand from the per-turbative analysis given in Section 4.3. In the sequence ofintermediate states |ψα〉 contributing to each of the termsin Eq. (31) (one of them represented in Fig. 11) we read-ily see that at less than half filling it becomes possible tomove the particles one after the other on the chain withoutever having two particles on neighboring sites. These se-quences give boundary condition dependent contributionsto En without any dependence on U in the denominators(E0 − Eαi

). Since they are the only contributions whichsurvive in the large-U limit, they define the value at whichthe phase sensitivity saturates.

The same happens at more than half filling, when al-ready the ground state contains the minimum number

2N−M of nearest neighbors. Here, it is possible to trans-late all of the particles at constant interaction energythereby leaving U -independent contributions to (31).

5.3 Charge reorganization at high disorder

Repulsive interactions give rise to charge reorganizationswhen the non-interacting ground state fixed by the disor-der configuration is not well adapted (energetically) to a fi-nite value of U . Assuming a very strong random potential,such that the one-particle states are close to on-site Wan-nier states, the ground state of N particles on M ≥ 2Nsites (more than half filling can be treated similarly, us-ing the particle-hole symmetry) at U = 0 is given by theparticles occupying the N lowest sites. The positions ofthese N sites on the ring are random. No reorganizationof the ground state due to short-range repulsive interac-tion takes place when this configuration does not containany two particles on neighboring sites.

The probability for obtaining such a configuration canbe calculated as follows. One starts from the configurationwhere the particles occupy the N first odd sites of the ring1, 3, . . . , 2N−1 and each of the particles has ‘his’ emptyeven site on its right. Then, M − 2N empty sites are stillavailable which can be distributed among the N gaps onthe right hand side of the particles, and we obtain

Ω =M

N

(

M −N − 1M − 2N

)

(37)

different possible configurations without nearest neigh-bors. The factor of M/N arises because of the M -foldtranslational symmetry of the problem and the indistin-guishability of the N particles.

Since the total number of possibilities to place N spin-less fermions on M sites is given by

T =

(

MN

)

, (38)

the probability to have a configuration without nearestneighbors in a given sample is

P =Ω

T=M

N

(

M −N − 1M − 2N

)

(

MN

) . (39)

Introducing the filling factor x = N/M , and using Stir-ling’s approximation for the evaluation of the factorials,we find

P ≃ eMg(x) with g(x) = ln

(

(1 − x)2(1−x)

(1 − 2x)(1−2x)

)

.

(40)

Since g(x) is negative for all 0 < x ≤ 1/2, we obtain alwaysP = 0 in the thermodynamic limit M → ∞ at constantfilling x. Therefore, in the case of strong disorder (W ≫ t),


U

logD

Fig. 14. Phase sensitivity for ten samples with quarter-filling(N = 5, M = 20) and strong disorder (W = 9). Roughly halfof the samples exhibit a peak in D(U) (solid lines), while theother half yield a monotonously decreasing phase sensitivity.

the probability to obtain a charge reorganization due torepulsive interactions is one in the thermodynamic limit,at arbitrary filling.

In Fig. 14 we consider the case of strong disorder(W =9) and quarter filling (M =20, N =5). For large Uwe observe the saturation of D(U) described in the pre-vious chapter. In the regime of smaller U > 0 we observesamples for which there is a peak in D(U) (exhibitingalso a charge reorganization) and others in which D(U)is a monotonously decreasing function. In the former casethe U = 0 configuration has nearest-neighbor sites thatare occupied, while in the latter there is no occupancy ofnearest neighbors in the non-interacting problem, and byincreasing U we do not achieve any substantial charge re-organization. A systematic study over many samples withthe conditions of Fig. 14 yields a probability of about 0.5to obtain a sample which does not present a peak in D(U),while Eq. (39) predicts P ≈ 0.26. This discrepancy can betraced to the fact that Eqs. (39) and (40) are valid in thelimit of a very strong disorder W ≫ t. For the parametersused in Fig. 14 it is not always true that the N parti-cles occupy the N sites with lower electrostatic potential,since the non-zero kinetic energy can split the two levelsassociated with two neighboring sites if their on-site en-ergies are closer than t (i.e. when a local potential well isformed by two nearly-degenerate sites). This splitting re-duces the probability to have two adjacent sites occupiedin the non-interacting ground-state.

5.4 Low fillings - weak disorder

We have analyzed above the limit of very strong disorder,when the one-particle states are almost completely local-ized on one of the sites. We have seen that in this regimea short-range interaction leads to a charge reorganizationwhenever two neighboring sites are occupied at U = 0.Therefore, in finite size samples, the probability of having

0 2 4 6 8 10 U

−10

−8

−6

−4

−2

log

D

a

b

c

a’b’

c’

Fig. 15. Thin lines: three samples (a, b and c) with M = 40,N = 10 and W = 4. Thick lines: three samples (a’, b’ and c’)with M = 40, and N = 10 obtained by scaling the previousimpurity potentials from W =4 to W =5. Thick solid line anddots: average of log(D) for W =5.

peaks in D(U) is maximum at half filling. At weaker dis-order, and without interaction, the particles are localizedover several sites. The occupation of a site is no longer analmost good quantum number (0 or 1) and turning on ashort-range interaction does not always result in a chargereorganization.

In Fig. 15 we present the case of quarter filling (N=10,M = 40) for three samples (a, b and c) with a disorderW =4, and another three samples (a’, b’ and c’) obtainedfrom the first ones by scaling the impurity potentials toW = 5. For instance, the configurations of samples a anda’ are obtained from the same set of random numbers, andit is only the overall scale of the fluctuations that changes.For certain samples (like c) the non-interacting system hasall the electrons localized far away from each other andthe charge configuration does not change when turningon the interactions, resulting in a monotonously decreas-ing D(U). On the other hand, in other samples (like a andb) the non-interacting one-particle wave functions overlapconsiderably and the appearance of a short-range inter-action results in a charge reorganization. In all cases weappreciate the saturation ofD(U) for large U , as discussedin 5.2.

Going from W = 4 to W = 5 for each of the sampleschanges the details of the curves, but not their nature. Forinstance, the peaks of a’ and b’ are obtained for strongerinteractions than those of a and b. Such a behavior is un-derstandable since higher disorder within the same ran-dom configuration necessitates a stronger interaction toproduce a charge reorganization. If we keep increasing Wfor a sample exhibiting a peak, we obtain sharper reso-nances at increasing values of the interaction. Of course,for all samples, the typical values of D are reduced whenincreasing W . Decreasing the disorder below W = 4 re-sults in broader peaks, that disappear once the localiza-tion domain of the individual wave-functions becomes of


the order of the inter-particle distance. The presence ofpeaks in the phase sensitivity is therefore fixed by the im-purity configuration of the sample, and independent of thestrength of the disorder in some range of W .

The different behaviors obtained among the sampleswith a given value W of the disorder are responsible forthe fact that the average phase sensitivity (〈logD(U)〉,thick dots in Fig. 15) decreases monotonously with theinteraction strength, instead of exhibiting the broad max-imum obtained for half filling and high disorder. We haveseen that in the limit of very strong disorder it was possi-ble to give a crude estimation of the probability to have asample that presents a peak in D(U). Once the disorder isweak enough to have single-particle states localized overseveral sites, we know that such a probability decreases,but it is rather difficult to extend the previous analysisin order to give an estimation of it. The relevant param-eters are the one-particle localization length ξ1, fixed bythe disorder, and the inter-particle distance 1/kF, fixed bythe electron filling. As a function of these two parameterswe can clearly distinguish two cases:

(i) When ξ1 becomes much smaller than 1/kF, thecharge density without interaction in finite size samplesis more and more likely to consist of N distant peaks.Since such configurations are well adapted to the inter-acting case as well, the probability to find reorganizationsof the ground state due to the interaction is reduced.

(ii) In the opposite case, when ξ1 becomes much largerthan 1/kF, no distinct one-particle peaks are present inthe charge density at U = 0, and there are no regions ofthe sample with very small charge density. While increas-ing U , the total energy is minimized by gradually pushingthe particles away, but there are no sudden charge reorga-nizations accompanied with peaks of the phase sensitivity.This behavior leads to a smooth decrease of D(U).

Between the two previous cases, we find the optimalsituation, kFξ1 ≈ 1, to observe the delocalization effect ofinteractions. This rough criterium is based on mean values(filling N/M and disorder W ) and only fixes the proba-bility to obtain peaks in D(U). The occurrence, or not,of charge reorganizations depends on the specific sample.That is, on the random configuration of the impurity po-tential. We have seen that if a given sample exhibits a peakof D(U), such a behavior is maintained over some range ofvalues of the disorderW (small enough not to take us awayfrom the condition kFξ1 ≈ 1 and into the limits (i) and(ii) previously discussed). It is worth to notice that thedisappearance of charge reorganizations, that we obtainupon decreasing the disorder or the electron density, aretightened to the fact that we work with repulsive nearest-neighbor interactions. A long range interaction could beeffective in yielding charge reorganizations away from theoptimal conditions of half filling and strong disorder.

6 Conclusions

We have investigated spinless fermions in strongly disor-dered chains, as a function of the strength of a short-rangerepulsive interaction, mainly for densities corresponding

to half filling. Such a system is a Fermi glass (Andersoninsulator) for small values of the interaction (when the dis-order is the dominant energy scale), and a Mott insulatorwhen the interaction dominates over the disorder. Usinga powerful numerical method, the density matrix renor-malization group algorithm, we have been able to addressthe transition regime between these two previous limits,which is not accessible by perturbation theory or meanfield approaches. We have calculated the charge density ofthe many body ground state, as well as the dependence ofthe ground state energy on the boundary condition. Thislast property, quantified by the so-called phase sensitiv-ity, is related to the transport properties and also to thepersistent current in the chain. It has a small finite valuefor the Anderson insulator and decreases with increasinginteraction strength in the Mott insulator.

In striking contrast to the case of weak disorder, whererepulsive interactions always strengthen localization forspinless fermions in one dimension, we have shown thatin strongly disordered samples, and close to half filling,the ensemble average of the phase sensitivity can be en-hanced by a repulsive interaction U . It shows a maximumat a value UF ≈ t, that is, for interaction strengths of theorder of the kinetic energy scale. The study of the density-density correlations reveals that the ground state is themost homogeneous (liquid-like) for U ≈ UF, consistentlywith the maximum in the phase sensitivity.

An analysis of different system sizes shows that thesefeatures persist in the thermodynamic limit. Even thoughthe system stays an insulator for all values of the interac-tion strength, the size dependence of the phase sensitivityallows to extract a many-body localization length, whichis slightly enhanced by the interactions and maximum atUF.

While the interaction-induced enhancement of thephase sensitivity is a rather small effect for the ensembleaverage, it can reach several orders of magnitude in indi-vidual samples, at sample-dependent values Uc. At thesevalues of the interaction strength, abrupt reorganizationsof the many-body ground state structure occur. The tran-sition from the Anderson insulator to the Mott insulator,upon increasing the strength of the repulsive interaction,is typically made in two steps. In a finite size system, thefirst change of the ground state structure at Uc is followed,at much larger interaction values Um ∝W , by the installa-tion of the Mott insulator. The interaction strength Um di-verges in the thermodynamic limit, reflecting the fact thatthe perfect order of the Mott insulator is destroyed by anarbitrarily small amount of disorder in an infinite size sys-tem. We have found that the reorganizations of the groundstate structure correspond to avoided level crossings be-tween the many body ground state and the first excitedstate, as a function of the interaction strength. We haveused simple toy-models to clarify the relationship betweenavoided crossings, charge reorganizations, and peaks in thephase sensitivity.

The two limits of weak and strong interaction canbe understood using perturbative schemes. Recently pub-lished Hartree-Fock results [61] exhibit some qualitative


similarities with our quasi-exact results. However, theaverage phase sensitivity is strongly underestimated byHartree-Fock at strong interaction values. A direct com-parison of the phase sensitivity for individual samples[62] shows that Hartree-Fock is rather accurate for weakinteraction values up to the first charge reorganization(U < Uc), but fails for stronger interactions when theground state is very different from the non-interactingground state. In the regime of very strong interaction, aperturbative development starting from the Mott insula-tor can describe quantitatively the phase sensitivity forU > Um.

In contrast, the intermediate regime UF < U < UW

where both, interaction and disorder play an importantrole, is intrinsically non-perturbative and more difficult toanalyze. The interplay of interactions and disorder leads tostrong electronic correlations in the intermediate regime,resulting in a behavior qualitatively different from the twolimiting situations.

In the limit of strong disorder, we were able to pro-vide a rough estimate of the probability of observing acharge reorganization in a given sample, as a functionof the particle density and the system size. This showsthat for strong disorder, half filling is the optimal condi-tion to obtain interaction-induced charge reorganizations.At lower electronic densities, the probability in finite-sizesamples is reduced with respect to this optimal situation.However, when the disorder strength is reduced simulta-neously, such that the one-particle localization length isof the order of the inter-particle distance, the reduction ofthe probability is less important. We expect that the useof a long-range interaction could also favor the occurrenceof charge reorganizations.

We recall the fact that the delocalization effect of re-pulsive interactions has been predicted in models otherthan that of our work: Disordered Hubbard models in1D [23] and in 2D [63], systems with strong binary disor-der [44], rings coupled to a side stub [57], and interactingbosons in a disordered chain [64].

As discussed in the introduction, the understand-ing of the metal-insulator transition in disordered two-dimensional systems has been one of our motivations forthis study. Even if our model is much simpler than the re-alistic problem of interest, it contains non-trivial featuresthat may be useful to understand the transition, like theconcept of charge reorganization discussed above. The in-termediate regime that we find between the limits of weakand strong interactions can be considered as a precursorof the correlated phase found in numerical studies [13]of two dimensional disordered clusters (with long-rangeCoulomb interaction) when the Wigner molecule is aboutto be formed. Consistently with our results, the use of theKubo-Greenwood formula and an exact diagonalization ina truncated basis of Hartree-Fock states [65] yields an av-erage conductance which is slightly increased by a smallrepulsive interaction for spinless electrons in strongly dis-ordered 2D systems.

Recent experimental measurements of the local com-pressibility in the localized phase of a two-dimensional

system [66] yielded important spatial inhomogeneities andvery large fluctuations as a function of the carrier density,while the metallic phase appeared as spatially homoge-neous and less fluctuating. The strong fluctuations of thelocalized phase can be interpreted as charge reorganiza-tions, very much in line with the interpretation of peaksin the phase sensitivity that we have thoroughly discussedin our work.

Since the existing numerical work in two dimensionsrelies on some drastic approximations or is restricted tosmall systems, it could be useful to extend our numericaltechniques beyond the one-dimensional case. In addition,the electron spin has been shown experimentally to playa major role in the metal-insulator transition [9,67], andit should be interesting to relax the condition of spinlessfermions in order to approach the realistic case.

Acknowledgements

We thank G. Benenti, G.-L. Ingold, A. Kampf, D. Voll-hardt, and X. Waintal for useful discussions, and Ph.Jacquod for drawing our attention to Ref. [21]. We grate-fully acknowledge financial support from the DAAD andthe A.P.A.P.E. through the PROCOPE program and fromthe European Union through the TMR program.

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Delocalization effects and charge reorganizations induced by repulsive interactions in strongly disordered chains