Controlling the competition of superconducting and charge

UNIVERSIDAD DE LOS ANDES, BOGOTÁ, COLOMBIA

MASTERS THESIS

Controlling the competition ofsuperconducting and charge ordered states

by doping in Fermi-Hubbard systems

Author:Jesús David Rincón Puche

Supervisor:Prof. Dr. Ferney Rodríguez,

Universidad de los AndesDr. Juan José Mendoza Arenas,

Universidad de los Andes

A thesis submitted in fulfillment of the requirementsfor the degree of Masters in science

in the

Condensed matter GroupDepartment of Physics

Faculty of Science

January 22, 2020

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Declaration of AuthorshipI, Jesús David Rincón Puche, declare that this thesis titled, “Controlling the competition ofsuperconducting and charge ordered states by doping in Fermi-Hubbard systems” and thework presented in it are my own. I confirm that:

• This work was done wholly or mainly while in candidature for a research degree at thisUniversity.

• Where any part of this thesis has previously been submitted for a degree or any otherqualification at this University or any other institution, this has been clearly stated.

• Where I have consulted the published work of others, this is always clearly attributed.

• Where I have quoted from the work of others, the source is always given. With the ex-ception of such quotations, this thesis is entirely my own work.

• I have acknowledged all main sources of help.

• Where the thesis is based on work done by myself jointly with others, I have made clearexactly what was done by others and what I have contributed myself.

Signed:

Date:

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“The key of joy is disobedience.”

Aleister Crowley.

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Universidad de los Andes, Bogotá, Colombia

AbstractFaculty of Science

Department of Physics

Masters in science

Controlling the competition of superconducting and charge ordered states by doping inFermi-Hubbard systems

by Jesús David Rincón Puche

One of the most interesting and highly-debated effects in quantum materials is the competi-tion between superconductivity (SC) and charge density wave (CDW). An important exampleof this interplay has been reported in transition metal chalcogenides, which are materials ofgreat interest due to the variety of phases that emerge because of the strong correlations be-tween their components. In particular, it has been observed that electronic doping can enhanceone the above phases, weakening the other [1].

Recent theoretical research, based on mean field theory, has reported that the additionof disorder to the system weakens the CDW, generating some isles where the SC phase isstronger [2]. However, given the highly correlated nature of these systems, is necessary to gobeyond this approximation, which fails to characterize quantum criticality.

In the present work, we analyze the CDW-SC transition for various electronic fillings ina theoretical model of correlated fermions, defined by the one-dimensional extended Fermi-Hubbard Hamiltonian. This systematic study is performed by calculating the ground state foreach filling and different interaction strengths between fermions. To do so, we use the den-sity matrix renormalization group algorithm, which captures spatial correlations. We mainlycharacterize each phase by the means of the Luttinger parameter, which indicates whether thesystem is domain by attractive or repulsive interactions between fermions. We find that reduc-ing the filling induces SC while degrading the CDW phase, by evidencing an increase of theLuttinger parameter. Our results will allow to design experimental strategies to strengthen SCvia doping in quantum materials.

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AcknowledgementsIn these last two years I have grown both as a scientist and as person thanks to the support ofseveral people. First, I would like to thank prof Ferney Rodríguez for his constant support inthe project and constant advises allowing this project to achieved what was proposed.

Secondly, I would like to specially thank Dr. Juan José Mendoza-Arenas for his constantmotivation, discussion, advises, guidance, comprehension in both my academic and personallife. All his expectations and discussions motivated me to do my best and give my all despiteany problem that may come.

To Dr Fernando Gómez, I give my most sincere "Thank you" to all the help he providedduring the initial phase of this project. Without his patience during this phase, I may not havelearned the basic tools that I later developed on my own and with Juan Jose’s help.

To Fabio, I give my appreciation because of the laughs that we had during the courses wetook together and the discussions we had regarding the codes. These discussions allowed meto optimized some problems and improved in ways that I may had never think of.

I am grateful to all my friends during this process. First to Daniel Lozano with whom Ilearned the basics of this process and has always been there for me in both silly and seriousproblems. To Juan Felipe Méndez, his support and laughs during the short time we shared inthe office but to the great friendship we have built during all these years. To Santiago Cortés,for these seven years of friendship and the tips regarding the propper presentation of an aca-demic work. To Enrique Araujo and Daniel Moreno, my chilhood best friends, for givingafternoon and nights to spare my mind off and re-motivate myself.

Lastly, I want to thank my family. To my mother, Athia, and father, Luis Francisco, thatteach me every ethic and moral value I have, as well as to give everything I have in everythingI do. To my brother, Luis Felipe, that always gave me support during this time, as he has beendoing all our lives. To Martha Cavadía, who is like a second mother to me, who always cheeredme up.

To everybody that has been or was in this journey: Thank you. Honestly, words are notclosed to describe what you all mean to me.

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We would like to thank Colciencias for the economic support of this project under the name "Produc-ción y caracterización de nuevos materiales cuánticos de baja dimensionalidad: criticalidad cuántica ytransiciones de fase electrónicas." No 120480863414. Also, to the Faculty of science of the University forthe finantial support given via the project "Matrix product state analysis of control of superconductivityby doping." INV-2018-49-1351 .

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Contents

Declaration of Authorship iii

Abstract vii

Acknowledgements ix

1 Introduction 11.1 Project overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Extended Fermi-Hubbard model 72.1 Physical parameters for characterization . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Spin and charge gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Luttinger Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 The model and ground-state phase diagrams . . . . . . . . . . . . . . . . . . . . . 11

3 Methodology: DMRG in the MPS language 153.1 DMRG algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 MPS and MPO formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Singular Value Decomposition and Schmidt decomposition . . . . . . . . 163.2.2 Definition of MPS and MPO . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.3 Graphical representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Expected value calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Ground state calculations with MPS . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.1 MPO representation of the Extended Fermi-Hubbard Hamiltonian . . . . 233.4.2 Iterative ground state search . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Code details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Results and analysis 274.1 Benchmark: Half-filling characterization . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Charge gap results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1.2 Double occupation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1.3 Luttinger parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Results below Half-filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.1 Charge gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2.2 Double occupation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.3 Luttinger parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Phase-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 Preliminary studies below quarter . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Conclusions and perspectives 415.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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5.2.1 Theoretical perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2.2 Experimental perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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List of Figures

1.1 Cooper Pairs. This is the main constituent that generates SC. In singlet super-conductivity, electrons of opposite spin are coupled via an indirect interaction tocreate a pair of spin 0 (singlet). Figure taken from ref. [3]. . . . . . . . . . . . . . . 2

2.1 Phase Diagram of the Extended Fermi Hubbard Model at half-filling, as a func-tion of U and V for J = 1. Taken from Ref .[4]. . . . . . . . . . . . . . . . . . . . . 11

2.2 Graphical representation of a CDW. Here we can observe how charge groupsperiodically through the chain. This periodicity can vary for each configuration.Image taken from ref. [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Phase diagram of the extended Fermi-Hubbard model at quarter filling. Modi-fied from ref. [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Building blocks of a MPS. The number of free legs gives the rank of the tensor. . 203.2 Graphical representation of tensor products. a) is the dot product in graphical

notation and b) a matrix product in graphical notation. . . . . . . . . . . . . . . . 213.3 Representation of a MPS elements in a chain. a) represents the first row element,

A1 of the MPS; b) second, the bulk element, Aj , and c) the last column element,AL−1. The sub-indices aj represent the sub-indices in the matrices of the MPS. . . 21

3.4 Graphical representation of a MPS . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5 Graphical representation of a MPO . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6 Graphical representation of 〈ψ| Oj |ψ〉. . . . . . . . . . . . . . . . . . . . . . . . . . 223.7 Graphical representation of minimization process. The left hand represents 〈ψ| H |ψ〉

and the right part, λ 〈ψ|ψ〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.8 Graphical representation of one step of the minimization. In this case, we have

taken out one element A. From here, we built Heff and Neff . . . . . . . . . . . . 25

4.1 ∆c vs 1/L. The blue dots represent the values of ∆c in V = 0 for each size andthe yellow line, the quadratic fitting, which gives a very good fit, namely R2 = 1. 28

4.2 ∆c vs V in the TDL. The solid line is a guide to the eye. . . . . . . . . . . . . . . . 294.3 dj and d at half-filling and L = 128. (a) is the double occupation spatial profile

vs j for different V s and (b) is d vs V . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 N(q) and its zoom at half-filling for V = 0.06 and L = 80. (a) N(q) vs q (b) zoom

at q → 0. The blue points represent the values of N(q), whereas the red line tisthe linear fit. The latter is excellent, with R2 = 0.9994. . . . . . . . . . . . . . . . . 30

4.5 K vs V at half filling. L = 64. The solid represents the interpolation. The yellowmark represents the critical point at this system size. . . . . . . . . . . . . . . . . . 31

4.6 E vs ∆N . L = 10 and V = −3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.7 ∆c vs V in the TDL at f = 7/16. The red curve represents the fit with R2 = 0.9997. 334.8 dj and d at V = 2.5 and L = 64. a) represents the system when two particles

have been removed and b) represents the system when four particles have beenremoved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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4.9 dj and d at f = 7/16 and L = 128. a) is the double occupation spatial profile vs jat V = 0.17 and b) is d vs V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.10 dj and d at f = 4/16 and L = 128. a) is the double occupation spatial profile vs jat V = 0.51 and b) is d vs V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.11 Change of K for different fillings for L = 64. a) is K vs V for f = 5/16. b) isK vs V for f = 4/16. The solid represents the interpolation. The yellow markrepresents the critical point at this system size. . . . . . . . . . . . . . . . . . . . . 35

4.12 Vc vs 1/L. The blue dots represent the Vc (where K = 1) for each size and the redline the quadratic fitting. Again, the fitting is successful with a R2 = 0.9988. . . . 36

4.13 Phase diagram vs filling. The upper zone represents the CDW ground-state andthe lowe region, the SC state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.14 K vs V for f = 3/16 and L = 64. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.15 K vs V for f = 2/16 and L = 64. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.16 K vs V . L = 256 and 4 fermions in all the chain. . . . . . . . . . . . . . . . . . . . 38

xv

List of Tables

4.1 ∆c at f = 7/16, in the TDL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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Dedicated to my family and friends.

1

Chapter 1

Introduction

Since its discovery, superconductivity (SC) has been regarded as one of the most spectacular

quantum effects in condensed-matter physics. This phenomenon consists of a material pre-

senting no resistivity and expelling all the magnetic fields that act upon it, allowing the electric

current to flow without dissipation [7]. Thanks to this, it has a great potential for applications

such as levitating trains [8], magnetic resonance imaging [9], among many others. There are

several characteristics of the materials that feature this effect: First their transition tempera-

ture, Tc, to the superconducting phase is a fraction of the Fermi degeneracy temperature, TF ,

and also their behavior is governed by Cooper pairs, the fundamental elements for SC. These

consist of two electrons that form a bound state due to an effective attraction, which are dis-

tributed across the system, as seen in figure 1.1. Also, it is well known that the mechanism

underlying this indirect coupling, in several materials, is the exchange of virtual phonons,

while some researchers believe that for other complex systems it might be an interchange of

other excitations. Furthermore, several of them can be understood under the Bardeen, Cooper

and Schrieffer (BCS) theory [10].

Even though SC has been studied rigorously, its complete understanding is lacking. For

example, in strongly correlated systems, SC competes with other quantum effects generated

by correlations such as the Charge-density wave (CDW), which occurs when there is a charge

periodicity in the lattice (prominently in low dimensional systems [11, 12]) which weakens the

SC phase. When enhanced by means of electronic doping (changing the filling in the system,

i.e the total number of charge carriers) or disorder, SC can grow while CDW is destroyed. This

impact of doping has not been studied in detail, leaving it as an open problem.

Some of the correlated materials that have received a lot of attention recently, where the

SC and CDW competition has been studied extensively, are the transition metal chalcogenides

2 Chapter 1. Introduction

FIGURE 1.1: Cooper Pairs. This is the main constituent that generates SC. In singletsuperconductivity, electrons of opposite spin are coupled via an indirect interaction

to create a pair of spin 0 (singlet). Figure taken from ref. [3].

(TMC). These are constituted by one transition metal and a number of chalchogens (two chal-

chogens make a dichalcogenide, for example). They are interesting materials for the study of

optical physics in the field of photonics and optoelectronics applications as well as nanoelec-

tronics [13, 14]. In addition, these materials are extremely sensitive to external perturbations

which leads to large changes in their physical properties [13, 14]. This make them very attrac-

tive systems to understand the competition that occurs among the mentioned phases [13, 14].

Since they do not have magnetic states, and possess a low critical temperature [13, 14], the

study of this competition becomes simpler when compared to that in more complex materials

such as cuprates, which are high temperature superconductors but present phases with mag-

netic order.

An example of this competition can be seen when adding pressure to the chalcogenide in or-

der to suppress CDW [15]. Another example is observed when the dimensionality of 2h−TaS2

is reduced, increasing SC. This reduction consists of removing layers until reaching an almost

one dimensional material, which increases Tc from 0.8K to 3.4K [16]. Additionally, disorder

in chalcogenides can enhance SC by weakening CDW, as mentioned above [17], where Tc has

been increased to 2.9K. Furthermore, some experimental studies have shown that when the

dichalcogenide is doped, the CDW or the SC get stronger, depending on the type of doping [1].

This could be done by maintaining the same number of charge carriers by changing the original

chalcogen (electronic doping), or alternatively replacing the metal for one of identical valence

(isoelectronic substitution). Additionally, recent experiments doping 2H − TaSe2−xSx with S

Chapter 1. Introduction 3

have increased SC in the material, while weakening CDW [18].

Besides the dichalcogenide, there are several studies involving trichalcogenides materials.

They corresponds to quasi 1D layered materials that have an unique chain-like structure [19]

and present most of the properties of the TMDs, as well [13, 14]. These are also attractive for

applications in field-effect transistors, infrared, visible and ultraviolet photodetectors, and for

the next generations in electronics [19]. Since they are 1D materials, they can present unique

properties such as the so-called Luttinger Liquid or Spin Charge Separation [20], resulting in

very rich and interesting effects that are not present in higher dimensions.

In addition to experiments, there have been many efforts on understanding, from a theoret-

ical perspective, the interplay between the SC and CDW phases. Some of these studies about

this phenomenon have been recently performed with mean field theory [2], which neglects

correlations. However, it is necessary to to incorporate them, due to the correlated nature that

many SC materials posses. For that reason, it is important to have simple models that ex-

hibit said correlations. One of is the Extended Fermi-Hubbard model (EFH), which is believed

to contain the key to understanding SC in correlated systems, since it possesses the competi-

tion between hopping of Fermions and their on-site and nearest neighbours interactions. This

Hamiltonian presents phenomena such as SC, CDW, spin-density wave (SDW), among oth-

ers [4, 6]. Since CDW and SC are neighboring phases, it constitutes an ideal starting point for

establishing strategies of enhancing one of the phases, by weakening the other. This model

possesses analytical solutions when there is no interaction between neighbours, which can be

found by the means of the Bethe ansatz [20]. However, most of its development has come from

numerical simulations.

In recent years, intense research has been performed in the half-filling (i.e, the same num-

ber of electrons as lattice sites) zero-magnetization EFH in order to obtain more information

on the phases present in the model. To characterize each phase, physical quantities such as the

charge gap and the spin gap are employed. Furthermore, correlation quantities such as order

parameters for CDW and SC can be applied, as well as the Luttinger Parameter [4, 6, 21]. Other

characterization techniques include quantum information tools, such as the Von Neumann En-

tropy, where most of the phase diagram has been reproduced [4, 21]. This quantity is used to

find the transition point between phases when a minimum or a discontinuity is found.


Another interesting case of study is the zero-magnetization quarter filling (i.e, the total

number of particles is equal to half the chain size) EFH. However, unlike the half-filled case,

there are very few studies on this scenario. One of the most important works constructed its

phase diagram via exact diagonalization. Here, using the charge gap, the spin gap and the

Luttinger parameter each phase was characterized [6]. Unfortunately, exact diagonalization

can only go as far as a chain of few sites and a reliable phase diagram in the thermodynamic

limit is not found in literature. A subsequent study using quantum Monte-Carlo [22] shows

the possible coexistence of phases (such as CDW and SDW). Also, new phases for some doped

systems were reported. In addition, this work also reported that for larger systems all critical

points decreased when compared to those of ref. [6]. However, this computational method is

limited by the Fermion sign problem, which causes an exponential increase of the computing

time with the number of particles [23, 24].

This model has been applied to study physical properties on real materials, such as the

TMD. Recently [2], a simulation based on mean field was made on the Extended Fermi Hub-

bard model, in order to compare with the experimental data on TMC. It was observed that by

simulating doping or disorder (at half-filling), the CDW phase is weakened, generating islands

where the SC phase is stronger, suggesting that both can enhance SC. Furthermore, a recent

study done with the so-called Bogoliubov de-Gennes formulation simulated disorder in an in-

tra and inter-orbital Hubbard model, at half-filling. This result shows how disorder increases

SC and CDW, that qualitatively fit with experimental data [25]. However, these studies are

limited due to its use of mean field theory, which does not take spatial correlations in consid-

eration, and going beyond this is important in order to understand how correlations affect this

interplay.

There are several numerical methods that take correlations in account. One numerical

method to stand-out is the Density Matrix Renormalization Group (DMRG). This is the most

powerful algorithm for studying one dimensional systems, allowing to find all the different

properties in the ground state with a highly efficient variotional process [26], by locating the

state on a fraction of the Hilbert space that contains all the relevant information of the system.

In recent years, this powerful tool has been formulated in the so-called matrix product state

(MPS) language. Here, the ground state is represented as a product of matrices that compactly

1.1. Project overview 5

incorporates the correlations in the system, thus the name given earlier. In fact, a recent study

presents results for a 2D modified Hubbard model using DMRG, finding phases such as CDW

and SDW [27]. For more information about the method see chapter 3 and reviews [24, 28, 29].

Although different numerical methods exists such as the one mentioned above, few stud-

ies have been performed changing the total charge carriers in the chain, giving only detailed

information about half-filling (one electron per site). Thus, the main purpose of this project is

to develop a systematic study of the CDW-SC transition varying the filling of the chain, which

serves as electronic doping. The analysis goes from half-filling to less than quarter filling, using

DMRG in the MPS language as the main calculation tool.

1.1 Project overview

In the present work, we study how electronic doping can enhance SC by weakening CDW, in

a EFH model. To do so, we consider a particular on-site interaction and then vary the parti-

cle number sector from Half-filling to quarter-filling, to observe how this electronic doping has

changed the critical point between CDW and SC for each filling. By obtaining the ground states

and correlations using DMRG in the MPS language, we proceed to calculate the charge gap, to

characterize whether the phase is metallic or insulating. Here, we have found that CDW is of

metallic behavior for fillings below the half-filled case. Afterwards, we proceed to calculate the

Luttinger parameter in order to determine the onset of SC. These results indicate how doping

changes the critical point for each filling. Finally, we extrapolate the results in order to have

a phase diagram in the thermodynamic limit (TDL) for each filling and present preliminary

results below quarter filling, information not reported in the literature.

The work is structured as follows.

• Chapter 2 presents the main features of the EFH model, which is the main object of study

in this project. We first explain how the model is constituted. We then explain the physical

quantities used to characterize each phase: the charge and spin gap, and the Luttinger

parameter. Following that, we explain the known phase diagrams for the half and quarter

filling cases.


• Chapter 3 presents a detailed explanation of DMRG in the MPS language. First, we intro-

duce the linear algebra concept that constitutes the core of the formalism: The singular

value decomposition. From here, we introduce the Schmidt decomposition from which

the MPS representation of the state is built. Afterwards, we explain in detail how the

ground state energy is calculated via a variational minimization.

• Chapter 4 shows our main results regarding charge gap, double occupation and Luttinger

parameter, for all the different fillings. With these results we have properly characterized

the SC and CDW phases and constructed the phase diagram. Finally, we provide prelim-

inary results for fillings below quarter filling.

• Chapter 5 contains the conclusions of the work, as well as the perspective and further

related work.

7

Chapter 2

Extended Fermi-Hubbard model

In solid state physics, there exist single particle models, such as the tight-binding model that

calculates the band structure and single-particle Bloch states of some materials. Beyond this

theory, the Fermi-liquid theory models correlated systems by means of weakly-interacting

quasi-particles, explaining several phenomena such as the normal state in Helium-3 and of

heavy Fermions [30]. However, in order to explain the physics underlying highly correlated

systems, simple models that contain strong correlations are necessary. The simplest and most-

known model is the Fermi-Hubbard Hamiltonian, which is the prototypical example of a

strongly correlated electron system. It features the competition between the kinetic hopping

and local interaction energy that leads to a enormous amount of interesting physics [31, 32], in-

cluding the Mott insulator state. In recent years, an extension that includes nearest-neighbour

coupling has been the object of intense research. This is the case not only because it results in

an even richer phase diagram, but also because it can model materials such as graphene [33]

and transition metal dichalcogenides [34].

Given our work is focused on characterizing one dimensional systems, at the beginning of

the present chapter we discuss how the different phases can be characterized. Afterwards, we

introduce the 1D model and we give a detailed explanation of the ground-state phase diagram

of half- and quarter-filling (the only known diagrams), focusing primarily in the regions that

contain the SC-CDW competition.

2.1 Physical parameters for characterization

In this section, we detail the physical parameters used to characterize each phase: the spin gap,

∆s; the charge gap, ∆c, and the Luttinger parameter, K. In this particular project we focus on

the characterization using ∆c and K, but mention ∆s for completeness.

8 Chapter 2. Extended Fermi-Hubbard model

2.1.1 Spin and charge gap

For a unidimensional system with ground-state energy, E(L;N↑, N↓) for N↑ particles with spin

up, N↓ particles with spin down on L sites, two quantities can be defined: The spin gap and

the charge gap. The spin gap, ∆s, is defined as the energy needed to create a spin excitation.

This quantity can be calculates as follows,

∆s = E(L;N + 1, N − 1)− E(L;N,N), (2.1)

where N represents the filling of particles of each nature in the lattice. The charge gap, ∆c, is

defined as the necessary energy to add or remove a particle in the lattice. When the system

presents insulating properties, ∆c > 0, while metallic behavior corresponds to ∆c = 0 [6, 35,

36]. Since we work with systems with zero magnetization, we calculate the double charge

gap [36], defined as

∆c = E(L;N + 1, N + 1) + E(L;N − 1, N − 1)− 2E(L;N,N). (2.2)

In this form we calculate energies to add or remove one particle of each spin orientation.

2.1.2 Luttinger Parameter

This section explains briefly the main characterization tool of the present work, The Luttinger

parameter, which allows to measure the nature of interaction between the particles in the chain,

namely attractive or repulsive. This quantity is introduced when studying the Luttinger model,

which is one of the simplest model known used to study interacting particles. This Hamil-

tonian is obtained when Fermionic annihilation and creation operators are approximated to

continuous operators in the non-interacting Fermi sea near the Fermi Level. Operators ψν rep-

resent the Fermions on the right side (ν = r) or the left side (ν = l) of non-interacting sea.

When we use these terms to create an interacting spinless model [32], the following Hamilto-

nian results

Hint =

∫ L/2

−L/2dx

[g4

2

∑ν

: (: ψ†νψν :)2 : +g2 : (: ψ†rψr :: ψ†lψl :) :

]. (2.3)

This is known as the Luttinger model. Here : ABCD : is the normal ordering for operators,

which consists of moving all the annihilation operators to the right, and multiplying by−1 each

2.1. Physical parameters for characterization 9

time an operator is exchanged with another due to the fermionic anti-conmutation relations.

The coupling g2 is the interaction term between ψl and ψr and g4 interaction for the same type

of field ψν . These constants are known as ”g-ology" [32].

Constants g2 and g4 ca be normalized as

g2 =g2

2πvf, g4 =

g4

2πvf, (2.4)

where vf is the Fermi velocity. After a lengthy normalization, which is presented in more

detail in refs [32, 37]. This normalization is used in order to re-write equation (2.3) as an

intermediate Hamiltonian,

Hint =πvfL

(1 + g4)

[∑ν

Nν2 + 2λNRNL

], (2.5)

where λ = g21+g4

, which is the new interaction parameter. From this we see that g4 only renor-

malizes vf , thus the nature of the interaction is defined by g2. It is easy to observe that when

g2 is negative, λ < 0 and when g2 is positive λ > 0, leading to attractive or repulsive interac-

tions, respectively. The hamiltonian in Eq. 2.5 is still not diagonal as it incorporates interactions

between different modes. To finally reach the diagonal form, we apply Bogoliubov transfor-

mations [32], which results in,

Hb = vf (1 + g4)√

1− λ2∑ν=1,2

∑q>0

qdν†q dνq + const (2.6)

where dνq represents the pseduo-Bosonic annihilation operator after applying the Bogoli-

ubov transformations and q the momentum vector. It is convenient to define the Luttinger

parameter, K1, as follows:

K =

√1− λ1 + λ

. (2.7)

From here we can conclude that K measures the attraction or repulsion of the Fermions

in the chain. If K > 1, the fermions have an attractive behavior; if K < 1, particles have a

repulsive interaction, and K = 1 indicates a non-interacting Fermi gas [32]. In addition, it is

known that the EFH model at quarter and half filling (one fermion per site), as well as the

1Normally, this factor is presented as Kρ for charge interactions, because there is also a Ks for spin changes.However, when only one type of interactions is presented, K is enough to represent it [38]. In this project, wecharacterize charge interactions.


Hubbard model at other fillings, can be approximated to a Luttinger liquid when ∆c is small,

allowing to use K as a means to characterize each phase [6, 22, 32]. Additionally, ref. [6] has

shown that in the weak-coupling limit, i.e small values of U and V , the model behaves as a

Luttinger liquid. To do so, they have calculated numerically the relation between the so-called

Drude weight, σ0, the compressibility, κ, the charge velocity, u0, and the total particle density,

n. When this relation satisfies

σ0

πn2κu20

= 1. (2.8)

the model behaves a a Luttinger liquid. Numerically this solution is close to one, proving that

for small V the EFH model is a Luttinger liquid.

Finally, we need a simple way to calculate K using the results of our simulations (details

are given in chapters 3 and 4). To do so, we define the local density operator as

nj = nj,↓ + nj,↑, (2.9)

Where nj,σ is the density operator of site j and spin σ. From here, the density-density

correlations are given by

Njk =< njnk > − < nj >< nk > . (2.10)

We can define the structure factor of these correlations as its Fourier transform, giving

N(q) =1

L

∑j,k

Njkeiq(k−j). (2.11)

Here, q represents the wave vector and is defined as

q =2π

Lj j = 0, 1, ... (2.12)

Finally, in the limit q → 0 it can be shown that [39]

N(q) ' K

πq. (2.13)

This means that when a narrow zone of N(q) is taken when q → 0, a linear approximation

gives K. This information can be acquired using DMRG, which is explained in chapter 3.

2.2. The model and ground-state phase diagrams 11

FIGURE 2.1: Phase Diagram of the Extended Fermi Hubbard Model at half-filling,as a function of U and V for J = 1. Taken from Ref .[4].

2.2 The model and ground-state phase diagrams

Our subject of study is the one-dimensional extended Fermi-Hubbard model (EFH) with open

boundary conditions. For a chain of L sites, the Hamiltonian is:

H = −JL−1∑i=1

∑σ=↑,↓

(c†i,σci+1,σ + h.c) + UL∑i=1

ni,↓ni,↑ + VL−1∑i=1

∑σ,σ′=↑,↓

ni,σni+1,σ′ , (2.14)

where σ represents the direction of the spin, either down (↓) or up (↑); ci,σ and c†i,σ represent

the annihilation and creation operators, respectively, for a fermion at site i with spin σ; h.c, the

hermitian conjugate and ni,σ, the local density operator defined as ni,σ = c†i,σci,σ. In addition, J

is the hopping between nearest neighbors; U , the on-site effective interaction between particles

on each site and V , the effective nearest neighbor interaction. U and V are effective interac-

tion strengths because they mask physical effects such as Coulumb forces, phonon-mediated

interaction and so on. When U, V < 0, the interaction between particles is attractive, whereas

U, V > 0 means a repulsive coupling.

The interplay between hopping, on-site and nearest-neighbour interactions leads to a com-

plex phase diagram. In order to properly characterize each phase, the quantities defined in the

last section are used: ∆c, K, and ∆s. Figure 2.1 shows the ground state phase diagram, at TDL,

of this Hamiltonian at half-filling (∑

i < ni,↓ >=∑

i < ni,↓ >= L/2), which is constituted as

follows:


FIGURE 2.2: Graphical representation of a CDW. Here we can observe how chargegroups periodically through the chain. This periodicity can vary for each configu-

ration. Image taken from ref. [5].

• The charge density wave (CDW) phase takes prominence in the first two quadrants. More

exactly in all second quadrant and V > 0 until V ' U/2, for the first quadrant, as fig-

ure 2.1 presents. This phase is characterized by a spatial charge modulation along the

system as depicted in figure 2.2. In the first quadrant, V dominates over U , so some sites

have double occupation. In the second quadrant, any neighboring repulsion naturally

double occupancies due to the on-site attraction [4]. Here, K < 1 and ∆c > 0. Since

∆c > 0, this CDW is an insulating phase.

• The singlet SC (SS) is located in a small fraction of the third quadrant for U < 0 and

V < 0, as seen in figure 2.1. Here the effective attractive interaction prompts the fermions

to form pairs, which emerge locally due to U being dominant. Thus, they must form

singlets. Here, ∆c = 0, K > 1 and ∆s > 0.

• The triplet SC (TS) is located in a small fraction of the third and fourth quadrant for U > 0

and U ∼ 0 and V < 0, as depicted in figure 2.1. The main difference between this phase

and the SS is that TS has finite magnetization (the pairs have spin 1) whereas SS has not.

Here, SC occurs due to nearest-neighbours interaction dominating over local interactions,

allowing nearest-neighbour triplet pairs to be created. Here, ∆s = ∆c = 0 and K > 1.

• The phase Separation (PS) is located in most parts of the third and fourth quadrants, as

observed in figure 2.1. In the third quadrant, both U and V are highly attractive making

all the electrons cluster in a small region of the chain. In the fourth quadrant, U is re-

pulsive, but V >> U , generating the same effect as in the third quadrant. This phase is

characterized by having no translational invariance.

2.2. The model and ground-state phase diagrams 13

SDW

TS

CDW

SS

Insulator

PS

SS+no pairedcarriers

PSdomain + no pairedcarriers

FIGURE 2.3: Phase diagram of the extended Fermi-Hubbard model at quarter fill-ing. Modified from ref. [6].

• The bond Order Wave (BOW) is located in a small fraction of the second quadrant for V '

U/2 and U > 0, as reported in figure 2.1. This phase possess alternating strengths of the

expectation value of the kinetic energy operator on the bonds. It is worth mentioning that

this has been a controversial phase, as it location has not been definitely established [40].

• The spin Density Wave (SDW) is located in the rest of the first and fourth quadrant, i.e,

U > 0, |V | ≤ U/2, as shown in figure 2.1. Since the local repulsion takes prominence over

the neighbour coupling |V |, this phase has an antiferromagnetic ordering. Here, ∆c > 0,

K < 1, and ∆s = 0. Since ∆c > 0, thi SDW is an insulating phase.

The EFH model at half-filling is very well known. However, for different fillings the be-

havior of this competition is almost entirely unknown, where only few systematic studies have

been performed for quarter-filling (∑

i < ni,↓ >=∑

i < ni,↓ >= L/4), in very small systems.

An exact diagonalization phase diagram, developed for a 12 chain sites, for quarter filling is

presented in figure 2.3. In this case, we list the most important phases.

• SS is located in a small fraction of the third and second quadrant as seen in figure 2.3.

Here, ∆c = 0, K > 1 and ∆s > 0.


• TS is located in a small fraction of the fourth quadrant as observed in figure 2.3. Here,

∆c = ∆s = 0 and K > 1.

• CDW is located mostly in the second quadrant, as depicted in figure 2.3. Here K < 1,

∆c = 0 and ∆s > 0. Thus, this CDW has a metallic nature, in contrast to half-filling.

• SDW is located mostly in the first quadrant, as presented in figure 2.3. Here, ∆c = ∆s = 0

and K < 1, therefore SWD has a metallic nature.

Importantly, figure 2.3 shows for fixed U < 0, that the transition between CDW and SS

occurs at V > 0, indicating that changing the filling increases the critical nearest-neighbour

coupling, Vc, between SC and CDW. Since the phases of our interest (SS and CDW) do have

∆s > 0, this quantity cannot be used to study criticality between both. Such analysis must be

performed using the Luttinger parameter.

15

Chapter 3

Methodology: DMRG in the MPS language

3.1 DMRG algorithm

The Density Matrix Renormalization Group (DMRG) is the most powerful numerical method

to study strongly correlated 1D systems. This revolutionary method was initially developed

by Stephen White [26, 41]. However, it has been reformulated in the matrix product state lan-

guage (MPS) since its formulation becomes more natural. In this chapter, we explain it in said

language.

The purpose of the algorithm is to find the ground state of the system of interest by minimiz-

ing the energy via an iterative process [24]. The main advantages of this method are the ability

to simulate large systems, something that cannot be done using exact diagonalization; that

it does not have the fermionic sign problem, which occurs in quantum Monte Carlo [23, 24];

and that it captures correctly spatial correlations going beyond the capabilities of mean field

theory, allowing to find information about the behavior of the system, including quantum crit-

icality [24].

In a many-body system, the Hilbert space grows exponentially with its size as dL, where d

constitutes the dimension of the on-site Hilbert space. DMRG truncates the Hilbert space in

order to find all the relevant information associated to the state of interest, which usually is

the ground state. To illustrate this point, consider a 1D system divided into two subsystems, A

and B. The entanglement entropy, also known as the von Neumann entropy, is used as proxy

quantity to determine the quantum correlations between the partitions. This is defined as [24]

S = −∑i

wiln(wi), (3.1)

16 Chapter 3. Methodology: DMRG in the MPS language

where wi are the eigenvalues of the reduced density matrix of one of the subsystems. It can

be shown that wi is the same for both partitions, so this depends on what the subsystems share,

which is their boundary. Thus the entanglement entropy grows proportional to the surface of

the subsystems, rather than their volume [24, 28]. This is the so-called area law. In 1D, this

surfaces corresponds to a point, therefore, the entropy at the boundary is constant. This means

that correlations across the system do not grow arbitrarily, but are bounded, and thus can

be characterized efficiently without any sort of exponential growth [24, 28]. Because of this,

DMRG is the most successful method for 1D systems, away from criticality. At criticality the

area law is no longer fulfilled [42, 43], but the entropy only grows logarithmically with the

system size, so it can still be studied with this method [24, 44].

3.2 MPS and MPO formalism

First, we describe the basic ideas of the language of Matrix Product States (MPS) and Matrix

Product Operators (MPO). For this, some linear algebra concepts need to be introduced.

3.2.1 Singular Value Decomposition and Schmidt decomposition

For any matrix C of size m× n, the singular value decomposition is defined (SVD) as:

C = USV †, (3.2)

with the following properties [24]:

1. U is a matrix of dimension m × p, where p is defined as min(m,n) and has orthonormal

columns, which means that U †U = I , where I is the identity matrix. If m ≤ n it implies a

unitary matrix and also UU † = I .

2. S is a square matrix of dimension p × p, with non-zero positive diagonal entries defined

as Sa, known as the singular values. The (Schmidt) rank of this matrix is the num-

ber r of non-zero elements in the diagonal. We assume the have a descending order:

S1 ≥ S2 ≥ S3... ≥ Sr.

3.2. MPS and MPO formalism 17

3. V † is a matrix of dimension p× n and has orthonormal rows, which means that V †V = I .

If m ≥ n it implies a unitary matrix and also V V † = I .

Now we use these concepts for describing a pure state |ψ〉. Here, a bipartition A-B is con-

sidered, where A has l site and B has N − l. In this situation, this state can be written as

|ψ〉 =

NA∑i=1

NB∑j=1

Ψij |i〉A |j〉B , (3.3)

where |i〉A and |j〉B, with dimensions NA = dl and NB = dN−l respectively, represent the

orthonormal bases for each partition; the coefficients Ψi,j are the amplitude of probability of

each combination of states and can be written as the elements of a matrix Ψ. When applying

SVD to this matrix, it is obtained that

Ψi,j =

min(NA,NB)∑a=1

Ui,aSaVTa,j. (3.4)

This allows us to re-write the state as

|ψ〉 =

min(NA,NB)∑a=1

Sa |a〉A |a〉B . (3.5)

These diagonal elements, Sa, are known as Schmidt coefficients, and we define the vectors for

subsystems A and B as

|a〉A =dl∑i=1

Ui,a |i〉A |a〉B =dN−l∑j=1

V Ta,j |j〉B . (3.6)

Equation 3.5 is known as the Schmidt decomposition. Here, each squared value S2a represents

one eigenvalue of the bipartitions, wi. The normalization of the state implies that

min(NA,NB)∑a=1

S2a = 1. (3.7)

Since we have ordered the Schmidt coefficients, equation 3.5 can be approximated by truncat-

ing the number of singular values in the expansion to a new state, |ψ′〉. Namely, |ψ〉 ' |ψ′〉,

where |ψ′〉 is defined as

|ψ′〉 =

χ∑a=1

Sa |a〉A |a〉B , (3.8)


with χ < min(NA, NB). From equation 3.7, we can obtain the error ε associated to the trunca-

tion, as follows,

1 = 〈ψ|ψ〉 = 〈ψ|ψ′〉+ ε. (3.9)

Thus ε can be defined simply as,

ε ≡min(NA,NB)∑

i=χ+1

S2a = 1− 〈ψ|ψ′〉 , (3.10)

It turns out that this truncation can be performed by keeping a small number χ of co-

efficients with a very low error in 1D systems with short-range interactions, because of the

bounded behavior of correlations resulting from the area law, as mentioned before.

3.2.2 Definition of MPS and MPO

Now, we apply the concepts of Schmidt decomposition and truncation to a general many-body

state of the form

|ψ〉 =d∑

n1,...,nL=1

Cn1,...,nL|n1, ..., nL〉 , (3.11)

where Cn1,...,nLis the amplitude of probability for each many-body basis state and |ni〉 is the

basis for each site i in the lattice. In order to apply the concepts mentioned earlier, consider

each Cn1,...,nLas an element of a matrix. This matrix of dimensions d × dL−1, with n1 the row

index, and (n2, ..., nL) the column index, is defined as

Cn1,...,nL= Ψn1,(n2,...,nL).

We apply a SVD obtaining

Ψn1,(n2,...,nL) =

r1∑a1=1

Un1,a1Sa1,a1VTa1,(n2,...,nL).

Here, n1 has internal dimension d, thus r1 ≤ d. Now, we renameUn1,a1 asAn1a1

and Sa1,a1V Ta1,(n2,...,nL) =

Ca1,n2,n3,...,nLwhich allows to write Ψ(n1),n2,...,nL

in the following way

Ψn1,(n2,...,nL) =

r1∑a1=1

An1a1Ca1,n2,...,nL

.


From here, we can represent Ca1,n2,...,nLas r1d×dL−2 matrix, with row index (a1, n2) and column

index (n3, ..., nL). By applying SVD to the matrix, we obtain,

Ca1,n2,...,nL=

r2∑a2=1

U(a1,n2),a2Sa2,a2VTa2,(n3,...,nL).

As we did earlier, we define U(a1,n2),a2 = An2a1,a2

and Sa2,a2VTa2,(n3,...,nL) = Ca2,n3,...,nL

. Then:

Ψn1,(n2,...,nL) =

n1∑a1=1

n2∑a2=1

An1a1An2a1,a2

Ca2,n3,...,nL.

By repeating this process with the rest of the coefficients, Cn1,...,nLis expressed as,

Cn1,...,nL=∑a1,..,aL

An1a1An2a1,a2

An3a2,a3

...AnL−1aL−2,aL−1

AnLaL−1

.

This is a matrix product that we can simply re-write as:

Cn1,...,nL= An1

1 An2 ...ANL

L .

With these, our many-body state is represented as:

|ψ〉 =d∑

n1,...,nL=1

An11 ...A

nLL |n1, ..., nL〉 . (3.12)

This structure is known as a Matrix Product State (MPS), because now the total amplitude

of probability is represented by the product of different matrices [24]. Here, each site has its

own group of matrices Anii with an internal physical dimension (local basis of each state), di.

The first element, An11 , is a group of row vectors and AnL

L is a group of column vectors. Thus

contracting all the elements, we obtain a complex number. Up until now, we have only rewrit-

ten the state in an alternative form. However, the power of this formulation is that it naturally

suggests that each SVD can be used to truncate the state, and reduce the number of elements

from O(dL) to O(Ldχ2).

These ideas can also be applied to an operator, written in general

O =∑

n1,...,nL,n′1,...,n

′L

Dn1,...,nL,n′1,...,n

′L|n1, ..., nL〉〈n′1, ..., n′L| , (3.13)


resulting in a Matrix Product State (MPO)

O =∑

n1,...,nL,n′1,...,n

′L

Wn1,n′11 ...W

nL,n′L

L |n1, ..., nL〉〈n′1, ..., n′L| . (3.14)

In this MPO representation, Dn1,...,nL,n′1,...,n

′L

, is again a matrix product. The difference between

MPS and MPO relays in that the MPO has two internal physical dimensions, instead of one,

corresponding to income and outcome of basis states.

3.2.3 Graphical representation

The MPS and MPO description have a nice graphical representation, whose building blocks

are depicted in figure 3.1. This leads to a better understanding of equations 3.12 and 3.14, and

helps implementing the required software in an intuitive way.

Rank 0 Rank 1 Rank 2 Rank 3

FIGURE 3.1: Building blocks of a MPS. The number of free legs gives the rank ofthe tensor.

.

We define a tensor, as seen in figure 3.1, as any geometrical shape of our choice, but to il-

lustrate the idea we will use circles. Each of these circles has a specific number of legs that

represents the rank of the tensor. For example, a circle with no legs is a rank zero tensor, i.e,

a scalar; a circle with one leg is a rank one tensor, which is a vector; a circle with two legs is a

matrix or a rank two tensor, and so on [24].

Tensor products can be illustrated using this representation, as seen in figure 3.2, by con-

necting legs of a tensor to those of another one. Figure 3.2a represents a dot product between

a row and a column vector, resulting in a scalar (a circle with no free legs); figure 3.2b, the

product between three matrices and since the resulting tensor has two free legs, it is a matrix.

This can be done to create higher rank tensors or lower rank tensor, depending on the number

of remaining free legs. As mentioned above, a MPS structure consists of different matrices that


a)

b)

FIGURE 3.2: Graphical representation of tensor products. a) is the dot product ingraphical notation and b) a matrix product in graphical notation.

contain the amplitude of probability of each state when multiplied with each other. For that

reason, the main elements of a MPS are represented as in figure 3.3.

n1

a1

a)nj

aj-1

aj

b) nL

aL-1

c)

FIGURE 3.3: Representation of a MPS elements in a chain. a) represents the firstrow element,A1 of the MPS; b) second, the bulk element,Aj , and c) the last columnelement, AL−1. The sub-indices aj represent the sub-indices in the matrices of the

MPS.

Figure 3.3a is the first element, An11 , which is a collection of row vectors, thus a rank 2

tensor. The horizontal leg represents the rows and the upper leg, its physical dimension, n1.

Figure 3.3b is the bulk element of a MPS. The horizontal legs correspond to rows and columns

of each matrix associated to Anii and the upper leg is the physical dimension. Lastly, we have

Figure 3.3c, which is the column vectors at the end of the lattice, AnLL . The column vectors are

depicted as the right leg coming out of it and the upper leg represents the physical dimen-

sion. The product of these elements is shown in figure 3.4. Here all the free legs correspond to

the physical dimension for each site. Finally, figure 3.5 gives the graphical representation of a

MPO. The MPO has two physical indexes per site, rather than one, which represent the local

basis of each state.


...

FIGURE 3.4: Graphical representation of a MPS.

...

FIGURE 3.5: Graphical representation of a MPO.

3.3 Expected value calculation

An expected value for a site j is defined as,

< Oj >= 〈ψ| Oj |ψ〉 , (3.15)

where Oj is an operator on a site. We present this operation in graphical form in figure 3.6, in

order to better illustrate what occurs inside the algorithm

...

...

...

...

1 2 j j+1

FIGURE 3.6: Graphical representation of 〈ψ| Oj |ψ〉.

As observed, only in site j the contraction takes place, while in the rest of the sites, the

states contract with no operator. From here, it is straightforward how to calculate correlations

for operators, Oj and Ok, where k 6= j (< OjOk >).

3.4. Ground state calculations with MPS 23

3.4 Ground state calculations with MPS

3.4.1 MPO representation of the Extended Fermi-Hubbard Hamiltonian

In section 3.2.3, the graphical representation of a general MPO has been presented. However,

for simple Hamiltonians with nearest-neighbour interactions, the corresponding MPO has a

very simple form [24]. Here, the elements in the MPO for the EFH are presented. First, it

is important to recall that the summation in the Hamiltonian represents an operation in the

Hilbert space of the entire system. Namely, the Hamiltonian of equation (2.14), represents the

operation

H = −Jc†1,↑ ⊗ c2,↑ ⊗ I ⊗ I...− Jc1,↑ ⊗ c†2,↑ ⊗ I ⊗ I..+ Un1,↑n1,↓ ⊗ I ⊗ I...+ V n1,↑n2,↓ ⊗ I ⊗ I...,

where ⊗ represents a direct product and I is the on-site identity. It is easy to express it as a

simple product, such that

O = W [1]W [2]...W [L]. (3.16)

Here each block W [i] acts on its respective Hilbert space at site i, whose tensor product gives

the global Hilbert space. Then the multiplication of these operator-matrices provides the total

Hamiltonian.

The construction of the Hamiltonian used in this project will have the following represen-

tation: For sites in the bulk,

W [i] =

I 0 0 0 0 0 0

c†↓ 0 0 0 0 0 0

c↓ 0 0 0 0 0 0

c†↑ 0 0 0 0 0 0

c↑ 0 0 0 0 0 0

n↑ 0 0 0 0 0 0

Un↑n↓ −Jc↓ −Jc†↓ −Jc↑ −Jc†↑ V n↓ I

(3.17)


Here the first column and the last row represent the first and final elements of the MPO,

which are given by

W [1] =[Un↑n↓ −Jc↓ −Jc†↓ −Jc↑ −Jc

†↑ V n↓ I

](3.18)

W [L] =

I

c†↓

c↓

c†↑

c↑

n↑

Un↑n↓

. (3.19)

By performing the matrix product of equation 3.16, the Hamiltonian is readily obtained.

3.4.2 Iterative ground state search

The main purpose of DMRG is to find the ground state of a Hamiltonian, via an iterative

variational process. In particular, we must find the MPS |ψ〉 that minimizes

E =〈ψ| H |ψ〉〈ψ|ψ〉

. (3.20)

To minimize the energy we utilize Lagrange multipliers, in order to ensure normalization.

Namely, we minimize 〈ψ| H |ψ〉 under the constrain 〈ψ|ψ〉 = 1. Thus, this quantity of interest is

〈ψ| H |ψ〉 − λ 〈ψ|ψ〉 . (3.21)

In this process, |ψ〉 is the desired ground state with energy λ. The graphical representation of

this operation is shown in figure 3.7.

Due to the bra-ket product, this is a highly non-linear problem. To solve this minimization,

we use the so-called alternating least square (ALS) method. This consists of making an initial

3.4. Ground state calculations with MPS 25

...

...

... -...

...

1 2 j j+1

1 2 j j+1

FIGURE 3.7: Graphical representation of minimization process. The left hand rep-resents 〈ψ| H |ψ〉 and the right part, λ 〈ψ|ψ〉

guess over all matricesA of the MPS; then keepA2, ..., AL fixed and optimizing overA1 [24, 44].

This transform equation 3.21 into:

minA1

(A1†HeffA1 − λA1†NeffA

1). (3.22)

Here, Heff and Neff are effective Hamiltonian and normalization matrices that depend on

the rest of the elements. The next step consists of deriving equation 3.22 with respect to A1†, as

follows

∂

∂A1†

(A1†HeffA

1 − λA1†NeffA1) = HeffA

1 − λNeffA1 = 0. (3.23)

This is simply a generalized eigenvalue problem, sketched in figure 3.8 for a bulk site, which

can be solved using standard linear algebra libraries. This eigenvalue problem first gives us

a state that lowers the energy but not the optimal one. This process is repeated for all sites,

from left to right and then from right to left, until the energy converges. In order to optimize

the problem, we apply the truncation method to a small value, χ. In that way, the truncation

will give us only a few number of eigenvalues that contain all the relevant information of the

system. Convergence is achieved for sweep k when Ek −Ek−1 < ε, where ε is around the order

of O(10−8). If convergence is not achieved, the routine will increase the value of χ repeating

the process. The increases of χ are done until convergence is acquired or until χ reaches a

...

...

... -...

...

1 2 j j+1

1 2 j j+1

=0

FIGURE 3.8: Graphical representation of one step of the minimization. In this case,we have taken out one element A. From here, we built Heff and Neff .


maximum value set by the user.

3.5 Code details

We use the Tensor Network Theory library (TNT) [45] developed by Oxford University which

proves as a powerful tool to solve the lineal algebra related to the MPS formalism including

the variational routine stated as the tool to find the ground state. For that reason we have de-

veloped a code in Matlab that create initialization files and a routine in C that calculates the

ground-state using the TNT library functions. Additionally, we also use Math Kernel Library

(MKL) developed by Intel as well as the linear algebra libraries developed by the National

Algorithm Group (NAG) in order to optimize the linear algebra problem associated. In small

systems we utilize ARPACK to optimize the linear algebra problem. Additionally, our simula-

tions use symmetries (conservation of the number of particles with spin up and down). This

allows us to calculate the ground state with the number of particles required by each simula-

tion.

27

Chapter 4

Results and analysis

In this chapter we present the main results of the analysis of the EFH at various fillings, with

open-boundary conditions, using the physical properties presented in chapter 2, namely the

charge gap, ∆c and the Luttinger Parameter, K for a fixed attractive on-site interaction U (U =

−2) and different nearest-neighbour interactions, V . We set the energy scale by taking J = 1. In

the first section, we discuss benchmark results by characterizing the transition and the phases

at half-filling, in order to test the accuracy of our simulations. Here, we present ∆c, the double

occupation for each site as well as its mean value, and K. Section 4.2 focuses on observing

how doping affects the SC-CDW transition by characterizing different fillings in the same way

as half-filling. We observe a metallic behavior for both CDW and SC, and thus argue that the

criticality between both phases cannot be indicated by ∆c, but rather by K. Subsequently, we

localize the critical points between CDW and SC for each filling and build our most important

result: the corresponding phase diagram. Finally, this chapter presents a first study for fillings

below quarter filling. We have simulated chains of sizes L = 32, 48, 64, 80, 96, 128; an initial

χ = 500 and final χ = 1500. Approximately, from 3 (for small systems) to 10 (for large lattices)

swepts are made in each simulation.

4.1 Benchmark: Half-filling characterization

We start by discussing our results of half-filling, that serve as a benchmark for our simulations.

To do so, we first present the results of charge gap in TDL; then, we present the double occu-

pation profile for certain fillings as well as its average. Finally, we observe the difference of the

Luttinger parameter in CDW and SC states.

28 Chapter 4. Results and analysis

4.1.1 Charge gap results

First, we calculate ∆c for different system sizes according to equation 2.2. Then we perform an

extrapolation. The most common form is given by [6]

∆c(L) =A

L2+B

L+ C, (4.1)

where A, B and C are constants and C = limL→∞∆c(L). An example of this extrapolation

is displayed figure 4.1 for V = 0. For this particular case, we obtain C = −7(95)× 10−4. In this

case, we can observe how the uncertainty is larger than the value of C, so we can conclude that

this value is indeed C = 0. Similar extrapolations are performed for other parameters of the

model.

0.000 0.005 0.010 0.015 0.020 0.025 0.030

1/L

0.00

0.05

0.10

c

FIGURE 4.1: ∆c vs 1/L. The blue dots represent the values of ∆c in V = 0 for eachsize and the yellow line, the quadratic fitting, which gives a very good fit, namely

R2 = 1.

With this in mind, we calculate ∆c in the TDL for several nearest-neighbors couplings. Fig-

ure 4.2 shows the behavior of ∆c as a function of V , for half-filling. As reported in ref .[4], the

CDW is an insulator. This is evidenced by the fact that ∆c > 0, which takes place for all V > 0.

Thus, our results are consistent with the literature.

4.1.2 Double occupation

The double occupation, dj , is a quantity that defines the average occupation of spin up and

down particles on the same site. It is defined as:

dj =< nj,↑nj,↓ >, (4.2)

4.1. Benchmark: Half-filling characterization 29

0.0 0.2 0.4 0.6 0.8 1.0

V

0.0

0.2

0.4

0.6

0.8

1.0

c

FIGURE 4.2: ∆c vs V in the TDL. The solid line is a guide to the eye.

where nj,↑ and nj,↓ represent the density operators defined in equation (2.14). Also, we consider

the average of dj across the whole chain [40]. This is given by:

d =1

L

∑j

dj. (4.3)

These two quantities provide an idea of the spatial profile of the charge along the chain. Fig-

ure 4.3 shows dj and d for several values of V ≥ 0. First, figure 4.3a presents two types of

periodicity along the chain. The first one is a two-site periodicity which alternates between

low and high values of dj , and shifts at the center of the chain to the opposite pattern. The

second periodicity modulates this behavior and has the period of the whole chain. This pe-

culiar periodicity occurs due to the open boundary conditions of the system, so the left and

the right halves are symmetric with respect to the center. Additionally it can be observed how

the charge modulation is degraded when decreasing V , until it vanishes when reaching the

SC state at V = 0. This general trend is shown in a more compact way in figure 4.3b, which

manifests the monotonic growth of d with V .

4.1.3 Luttinger parameter

Here we present an analysis for the results of K at half-filling. As mentioned in chapter 2,

K correctly characterizes the phase depending on its value, and when K = 1 the transition

CDW-SC takes place. To discuss our results, we need to explain how K is obtained, following

the description of sec. 2.1.2. First we present the structure factor of equation 2.11, N(q), in

figure 4.4. Here we observe at maximum at q/(2π) = 1/2 and a symmetrical behavior. From


0 20 40 60 80 100 120

j a)

0.3

0.35

0.4

0.45

dj

V=0.00

V=0.02

V=0.04

V=0.06

V=0.08

V=0.10

0.00 0.02 0.04 0.06 0.08 0.10

V b)

0.326

0.328

0.330

0.332

FIGURE 4.3: dj and d at half-filling andL = 128. (a) is the double occupation spatialprofile vs j for different V s and (b) is d vs V .

here, we zoom in the regime q → 0, as depicted in figure 4.4b, where we apply a simple linear

fit. As mentioned in section 2.1.2, we have:

N(q) =K

πq.

Thus K is given by the slope of the fit.

0.0 0.2 0.4 0.6 0.8 1.0

q/(2 )

0.0

0.5

1.0

1.5

2.0

2.5

N(q

)

a)

0.0 0.1 0.2 0.3 0.4

q/2

0.00

0.02

0.04

0.06

0.08

0.10

0.12

N(q

)

b)

FIGURE 4.4: N(q) and its zoom at half-filling for V = 0.06 and L = 80. (a) N(q) vsq (b) zoom at q → 0. The blue points represent the values of N(q), whereas the red

line tis the linear fit. The latter is excellent, with R2 = 0.9994.

By applying this linealization, we obtain K for the various values of V considered. Ad-

ditionally, to properly identify the critical points (at which K = 1), we interpolate our data

using another simple fit. This is depicted in figure 4.5 from which we identify the critical point

Vc = 0.076.

The figure also shows that for V > 0, K < 1, so the system remains in the CDW phase.

By comparing to the information provided by the charge gap, we have correctly identified that

4.2. Results below Half-filling 31

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

V

0.98

1.00

1.02

1.04

1.06

1.08

K

FIGURE 4.5: K vs V at half filling. L = 64. The solid represents the interpolation.The yellow mark represents the critical point at this system size.

K < 1 characterizes the CDW and that K > 1 corresponds to the SC state.

Now, we introduce a simple extrapolation to obtain the critical point between SC and CDW

in the TDL. Similarly to the charge gap, a common scheme is given by [46]

Vc(L) =A

L2+B

L+ C, (4.4)

where A, B and C are constants of the extrapolation and C = limL→∞ Vc(L). In this extrapola-

tion Vc = 1(60) × 10−3; since the uncertainty is larger than the TDL value, it can be taken as 0.

This is consistent with the critical point V = 0 known from previous research [4].

4.2 Results below Half-filling

Now we discuss our main results, namely the properties of the model below half-filling. Before

presenting the results, it is important to explain that this model has a particle-hole symmetry.

This indicates that the physics of the system when increasing the filling ( i.e, adding particles)

from half-filling is equivalent to that when reducing the filling (i.e, adding holes), so only one

needs to be studied. We present this symmetry in two ways: energy gap and number operators.

The first one is related to the equality

∆E(N + 1) = −∆E(N − 1), (4.5)


which means that the energy increases or decreases in the same amount when a particle is

added or removed, respectively. This is depicted in figure 4.6 where we note that equation 4.5

holds exactly.

2 2.5 3 3.5 4 4.5 5 5.5 6

N

-120

-100

-80

-60

-40

-20

0

E

Particles added

Holes added

Ground state

FIGURE 4.6: E vs ∆N . L = 10 and V = −3

The second form corresponds to the relation between creation and annihilation operators,

given by [47]:

c†hole,i = (−1)icparticle,i, (4.6)

for site i. From here, it can be proved that

c†hole,ichole,i = 1− c†particle,icparticle,i [47] (4.7)

Thus the population of holes can be directly obtained from that of particles and vice-versa.

Similar reasoning applies to the various correlation functions of the model. Given this sym-

metry, we restrict to fillings below half-filling. We emphasize that few previous studies of the

EFH model exist in these regimes, which consider mostly quarter-filling at small system sizes.

4.2.1 Charge gap

In this section we present ∆c for regimes below half-filling. The results presented here are

obtained in the same way as those presented in section 4.1.1. First, we need to properly define

the concept of filling, namely


0.000 0.005 0.010 0.015 0.020 0.025 0.030

1/L

0.00

0.02

0.04

0.06

0.08

0.10

0.12

c

FIGURE 4.7: ∆c vs V in the TDL at f = 7/16. The red curve represents the fit withR2 = 0.9997.

f =1

L

∑i

< ni,↑ >=1

L

∑i

< ni,↓ >, (4.8)

where ni,↓ and ni,↑ are again the density operators for spin up and down at site i. Electronic

doping occurs when we reduce f below 1/2. With this in mind, we calculate equation (4.1) for

different nearest-neighbor interactions V and several system sizes just below half-filling. One

case is exemplified in figure 4.7. Here we can observe a very small gap, that tends to 0 when

we increase the size of the chain. Table 4.1 depicts two points in the TDL for CDW and SC

phases (determined with K as seen in section 4.1.3) at f = 7/16. We observe that both ∆c are

much lower than their uncertainty, so they effectively vanish. Thus we conclude that for this

filling both phases are metallic. We made a similar observation for other fillings. Thus we can

conclude that both phases are metallic below half-filling; this has been previously observed

only at quarter-filling [6]. Our novel observation is the fact that even a filling slightly below

half-filling already generates a metallic CDW. This result connects with the fact that the V = 0

(Hubbard) model is known to be metallic away from half-filling [37], which is only insulating

at half-filling (Mott insulator) and maximal and zero filling (band insulator). Now we can con-

clude that this behavior is also true for the EFH model.

Phase V ∆c

CDW 0.17 4(90)× 10−2

SC 0.00 5(39)× 10−3

TABLE 4.1: ∆c at f = 7/16, in the TDL.


4.2.2 Double occupation

In addition to the information given by ∆c, we present the double occupation spatial profile,

in figures 4.8, 4.9 and 4.10 and its average for some fillings in figures 4.9 and 4.10. As it

can be observed in figure 4.8, when two particles are removed, two more nodes appear in the

chain, in addition to the single node observed at half-filling (see figure 4.3). Note that each

node corresponds to a shift of the two-site periodicity of the charge. Additionally, when the

number of particles removed increases, so does the number of nodes (as seen in figure 4.9a),

until no more nodes can be introduced along the chain, as figure 4.10a depicts. Again, all of

these spatial profiles emerge due to open boundary conditions in order to keep the left-right

symmetry in the chain. In contrast to half-filling, the decreasing of the amplitude of the CDW is

almost nonexistent when V decreases, as seen in figures 4.9b and 4.10b, by very small changes

of d. Overall, it can be concluded that dj cannot distinguish the difference between CDW and

SC away from half-filling.

0 10 20 30 40 50 60

j

0.0

0.2

0.4

0.6

0.8

1.0

dj

a)0 10 20 30 40 50 60

j

0.0

0.2

0.4

0.6

0.8

1.0

dj

b)

FIGURE 4.8: dj and d at V = 2.5 and L = 64. a) represents the system when twoparticles have been removed and b) represents the system when four particles have

been removed.

4.2.3 Luttinger parameter

Now we present our most important characterization: The changes in the Luttinger parameter,

K, due to the variations in the filling. As mentioned before, the value ofK determines in which

phase the system is located.

Figure 4.11a shows the same behavior as the one presented in half-filling: the increasing in

V decreasesK. However, in contrast to the half-filling case, the most important result to note is

that the critical point is Vc > 0, more exactly Vc = 0.268. This is a first indicator of how doping


0 20 40 60 80 100 120

j

0.15

0.20

0.25

0.30

dj

a)0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18

V b)

0.272

0.273

0.274

0.275

0.276

0.277

0.278

FIGURE 4.9: dj and d at f = 7/16 and L = 128. a) is the double occupation spatialprofile vs j at V = 0.17 and b) is d vs V .

0 20 40 60 80 100 120

j

0.05

0.10

0.15

0.20

dj

a)0.42 0.44 0.46 0.48

V b)

0.1320

0.1325

0.1330

0.1335

FIGURE 4.10: dj and d at f = 4/16 and L = 128. a) is the double occupation spatialprofile vs j at V = 0.51 and b) is d vs V .

0.25 0.30 0.35

V

0.97

0.98

0.99

1.00

1.01

K

SC

CDW

a)0.4 0.42 0.44 0.46 0.48 0.5

V

0.980

0.985

0.990

0.995

1.000

1.005

1.010

1.015

K

b)

FIGURE 4.11: Change of K for different fillings for L = 64. a) is K vs V for f =5/16. b) is K vs V for f = 4/16. The solid represents the interpolation. The yellow

mark represents the critical point at this system size.


helps the SC phase take prominence over CDW, as now the SC state is in a region where CDW

is present at half-filling.

Figure 4.11b, which corresponds to quarter-filling, presents a similar behavior to figure 4.11a.

It can be observed that this filling has a higher impact on the location of the transition, since it

has a larger critical point, Vc = 0.44. This value is notably lower than the one reported ref. [6]

(see figure 2.3), showing the importance of taking large system sizes to properly identify the

quantum phase transitions of the model.

4.3 Phase-diagram

Now that we have a clear idea of how doping affects the system, we present the most important

result in our work: The phase diagram for different fillings. To create this diagram, we perform

the extrapolation of equation 4.4 to obtain the critical point for different fillings between SC and

CDW in the TDL. Figure 4.12 shows this behavior for quarter filling.

0.010 0.015 0.020 0.025 0.030

1/L

0.45

0.50

0.55

Vc

FIGURE 4.12: Vc vs 1/L. The blue dots represent the Vc (where K = 1) for eachsize and the red line the quadratic fitting. Again, the fitting is successful with a

R2 = 0.9988.

When the extrapolation is applied to different Vc and fillings, figure 4.13 is obtained. It

nicely shows the behavior mentioned in the last section: The lower the filling, the higher the

critical point. First, for half-filling, the transition is consistent with theory, and when V > 0,

4.4. Preliminary studies below quarter 37

CDW is more prominent compared to the SC phase. Then we can observe a monotonous in-

crease of Vc when lowering the filling up to quarter-filling, thus the region of SC increases con-

siderably when compared to CDW, confirming how doping has destroyed CDW and enhanced

SC, reaching a maximum criticallity at Vc = 0.4059.

CDW

SC

FIGURE 4.13: Phase diagram vs filling. The upper zone represents the CDWground-state and the lowe region, the SC state.

4.4 Preliminary studies below quarter

In this section we present some preliminar studies for fillings below quarter filling. It is worth

noting that unlike quarter filling there is no information in the literature regarding this regime,

as explained by the higher computational effort requiered to perform successful simulations.

To characterize this region, we focus on the Luttinger parameter.

0.46 0.48 0.50 0.52 0.54 0.56

V

1.065

1.070

1.075

1.080

1.085

1.090

1.095

K

FIGURE 4.14: K vs V for f = 3/16 and L = 64.


0.56 0.58 0.60 0.62 0.64 0.66

V

1.200

1.205

1.210

1.215

1.220

1.225

K

FIGURE 4.15: K vs V for f = 2/16 and L = 64.

First we consider cases not far from quarter-filling, namely f = 3/16 and f = 2/16. At first

glance, figure 4.14 suggests that more doping increases Vc, which might be around Vc = 0.70,

as obtained from a linear extrapolation. One possible explanation for this large increase of Vc

is that since there are few particles, the density-density interactions are weak when compared

to the hopping and thus it becomes easier for the particles to move across the chain. However,

it is necessary to simulate larger chains since reducing of finite size effects could give a better

insight on the origin of this effect. These same insights can be applied to figure 4.15, because

the transition occurs at a larger critical point than in figure 4.14, around V ≈ 0.90.

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

V

0.18

0.20

0.22

0.24

0.26

0.28

K

FIGURE 4.16: K vs V . L = 256 and 4 fermions in all the chain.

However, in the diluted limit, (f → 0), the opposite behavior should be expected, i.e, a

decrease of K when more particles are removed from the system, since very few Cooper pairs

might be formed. As observed in figure 4.16 for an example in this regime, K is significantly

lower than 1 and decreases as V increases. These results are consistent with this expectation.

4.4. Preliminary studies below quarter 39

This suggest that there must be a filling where doping starts affecting SC negatively instead of

increasing it. It is important to mention that in order to observe this effects we need to simulate

chains of very large sizes (of hundreds of sites) which becomes very demanding computation-

ally. This corresponds to work to be performed in the near future.

41

Chapter 5

Conclusions and perspectives

5.1 Conclusions

We have studied the EFH model at different fillings using DMRG in the MPS language. In

particular, we have focused on the impact of filling on the quantum phase transition between

the SC and CDW states. For this we have considered the charge gap, the spatial profile of the

double occupation and the Luttinger parameter, for a fixed on-site attraction, U < 0.

• We have correctly characterized the CDW-SC transition at half filling by means of the

charge gap ∆c and Luttinger parameter K. First, when ∆c > 0, the system is in an in-

sulating CDW phase, whereas when ∆c = 0, the system is a SC. The transition is thus

observed at V = 0 as expected from previous studies [4]. This is further confirmed using

the Luttinger Parameter, where K > 1 indicates a SC phase, K < 1 a CDW, and the phase

transition at K = 1. Furthermore, we have observed how the spatial periodicity of the

double occupation decreases with V and the SC phase becomes homogeneous.

• We also confirmed the particle-hole symmetry of the model respect to half-filling, by

observing that the difference of energy when adding a number of particles is the same as

when removing the same number of particles (i.e, adding holes) from the system.

• At quarter-filling we have confirmed that both the SC and the CDW phases are metallic,

as indicated by a vanishing charge gap in the TDL. Additionally we have found that, in

contrast to half filling, the spatial profile of the double occupation is similar for both SC

and CDW. This is also shown by an almost constant average of this profile. This suggests

that the best way to characterize the quantum phase transition between the CDW and SC

phases at this filling is by the means of the Luttinger parameter.

42 Chapter 5. Conclusions and perspectives

• Between half and quarter filling, a regime with no reported studied, we have found that

CDW and SC have a metallic behavior, just like at quarter-filling. Even though the double

occupation changes slightly more than at quarter-filling, as reported by our results using

the average of this quantity, it still remains in the SC phase. Therefore, as in quarter

filling, the best way to characterize the CDW-SC phase transition at each filling is by the

means of the Luttinger parameter.

• By removing particles from the system we have observed the appearance of more nodes

in the charge modulation along the chain. These nodes keep growing until they reach

quarter filling, where there are no nodes in all the spatial profile.

• We have observed that changing the filling of the lattice shifts the critical point between

SC and CDW. Namely, the lower the filling, the higher the critical point, going from

Vc = 0 at half-filling to Vc = 0.4059 at quarter filling. Thus, the doping enhances SC at the

expense of CDW. This systematic study gives a first insight into the physics of the EFH

model at different fillings regarding the competition of these phases.

• By a simple extrapolation, we are able to find the critical point for different fillings in the

TDL. The ground-state phase diagram is depicted in figure 4.13, which clearly shows that

the region of SC increases, while CDW decreases.

• For fillings lower than quarter filling, but not in the diluted limit, results suggest that Vc

is still increasing. This behavior might be a consequence of the few fermions in the chain

making the density-density interactions less important, and since there is no literature

about fillings below one quarter, this behavior needs to be explored more deeply.

• In the diluted limit, we have observed a decrease in the Luttinger parameter as V in-

creases. This behavior is expected since for zero particles in the system, it is a band

insulator. However, the actual form in which this occurs still needs to be established due

to the computational difficulties associated to these simulations.

5.2 Perspectives

All of the obtained results motivate us to consider the following alternatives for future research

paths.

5.2. Perspectives 43

5.2.1 Theoretical perspectives

1. Our results might be extended by finding the transition SC-PS, in order to properly de-

limit the SC zone for each filling.

2. The calculated ground-states can be used as a starting point for dynamics studies to con-

trol the CDW-SC competition in out-of-equilibrium configurations by doping. Most dy-

namical studies have been performed in the (V = 0) Hubbard model [48–52] using the

time dependent DMRG method. However, very few focus on the EFH. Examples in-

clude a recent study in the dynamics of doublons [53] and dynamical control of the BOW

phase [54] using exact diagonalization.

3. Our investigation can be compared to other schemes of doping, such as the introduction

of disorder (simulating isoelectronic doping), in order to establish which of these forms

can optimally enhance SC [2, 17].

4. Our ground-state results could be a starting point to develop finite temperature studies,

to observe criticallity [55]. This can be done with similar methods to DMRG such as

dynamical study of mixed states using imaginary time evolution [56] and the so-called

Minimally entangled typical thermal state algorithms (METTS) [57].

5. The results obtained can be used as a benchmark for studying doping effects in more

complex systems such as ladders [58, 59] and lattices with electron-phonon interactions

like the Hubbard-Holstein model [60].

5.2.2 Experimental perspective

1. Experimental physicists can design experimental protocols bases on our results to control

materials such as TMD via electronic doping [13, 14].

2. Our results could also motivate the implementation of the EFH model in optical lattices,

where the Hubbard model has already been implemented, but the nearest-neighbor in-

teractions have not [61, 62]. This would constitute an important advance and attractive

plataform for testing our predictions since these systems possess atomic resolution and

capacity of control and are well isolated from the environment [63].

45

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