Liquid and Ordered Phases of Geometrical FrustratedCharges

A Monte Carlo study of the Falicov-Kimball model on the triangularlattice

Miguel Moreira de Oliveira

Thesis to obtain the Master of Science Degree in

Engineering Physics

Supervisors: Prof. Pedro José Gonçalves RibeiroProf. Stefan Kirchner

Examination CommitteeChairperson: Prof. Pedro Domingos Santos do Sacramento

Supervisor: Prof. Pedro José Gonçalves RibeiroMember of the Committee: Prof. Eduardo Filipe Vieira De Castro

November 2017

Acknowledgments

First and foremost I would like to thank my parents. They are the ones who always worked really hardto imbue in me good values, to provide for everything that was required and to allow me to study andpursue my dreams. For that sacrifice I shall be eternally grateful.

To my brother I leave some words of encouragement. When the time comes for you to be havinga hard time writing your own thesis just remember, if I could do it so can you. Hope it helps. You’rewelcome!

I thank my friends for all the good times we had. Without those distractions the arduous path tocomplete this work would have been that much harder.

I specially thank Sara Récio for being someone I did not think existed and for being next to me duringthe elaboration of this project.

I also give special thanks to my advisers Pedro Ribeiro and Stefan Kirchner. They provided tremen-dous help and guideness throughout all the work. I greatly appreciate their dedication and their willing-ness to help in reviewing the text. This thesis would not exist without them.

I would also like to thank Andrey Antipov for his help with some of the numerical aspects of this work.

I thank Pedro Bicudo and Nuno Cardoso for lending us a GPU for testing purposes.

I thankfully acknowledge the computer resources, technical expertise and assistance provided byCENTRA/IST. Computations were performed at the cluster “Baltasar-Sete-Sóis” and supported by theH2020 ERC Consolidator Grant "Matter and strong field gravity: New frontiers in Einstein’s theory" grantagreement no. MaGRaTh-646597.

I also acknowledge partial support by FCT through Pedro Ribeiro’s Investigador project IF/00347/2014and Grant No. UID/CTM/04540/2013 and by the National Science Foundation of China through StefanKirchner’s grant No. 11474250.

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Resumo

Fases-líquidas são estados da matéria que não demonstram ordem a longo-alcance, mesmo a tem-peraturas bastante baixas, onde por norma se esperam fases com alguma simetria quebrada. Pensa-seque estas fases estão associadas a fenómenos peculiares.

Neste projecto estudamos sistemas de interação de eletrões numa rede triangular com diferentesdensidades, o que produz diferentes graus de frustração-geométrica. Para frustração elevada é esper-ado comportamento semelhante ao de fases-liquidas a temperaturas baixas. Para tentar responder aesta possibilidade cosideramos o modelo de Falicov-Kimball: um dos mais simples para sistemas deeletrões correlacionados, para o qual existe um algoritmo (clássico) de Monte Carlo eficiente.

Para 1/3 de ocupação, onde a frustração é reduzida, foi determinado que a alta temperatura háfases desordenadas que apresentam comportamento diferente conforme a energia de interação: paraU grande há um isolador de Mott; para U intermédio um isolador de Anderson; para U baixo umafase fracamente localizada. Para baixa temperatura há uma fase ordenada com simetria Z3, que paraU grande leva a que a transição entre a fase ordenada e desordenada pertença a mesma classe deuniversalidade que o modelo de Potts para q = 3.

Para meia ocupação, onde a frustração é elevada, os resultados para temperatura alta são semel-hantes aos para 1/3 de ocupação. Para temperatura baixa, os regimes encontrados são maioritaria-mente inconclusivos, mas resultados perliminares parecem apontar para uma potencial ausência deordem. Temos esperança que a continuação do trabalho aqui apresentado possa levar a resultadosque consigam guiar a procura por materiais onde fases-liquidas existam.

Keywords: Modelo de FK, Monte Carlo, Frustração-geométrica, Fases-líquidas

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Abstract

Liquid phases are states of matter that display no long-range order even at very low temperatures,where usually some kind of symmetry-broken phase is expected. These phases are thought to possessunusual phenomena, such as emergence of effective gauge fields and fractionalization of the originaldegrees of freedom.

In this project we study interacting electronic systems on the triangular lattice with different fillingfractions, which produce different degrees of geometric frustration. For increased frustration one expectsliquid-like behavior at low-temperatures. To address this possibility we consider the Falicov-Kimballmodel: one of the simplest models of strong electron correlations, for which an efficient (classical)Monte Carlo algorithm exists.

At 1/3 filling, where frustration is reduced, we find that for high-temperature there are disorderedphases that behave differently according to the interaction strength: for large U there is a Mott insulator;for intermediate U an Anderson insulator; and for low U a weakly-localized phase. For low-temperaturethere is an ordered phase with a Z3 symmetry, which for high U leads the order-disorder transition tobelong to the same universality class of the two-dimensional Potts model with q = 3.

At half-filling, where frustration is high, the results for high-temperature are similar to the ones for 1/3

filling. For low-temperature the regimes found are still mostly inconclusive, but preliminary results seemto point at a potential absence of order. We hope that continuing the work presented here can lead toresults that can guide the search for compounds where liquid phases are realized.

Keywords: FK model, Monte Carlo, Geometrical frustration, Liquid phases

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Contents

List of Tables ix

List of Figures xi

1 Introduction 11.1 Ordered Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Geometrical Frustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Classical Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Quantum Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Falicov-Kimball Model 52.1 Falicov-Kimball Model: Definition and Exact Results . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Symmetries of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Bipartite Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Existence of a spectral gap and complementary filling conditions . . . . . . . . . . 72.1.3 Effective Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.4 Dynamical Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Simulating the FK Model 133.1 Why Can We Use Classical Monte Carlo? . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Thermodynamic and f-electron Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Order Parameter and Its Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.2 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.3 Other Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 c-electron’s Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.1 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.2 Inverse Participation Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Tools to Locate Candidates for Liquid Phases . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.1 Charge Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.2 Principal Component Analysis (PCA) . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Results 214.1 1/3 Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.1 Chemical Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1.2 Z3 Order Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1.3 Order-disorder Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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Critical Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.1.4 Mott and Anderson Insulator Phases and Respective Transition Lines . . . . . . . 284.1.5 Inconclusive Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1.6 Charge Susceptibility and PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Half-Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2.1 Mott and Anderson Insulator Phases and Respective Transition Lines . . . . . . . 354.2.2 PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Conclusion 39

Bibliography 40

A Monte Carlo Method 45A.1 Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.2 Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47A.3 Metropolis-Hasting Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48A.4 Computing Average Values of Observables . . . . . . . . . . . . . . . . . . . . . . . . . . 49A.5 Correlations Between Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50A.6 Estimation of Correlation Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

A.6.1 Binning Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52A.6.2 Jackknife Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

A.7 Cluster Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53A.8 Finite Size Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

A.8.1 Necessary Thermodynamic Quantities . . . . . . . . . . . . . . . . . . . . . . . . . 54A.8.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55A.8.3 Example: Ising model on a two-dimensional square lattice . . . . . . . . . . . . . . 55

B Newton’s Method For Fixing the Fillings 59B.1 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59B.2 Application to the filling fixing problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

B.2.1 Computing the derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

C Spline Functions and Error Estimation 63C.1 Determining the Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63C.2 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

C.2.1 Error in xmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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List of Tables

4.1 Values of ν determined using different quantities. The average value weighted by theerrors is also provided. The value of γ determined using the maxima of χ is shown as well. 27

4.2 Values of βc determined using different quantities. The average value weighted by theerrors is also provided along with the value determined via the crossing of the Bindercumulants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

A.1 Values of ν determined using different quantities. The average value weighted by theerrors is also provided. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

A.2 Values of βc determined using different quantities. The average value weighted by theerrors is also provided. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

A.3 Value of γ determined using the maxima of χ. . . . . . . . . . . . . . . . . . . . . . . . . . 57

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List of Figures

2.1 T−U phase diagram of the FKM on the square lattice at half-filling (image taken from [27]).Continuous lines correspond to second order phase transitions and dashed lines to firstorder. The inset corresponds to the estimation of the phase diagram after taking thethermodynamic limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.1 Phase diagram for the Falicov-Kimball model on the triangular lattice with 1/3 filling in theT − U plane, composed of five different regions: a charge density wave (CDW) orderedphase at low temperature and U ≥ 7. For high temperature there is a Mott insulator (MI)for large values of U , an Anderson insulator (AI) for intermediate ones and a metallic-like(WL) phase for low U . For low temperature and U < 7 the results are still inconclusive. . . 22

4.2 Specific heat as a function of the temperature for (a) U = 7 and (b) U = 20 for differentsystem sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Ground state configuration of the f-electrons at 1/3 filling for the CDW phase. . . . . . . . 254.4 (a),(c) Binder cumulants as a function of the temperature for several system sizes L and

two different interaction strengths U . (b),(d) Integrated correlation time τ|φ|,int for severalsystem sizes L and two different interaction strengths U . . . . . . . . . . . . . . . . . . . . 26

4.5 (a) Plot of χ(β) for different system sizes. (b) Integrated correlation time τ|φ|,int shown fordifferent system sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.6 Plot of the maxima of d log〈|φ|p〉dβ with p = 1, 2 and dU2

dβ . The lines are the fits of equations(A.45) (with a log− log scale) to the maxima and are used to extract the critical exponentsν. This scale was chosen so that the fits could be linear. The same thing for χφ(β) is alsodone here in order to extract the critical exponent γ. . . . . . . . . . . . . . . . . . . . . . 27

4.7 (a) Plot of the Binder cumulant U4 for different system sizes. The inset shows a close upof the crossing point at the critical temperature. (b) Plot of the pseudo-transition pointsβmax for the same quantities as in Fig.4.6. Equation A.46 is fitted to the data, where wetake ν to be the value estimated before. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.8 DOS(ω) and IPR(ω) computed via a histogram technique. All of the graphs are forT = 0.03. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.9 (a) DOS(ω = 0). (b) IPR(ω = 0). These quantities are used to determine the transitionline from Anderson to Mott insulator and from weakly localized to Anderson insulatorrespectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.10 Ground state configuration of the f-electrons at 1/4 filling. . . . . . . . . . . . . . . . . . . 314.11 Projection of the configurations in the subspace of the first two principal components y1

and y2. The color represents the temperature of a given configuration. The temperaturerange used includes the critical temperature of the phase transition. (a) L = 12 and (b)L = 10 both for U = 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.12 Plot of the projection of the configurations for different temperatures in the subspace ofthe two principal components determined by the PCA method for (a) L = 12 and (b) L = 10. 32

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4.13 Phase diagram for the Falicov-Kimball model on the triangular lattice with half-filling in theT − U plane, composed of four different regions: For high temperature there is a Mottinsulator (MI) for large values of U , an Anderson insulator (AI) for intermediate ones anda metallic-like (WL) phase for low U . For low temperature it is hard to get a definite pictureof the phases present. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.14 Specific heat as a function of temperature for different values of U and different systemsizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.15 Histograms for the number of f-electrons, the energy distribution of the Ising term and forthe triangular plaquettes term. These were computed using the configurations sampledvia Monte Carlo for different values of U and T . The temperatures chosen were the onesbefore and after the bumps in specific heat. . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.16 DOS(ω) and IPR(ω) computed via a histogram technique. All of the graphs are forT = 0.03. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.17 (a) DOS(ω = 0). (b) IPR(ω = 0). These quantities are used to determine the transitionline from Anderson to Mott insulator and from weakly localized to Anderson insulatorrespectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.18 Projection of the configurations in the subspace of the first two principal components y1

and y2 for U = 5. The color represents the temperature of a given configuration. Thetemperature range used includes the first bump in the specific heat. . . . . . . . . . . . . 36

A.1 (a) Integrated correlation time for the energy and (b) its binning analysis. The results wereobtained for the Ising model on a 16 × 16 square lattice simulated with the Metropolis-Hasting algorithm with sequential updates close to βc = log(1 +

√2)/2. . . . . . . . . . . 52

A.2 Plot of χ(β) for different system sizes. A logarithmic scale was used to help with visual-ization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A.3 Plot of the maxima of d log〈|m|p〉dβ with p = 1, 2 and dU2

dβ . The lines are the fits of equations(A.45) (with a log− log scale) to the maxima and are used to extract the critical exponentν. This scale was chosen so that the fits could be linear. We also do the same thing forχ(β) here in order to extract the critical exponent γ. . . . . . . . . . . . . . . . . . . . . . 56

A.4 (a) Plot of the Binder cumulant U4 for different system sizes. The inset shows a close upof the crossing point at the critical temperature. (b) Plot of the pseudo-transition pointsβmax for the same quantities as in Fig.A.3. Equation A.46 is fitted to the data, where wetake ν = 1 (the exact value). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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Chapter 1

Introduction

This thesis overreaching theme is concerned with liquid phases. Such phases of matter are expectedto display some quite unusual phenomena, including emergent gauge fields and fractionation of theoriginal charge and spin degrees of freedom. We hope the work developed here opens up a new andefficient route to study them. The Falicov-Kimball (FK) model is used in this approach since this isone of the simplest models for correlated electron systems. We hope that this simple model appliedto a frustrated lattice is enough to generate a system where liquid phases are possible. In addition tothis, the FK model can be simulated efficiently using a classical Monte Carlo algorithm. This is a veryattractive characteristic of the model. Usually many body problems require using a quantum Monte Carloalgorithm that in the case of fermions, due to the negative sign problem, has a complexity that scalesexponentially with the system’s volume V [1]. For the FK model the complexity of a single step of thealgorithm scales with O(V 3), which is much more manageable.

We start by presenting a brief discussion of some subjects pertinent to the overall work and thenafterwards state the project’s objectives.

1.1 Ordered Phases

At low enough temperatures, materials are typically found in ordered phases that spontaneously breaksome symmetry of the underlying microscopic system. For example, a material that possesses magneticmoments at the microscopic level, at high temperature might be in a paramagnetic phase, that is, itsmagnetic moments are not aligned and therefore there is no net magnetization. Such system has nopreferred magnetization direction and it is rotational, or SO(3), symmetric. If we lower its temperaturebelow a certain critical value, magnetic moments spontaneously align in some direction and the systemacquires a net magnetization. In this case, there is a preferred direction and so we say that the previoussymmetry has been broken.

Besides magnetic order, other examples of low temperature ordered phases are superconductor orcharge orders.

In the ordered phase, systems lower their energy by assuming a more structured configuration andacquiring a finite energy gap with respect to some kinds of excitations. This typically renders orderedphases more robust than their disordered counterparts. The theoretical framework used to deal withthis phenomena is the Landau-Ginzburg theory of phase transitions. Within this framework it was un-derstood that the appearance of ordered phases depends on the dimensionality of the system. Forlow dimension systems the thermal and quantum fluctuations tend to destroy the long range order. Onthe other hand for high dimensions, each element of the system interacts with more neighbor elementswhich counterbalances the effect of fluctuations and allows for ordered phases. An example of this is the

1

Ising model which has no ferromagnetic phase for 1D, but for 2D exhibits a phase transition at a criticaltemperature Tc under which the system acquires a net magnetization.

1.2 Geometrical Frustration

In addition to dimensionality, geometrical frustration adds a layer of complexity to the picture that makesit even more interesting. Geometrical frustration happens when there are competing interactions due tothe system’s geometry that cannot be simultaneously satisfied. The classic example is the Ising modelwith an antiferromagnetic interaction on a 2D triangular lattice. If we look at a single triangle we see thattwo of the vertices can be anti-aligned with each other but the remaining one has the same orientation asone of the first two. Geometrical frustration may cause a huge degeneracy of the ground state [2], whichcan be easily seen in the previous example. Note that if we instead had considered a ferromagneticinteraction on the triangle, there would only have been two possible ground states, one with every spinpointing up and another with every spin pointing down, therefore, if we were to extend the triangle to atriangular lattice we would still only have the same two possible ground states. Going back to the trianglewith antiferromagnetic interactions, there are six possible ground states, and if we have a lattice madeof such triangles the number of ground states becomes enormous.

1.3 Classical Liquids

On a classical frustrated system at low temperature, thermal fluctuations allow the system to shift be-tween ground states. As we further lower the temperature, those fluctuations start to decrease andwhen the energy kBT becomes very small the system freezes in a certain ground state configuration.Due to the huge degeneracy of the ground state there can be no long range order [3]. We can see thisin the following way: imagine a certain ground state composed of patches with different orderings; thisground state can be such that the regions where those different patches overlap have the same energyper degree of freedom as the patches themselves; therefore there is no incentive to keep a single orderand the system ends up having none. It is in this low temperature regime that we can find what is calleda classical liquid.

Spin liquids are a type of liquid phases, with spin ice being a prime example of their classical coun-terpart [2]. Spin ice can develop in materials where the only magnetic components live on the vertices ofa pyrochlore lattice. This is a lattice made of tetrahedra, where each lattice point belongs to two of thosetetrahedra. This system is classical because the spins involved are considerably larger than 1/2 andsince they are localized, their interaction can be regarded through the classical Heisenberg model witha ferromagnetic exchange energy Jeff . The ground state of a single tetrahedra is again degenerate dueto frustration and consists of two spins pointing inward and two outward in what is called the tetrahedronrule, which is also valid in the full pyrochlore lattice. This is similar to what happens in water ice hencethe name spin ice. When kBT � Jeff the thermal fluctuations predominantly make the system shiftbetween configurations respecting the tetrahedron rule.

The fundamental difference between spin ice and a normal paramagnet lies on the fact that spin icepossesses long range correlations between spins that follow a power law. This happens because thereis an "artificial" magnetic field associated with the spins that, due to the tetrahedron rule, is divergencefree. Even though the spins fluctuate, they will do so in such a way that the magnetic field lines willremain closed, forming large loops. These loops, in a way, allow the spins to be correlated over largedistances [4]. It has been shown that this artificial magnetic field fluctuates in equilibrium in the sameway a magnetic field does in vacuum and that this implies that the correlations of the spins follow a

2

power-law, even though the system is away from a critical point. Sometimes, even at kBT � Jeff ,there can occur the formation of magnetic field lines that are not closed, i.e. the formation of magneticmonopoles. This behavior has been recently observed in experiments [4].

1.4 Quantum Fluctuations

We now turn to the role of quantum fluctuations in a frustrated system. These effects are weak at theenergy scale of the classical Hamiltonian. However, they can play a significant role at low temperatures.

For a typical system, as temperature decreases the instabilities simply lead to some usual orderedphase. However, in systems with a large degeneracy of the classical ground state manifold, instabilitiesmay lead to new exotic phases with unusual emerging degrees of freedom. For example, in the the-oretical description of some spin liquids, fractional particle excitations and effective gauge fields wereproposed [4]. These gauge fields are the result of coherent collective "movements" of the individual spindegrees of freedom.

The search for spin liquid phases has been a topic of research in condensed matter physics in thelast few decades [5]. Besides their intrinsic fundamental interest, this search was fuelled by a propositionby Anderson [6] that a spin liquid state could be the parent state of high temperature superconductors.However, even if the search for liquid phases (in particular spin liquid) has been intense, what hasbeen found most of the times was that when magnetic order is destroyed some other types of orderingemerge [7]. Nonetheless the study of frustrated systems at low temperatures remains one of the rootsto new phases of matter.

Besides magnetic order, matter can also arrange itself into charge ordered phases [5]. Here, frustra-tion may induce charge liquids. In Refs. [7–9] it was proposed the existence of a phase composed of anordered "solid" part and a liquid part on the triangular lattice called pinball liquid. Here the fixed chargeswould occupy one of the 3 sublattices that make up the triangular lattice and the liquid part would movearound the remaining two. Charge liquids are also expected to possess fractional excitations and gaugefields [10].

1.5 Objectives

We are now in condition to state this project’s objectives. The main general goal is to use a semi-classical model on the triangular lattice in order to probe a phase diagram that might include some liquidphase. If this endeavor proves successful a new, and most importantly, efficient avenue for studyingsuch extraordinary phases of matter is open. The model chosen for this task is the Falicov-Kimball (FK)model due to the fact that allows to efficiently simulate correlated charge systems, which not only is auseful characteristic, but also a rare one.

It is also important to draw some kind of parallel with previous results obtained for simulations of thismodel in non-frustrated scenarios. Such studies were typically carried on the square lattice at half filling.In this thesis, the triangular lattice at 1/3 filling is used as the non-frustrated case and similar results tothe ones obtained for the square lattice are expected.

With the triangular lattice at half filling, geometrical frustration is present and therefore a differentpicture is expected to emerge.

With this in mind the particular objectives of this work are:

• Develop a Monte Carlo algorithm that can be used to efficiently simulate the Falicov-Kimball modelon the triangular lattice;

• Simulate the model in the non-frustrated case at 1/3 filling;

3

– Find ordered phases and determine the respective order parameters;

– Use the Density of States (DOS) and the Inverse Participation Ratio (IPR) to evaluate theconducting/insulating capabilities of different phases;

– Use those quantities to determine the transition lines and obtain a detailed phase diagramwith the different phases found.

• Simulate the model in the frustrated case at 1/2 filling;

– Use the same approach as before to find ordered phases and locate candidates for liquidphases;

– Use Principal Component Analysis (PCA) and the Charge Susceptibility to gather further evi-dence that strengthens the possibility that liquid phases were present in the phase diagram.

4

Chapter 2

Falicov-Kimball Model

In this chapter we provide a literature review on several aspects concerning the Falicov-Kimball. Westart by defining the model and its symmetries. Then, we present a detailed account of some exactanalytic results, followed by a set of more recent findings obtained via numeric simulations.

2.1 Falicov-Kimball Model: Definition and Exact Results

The Falicov-Kimball (FK) Model was introduced in 1969, in an article by Falicov and Kimball [11], todescribe metal-insulator transitions for rare earths and transition metals. It has later been proposed asa model for crystallization and for binary alloy systems [12]. It is one of the simplest models of stronglycorrelated electrons, featuring a set of phases only possible in the presence of interactions. The modelwas actually discovered earlier by Hubbard as a limiting case of the Hubbard model [13].

The FK model can be thought of as a model of delocalized electrons that move around on a latticeand interact with fixed classical charges. The former are designated as c-electrons and the later asf-electrons. The model is defined on top of a lattice Λ with V lattice sites, which in 2D is given byV = L × L, where L is the system’s linear size. The Hamiltonian of the simplest version of this model,featuring spinless particles, is given by:

H = −∑ij

tijc†i cj + U

∑i

c†i cinf,i − µfNf − µcNc , (2.1)

where c†i (ci) represent the creation (annihilation) operator of a c-electron on lattice site i and nf,i thenumber of f-electrons located there, which can be 0 or 1. The first sum is a tight binding term andrepresents the kinetic energy of the c-electrons, with tij being the hopping matrix element which is t fornearest neighbors and 0 otherwise. The second sum represents an on-site interaction between the twospecies of electrons which is present whenever one of each kind is at the same lattice site, with U beingthat interaction energy. µc and µf are the chemical potential of each species and fix their respectiveparticle number Nc and Nf , which are defined as:

Nc =∑i

c†i ci and Nf =∑i

nf,i . (2.2)

Depending on whether U > 0 or U < 0 the interaction is, respectively, repulsive or attractive, alludingto different physical interpretations. If U > 0, the f-electrons can be thought of as belonging to localizedf-orbitals of an atom located at a given lattice site. The c-electrons, on the other hand, belong to orbitalsthat strongly overlap and so they become delocalized and form a band. If U < 0, then what we call f-

5

electrons should be seen as a set of positively charged ions with the c-electrons moving and interactingwith the localized potentials generated by the positively charged ions. As shall be seen below thereis a symmetry of the model that allows results obtained in one case to be related to results obtainedin the other, which means that even though both scenarios are physically different, they are equivalentmathematically. From now on we shall just refer to them as c-electrons and f-electrons.

2.1.1 Symmetries of the Model

In this section we establish symmetry properties of the FK model that will turn out useful when general-izing some of the results obtained to other regions of the parameter space of the FK model.

We start by introducing a convenient change of variables in the occupation number of the f-electrons,si = 2nf,i − 1 (si = ±1). Thus, the f-electron occupation can be mapped to a spin-like or ’pseudo-spin’variable. The Hamiltonian is then rewritten accordingly:

H(s) =∑i,j

hij(s)c†i cj − (µc − U/2)Nc − µfNf , (2.3)

wherehij(s) = −tij +

U

2siδij (2.4)

is the single-particle Hamiltonian for a given f-electron configuration s and can also be written in matrixform

h(s) = T +U

2S ,

(T )ij = −tij , (S)ij = siδij . (2.5)

We also define µc = µc − U/2 . It is convenient to introduce an effective interaction energy F (s)

between the f-electrons for a particular configuration s. Such interaction can be defined in terms of thegrand canonical partition function [14]

Z =∑{s}

Tr[e−βH(s)

]=∑{s}

e−βF (s) , (2.6)

where the sum is over all possible configurations for the f-electrons and the trace is over the 2V dimen-sional c-electron Hilbert space. This effective energy F (s) is actually a function of temperature. For theground state properties one has to take the limit of β →∞ [14]:

E(s, µc, µf ) = limβ→∞

F (s) =∑

εν(s)<µc

[εν(s)− µc]− µfNf , (2.7)

where εν(s) are the eigenvalues of h(s).

Under a particle-hole transformation for the f-electrons, defined as nf,i → 1− nf,i, the spin variablesare fliped si → −si and the Hamiltonian transforms as [15]:

H(s, µc, µf , U) −→ H(−s, µc, µf , U) =∑i,j

hij(−s)c†i cj − (µc − U/2)Nc + µfNf − µfV

= H(s, µc,−µf ,−U)− µfV ⇔

⇔ H(s, µc, µf , U) = H(−s, µc,−µf ,−U)− µfV . (2.8)

This implies that, if s is a ground state configuration for µc, µf and U , then −s is a ground state config-

6

uration for µc, −µf and −U . This statement is valid for any lattice and shows how the results for U > 0

can be mapped to results for U < 0, which decreases the phase space to be studied.

Bipartite Lattices

A set of stronger results are available if one further restricts the properties of the lattice. Unlike theprevious result which is valid for any lattice, the following statements are only true for bipartite lattices,which are composed of two sublattices A and B. Lattice points that belong to A only have lattice pointsfrom B as nearest neighbors and vice-versa.

In this case, in addition to the transformation si → −si, we also consider a particle transformation inthe c-electrons: c†i → εici and ci → εic

†i , where εi = 1 if i ∈ A and εi = −1 if i ∈ B [15]. The Hamiltonian

is transformed as:

H(s, µc, µf ) −→H(−s, µc, µf ) =∑i,j

hij(−s)εiεjcic†j − (µc − U/2)(V −Nc)− µf (V −Nf )

= H(s, U − µc, U − µf )− V (µf + µc − U) ⇔

⇔ H(s, µc, µf ) = H(−s, U − µc, U − µf )− V (µf + µc − U) . (2.9)

The transformation for the free energy F is thus given as

F (s, µc, µf ) = F (−s, U − µc, U − µf )− V (µf + µc − U) , (2.10)

which allows one to relate expectation values for the particle numbers from one case to the other [16]:

〈Nc〉(µc, µf ) = V − 〈Nc〉(U − µc, U − µf )

〈Nf 〉(µc, µf ) = V − 〈Nf 〉(U − µc, U − µf ) . (2.11)

In particular, for µc = µf = U/2 we have that 〈Nc〉 = 〈Nf 〉 = V/2. This is a symmetry pointthat forces the system into half-filling. At this special point, the particle-hole transformation leaves theHamiltonian invariant [16]. In this regime, it can be shown for bipartite lattices with dimension d ≥ 2 thatthe system displays long range order at low temperature for all values of U [16]. In this phase, calleda charge density wave (CDW), the f-electrons organize themselves into a checkerboard pattern i.e. thef-electrons occupy only one of the two sublattices and the c-electrons the other. The standard argumentfor the occurrence of this phase is that the c-electron’s Fermi surface is susceptible to nesting. Thisphenomenon happens when there is a vector in momentum space that connects a large portion of statesat the Fermi surface. This leaves the system susceptible to perturbations (like the Peierls instability) thatmodulate both species of electrons and allow the c-electron states near the Fermi surface to lower theirenergy, gapping the spectrum everywhere on the surface [17] thus yielding an insulating phase.

For asymptotic large U the system can be mapped into an Ising model. Thus, for sufficiently largeU the phase transition obtained upon increasing the temperature, between the ordered CDW phaseand the disordered one, is of second order and belongs to the same universality class as the 2D Isingtransition [16].

2.1.2 Existence of a spectral gap and complementary filling conditions

We now turn to some results that will help to understand the gaped nature of the system in some regionsof phase space.

The following theorem, called Gershgorin circle theorem, can be used to bound the spectrum of h(s).In the following we state the theorem, provide a prove and explore its implications to the FK model.

7

Theorem: Let A be a complex n × n matrix with entries aij . Every eigenvalue of A belongs to atleast one of the disks D(aii, Ri) centered at aii with radius Ri =

∑j 6=i |aij |.

Proof: Let λ be an eigenvalue of A and x its corresponding eigenvector, which can be scaled in sucha way that xi = 1 and |xj | ≤ 1 for j 6= i. We have that Ax = λx, in particular:∑

j

aijxj = λxi = λ ⇔∑j 6=i

aijxj + aii = λ . (2.12)

Applying the triangular inequality one has

|λ− aii| =

∣∣∣∣∣∣∑j 6=i

aijxj

∣∣∣∣∣∣ ≤∑j 6=i

|aij ||xj | ≤∑j 6=i

|aij | = Ri . (2.13)

Let us now apply this theorem to gather information about the spectrum of h(s). The single particleHamiltonian h(s) matrix is an adjacency matrix of the lattice plus diagonal entries that can either be |U |/2or −|U |/2. This means that there are only two disk which can contain the eigenvalues, D1(|U |/2, zt) andD2(−|U |/2, zt), where z is the coordination number of the lattice. If one combines the largest end of D1

with the lowest of D2 one gets a range for the spectrum of h(s), spec h(s) ⊂ [−|U |/2− zt, |U |/2 + zt].Those disks can overlap for small values of U . However, if |U | > 2zt they are disjoint and a gap forms inthe interval ]zt− |U |/2, |U |/2− zt[ [18]. In such case the system is an insulator since a finite amount ofenergy is needed to add an electron to the system.

Following Ref. [14], we now use the last result to derive a strong result on the occupation numberof electrons of both species. In particular we show that for sufficiently large U > 0 the system has〈Nc +Nf 〉 = V . For that, it turns out useful to define the filling fractions of both species of electron as:

ρc =〈Nc〉V

and ρf =〈Nf 〉V

. (2.14)

Theorem: Let |U | > 2zt and µc be in the gap, i.e. |µc| < |U |/2 − zt. If U > 0 then ρc + ρf = 1. IfU < 0 then ρc = ρf [14].

Proof: We start with U > 0. Let n−(s) be the number of negative eigenvalues of h(s) and n+(s) thepositive ones. Since U > zt there are no eigenvalues equal to zero and so we have n−(s) + n+(s) = V .We introduce the support of s as supp s = {i ∈ Λ|si = 1}. We can then write a decomposition of thesingle-particle Hilbert space as:

H(Λ) = H(supp s)⊕H(Λ\supp s) . (2.15)

We now assume n−(s) < V − Nf . This means the set of eigenvectors with negative eigenvalue is notenough to generate the subspaceH(Λ\supp s). We can therefore have a vector ϕ ∈ H(Λ\supp s) whichis orthogonal to those eigenvectors. ϕ is given by some linear combination of eigenvectors with positiveeigenvalue therefore the following internal product has to be positive:

(ϕ, h(s)ϕ) > 0 . (2.16)

However one has

(ϕ, h(s)ϕ) = (ϕ, Tϕ) + U(ϕ, Sϕ) ≤ ||T || ||ϕ||2 − U ||ϕ||2 = (zt− U)||ϕ||2 < 0 , (2.17)

which is a contradiction. We therefore have n−(s) ≥ V −Nf . A similar argument can be used to showthat n+(s) ≥ Nf . This two inequalities taken together become equalities. The ground state is such that

8

only the c-electron eigenstates with negative eigenvalue are filled and so Nc = n−(s) and Nf = V −Nc.This is equivalent to ρc + ρf = 1.

A similar procedure can be used to prove that in the case U < 0 one has ρc = ρf .

2.1.3 Effective Model

For U > zt and µc within the spectral gap, a perturbative expansion in powers of t2/U can be set upwhich results in finite t corrections around the atomic limit.

To obtain a systematic expansion for T = 0 one starts by writing

∑εν(s)<µc

εν(s) =1

2[Tr(h(s))− Tr|h(s)|]

=U

2Nf −

1

2Tr|h(s)| − U

4V , (2.18)

which together with Nc = V −Nf allows one to write (2.7) as [14]:

E(s, µc, µf ) = (µc − µf )Nf −1

2Tr|h(s)| −

(µc +

U

4

), (2.19)

One can now expand Tr|h(s)| [19].

Tr|h(s)| = Tr[h2(s)]1/2 =U

2Tr[1 + ∆]2 , (2.20)

with∆ = 2U−1J + 4U−2T 2 and J = TS + ST . (2.21)

Since U is large enough, one can expand the trace in a power series of ∆ and collect together termswith the same power of 1/U .

For the square lattice (z = 4) one obtains the following spin-like effective model [18]

E(s, µc, µf ) = −1

2(µf − µc)

∑i

si −1

2(µc + µf + U/2)V +

∑|i−j|=1

[t2

2U− 9t4

2U3

]sisj+

+∑

|i−j|=√

2

3t4

2U3sisj +

∑|i−j|=2

t4

U3sisj +

∑P

5t4

2U3sP +O

(t6

U5

). (2.22)

Where the sum with |i− j| = 1 is over the interactions between nearest neighbors, that with |i− j| =√

2

is over elements diagonal to each other on the same square, and that with |i − j| = 2 runs over next-to-nearest neighbors. The sum over P denotes the sum over all plaquettes composed of four sites, withsP = sP (1)sP (2)sP (3)sP (4) being the product of the four spins that belong to it.

For the triangular lattice (z = 6) one obtains [18]

E(s, µc, µf ) = −1

2

(µf − µc +

6t3

U2

)∑i

si −1

2(µc + µf + U/2)V +

∑|i−j|=1

[t2

2U− 6t4

U3

]sisj+

+∑

|i−j|=√

3

3t4

2U3sisj +

∑|i−j|=2

t4

U3sisj +

∑∆

3t3

2U2s∆ +

∑P

5t4

2U3sP +O

(t5

U4

). (2.23)

Here, the sum with |i − j| =√

3 is over elements diagonal to each other on the same rhombus and thesum over ∆ denotes the sum over spins on each triangular plaquette: s∆ = s∆(1)s∆(2)s∆(3).

In Ref. [18] these expansions are used to determine ground state configurations for both lattice types

9

in the plane of the chemical potentials.

2.1.4 Dynamical Mean Field Theory

Dynamical Mean Field Theory (DMFT) [20] has become one of the standard approximations to ad-dress the properties of strongly correlated systems. In classical mean-field theory, interactions betweendegress of freedom are treated by placing the degrees of freedom instead in an effective field that, inturn, is the result of all the other lattice degrees of freedom. Thereby, the problem becomes essentiallynon-interacting but acquires a self-consistency condition. In the DMFT case, the system is mapped to animpurity model in contact with a bath of fermions that form a band [21, 22]. The lattice site interactionsare thus reduced to interactions with this bath of electrons. The self energy constitutes the contribu-tion that the interactions with the system give to a particle’s energy. It can be made local in the limitd→∞ because it is the result of an expansion, albeit non-analytic one, with a small parameter 1/d thatwill select only "local" Feynman diagrams. It is also local on the Betthe lattices, which are non-cyclicgraphs defined recursively. The fact that the environment of each lattice site looks the same makes theself-energy local and justifies the previously mentioned mapping.

One of the successes of DMFT is in understanding the metal-insulator transition that occurs as theinteraction strength increases. When the Fermi level lies in a band gap the material is expected tobehave as an insulator and when it lies inside a band it should be a conductor. According to simple bandtheory what decides which situation the system is in, is the number of electrons per lattice unit. Howeverthere are materials predicted by band theory to be conductors that are in fact insulators. This happensbecause the interactions between the electrons themselves were neglected. When they are taken intoaccount there is a complex interplay between the band’s contributions to the energy of the system andthe interaction between the electrons, which for some value of U may lead the system to open a gap.DMFT applied either to the Hubbard model or the FK model predicts this type of transition, which areknown as Mott transitions and the insulator phase is known as a Mott insulator.

In the limit of iinfinitely large coordination number z, i.e. for d → ∞, the results for the FK modelobtained with DMFT are exact [23,24].

2.2 Numerical Results

The previous section covered analytic rigorous results for the FK model, which are somewhat limitedand restricted to some of the regions of phase space of specific lattices. When analytic approaches fail,numerical methods can be employed to complete the picture and obtain new conclusions. This sectionfocuses on presenting an overview of the numerical results obtained for the FK model.

References [12, 25, 26] proposed a classical Monte Carlo method based on the Metropolis-Hastingalgorithm to simulate the FK model on a lattice. The method itself is explained in the next chapter andin Appendix A, here we state only its main results.

In Ref. [25], the method was applied on a 2D square lattice at half-filling. Some of the results men-tioned in the previous section were numerically verified, which also served as a validation of the nu-merical algorithm. It was reported that the system at low enough temperature indeed ordered in acheckerboard pattern. For large U the phase transition was confirmed to be second order. For smallU a first order transition was reported. The density of states for the c-electrons has been computed,revealing a gap at half-filing in the checkerboard phase that indicates that the system behaves as aninsulator. For small U and large T , after a phase transition, the gap was no longer found, and thus it wassuggested this phase was conducting. For large U the gap persists even in the disordered phase.

In Ref. [27] the model was simulated for the same lattice at half-filling, but the study unveiled some

10

additional phases. In the disordered phase, it was concluded that for large U the system was a Mottinsulator (MI), for an intermediate one it was an Anderson insulator (AI) and for small U a finite-sizemetal-like (WL) phase was reported, see Fig.2.1. Additionally, for U → 0 the system behaves, as ex-pected, as a normal Fermi gas (FG). There is strong evidence that the metal-like phase is indeed afinite size effect and is expected to vanish when the thermodynamic limit is taken. These phases werefound by computing the density of states, but also the Inverse Participation Ratio (IPR) and the opti-cal conductivity. The IPR, defined in the next chapter, is a quantity used for diagnosing the degree oflocalization of a wave function. When the electrons are completely itinerant the IPR scales with theinverse of the system’s volume and when they are localized it remains constant. Using these tools, itwas observed that the metal-like phase was neither gapped nor were the c-electrons localized, whichshows the system behaves as a conductor. The Anderson insulator does not have a gap in the densityof states, but the IPR shows that the electron states are localized and the real part of the static con-ductivity vanishes, indicating the system is indeed an insulator in this phase. Finally the Mott insulator ischaracterized by a finite gap at the Fermi level, confirming this is indeed an insulating phase.

Figure 2.1: T −U phase diagram of the FKM on the square lattice at half-filling (image taken from [27]).Continuous lines correspond to second order phase transitions and dashed lines to first order. The insetcorresponds to the estimation of the phase diagram after taking the thermodynamic limit.

In Refs. [12] and [26] similar studies were conducted away from half-filling. For all cases tested,the concentration of electrons per lattice site was the same for both species and the value was alwaysbelow 0.5 (half-filling). Here, for low U and low enough temperature, the system organized into stripesof f-electrons which the c-electrons tended to avoid. For large U the system formed completely separateclusters for both types of electrons in what is called phase separation. The density of states for thec-electrons was also computed. For small U and high temperature, the density of states resembledthat of free electrons, as in the case of half filling. As the temperature was lowered the density of statedeveloped some features but now no band gap was created. For large U there was always a bandgap as in the case of half-filling, but since the Fermi level was inside the lower band the system wassupposed to behave as a conductor.

There is also a similar work in a 3D cubic lattice [28] at half-filling. Here, it was observed that thecritical temperature between the ordered and disordered phase increases relative to the d = 2 case. Itwas also determined that the value Uc after which the transition starts being second order, increases aswell.

The method has also been applied to the triangular lattice [29–32]. Note that this lattice is non-bipartite, i.e. it cannot be decomposed in two sublattices and so frustration effects are expected. InRef. [30] the possible ground state phases of an extended FK model were studied. This version of themodel includes correlated hopping terms that made it so that the c-electrons take into account the f-electron occupations of the neighboring sites to perform the hop. The chemical potential is also fixed so

11

that Nc + Nf = V . Phase diagrams were obtained on the (t′-ρf ) plane for different values of U , wheret′ is the correlated hopping strength and ρf is the concentration of f-electrons. As expected there weredifferent types of possible ordered phases in contrast to the square lattice where only one unique phaseis found. Ref. [31] and [32] also used extended models, both with the Nc+Nf = V condition. In [31], thespin of the fermions is taken into account and in [32] two degenerate possible states for the f-electronsat each lattice site were considered. In both cases, it was found a complex set of ordered phases at lowtemperature, some of which have large lattice units.

For the triangular lattice, most numerical or analytic studies concern only ground state configurationsand are rather limited in scope, in particular to specific filling conditions. The present work aims atfurther completing the physical picture for this lattice, extending the predictions to finite temperature andstudying different fillings that are expected to enhance or suppress order.

12

Chapter 3

Simulating the FK Model

In the previous chapter we presented the state of the art regarding the FK model. The rather comprehen-sive physical picture for the 2D square lattice was obtained from a combination of exact and numericaltechniques. In particular, numerical simulations were instrumental in addressing the properties and fullphase diagram of the model. This is unusual since for correlated electron systems for d > 1, generic nu-merical methods are typically restricted to very small systems sizes due to the exponentially increasingsize of the Hilbert space with the volume. For the FK model, an almost complete numerical solution ispossible due to the specifics of the model.

In this chapter, we discuss the particular features that allow for an efficient simulation of this modeland introduce the relevant quantities used later to analyze the results. Those quantities are computedthrough a classical Monte Carlo method for the FK model. A reader unfamiliar with this method may findAppendix A useful where its general application is explained in detail. An example of the method usingthe Ising model on a square lattice is also provided.

3.1 Why Can We Use Classical Monte Carlo?

Typically, condensed matter physics is concerned with many body systems involving the interaction ofmany identical particles. This type of problem is of extreme importance to understand the properties ofmaterials that cannot be inferred by looking at their individual constituents separately. However, suchproblems are typically hard to solve analytically and even the best algorithms to tackle these problemsusually need an amount of resources that scale exponentially with the system size [1]. The FK modelat first glance might look like such a many body problem, but closer inspection reveals that an efficientalgorithm can be found. This is because there is an extensive number of conserved quantities [H,nf,i] =

0 for every i in the lattice, which means that the nf,i are good quantum numbers. As they are conservedthey behave as classical variables which can take the values nf,i = 0, 1. So, for each configuration ofthese classical charges, we can look at the Hamiltonian as that of a model of delocalized non-interactingelectrons that see scattering potentials across the lattice. To see this more clearly let us write theHamiltonian as

H =∑i,j

c†ihijcj − µcNc − µfNf , (3.1)

where hij can be written ashij(nf ) = −tij + Unf,iδij . (3.2)

13

We now can see that H is just a second quantized version of h(nf ), so the c-electrons just fill theeigenstates of h(nf ). The partition function can be written as

Z = Tr[e−βH

]=∑{nf}

Trc[e−βH

]=∑{nf}

(eβµfNfTrc

[e−βc

†[h(nf )−µc]c])

, (3.3)

where the sum is over all configurations of the f-electrons, c†h(nf )c is just the first term of (3.1) in matrixform and c (c†) is a column (row) vector whose elements are the annihilation (creation) operators for eachlattice site. The c-electron degrees of freedom can be integrated out by a suitable unitary operation

Trc

[e−βc

†[h(nf )−µc]c]

=∏ν

(Tr[e−β[εν(nf )−µc]c†ν cν

])=∏ν

(1 + e−β[εν(nf )−µc]

), (3.4)

where c = Uc, with U being an unitary transformation that diagonalizes h(nf ) and εν(nf ) are the eigen-values of h(nf ) for a particular configuration of the f-electrons. This finally allows us to define an elec-tronic free energy

F (nf ) = − 1

β

∑ν

log[1 + e−β[εν(nf )−µc]

]− µfNf , (3.5)

which in turn helps us to simplify the partition function

Z =∑{nf}

e−βF (nf ) . (3.6)

Looking at the partition function in this form one can clearly see how a classical Monte Carlo algo-rithm, like the Metropolis-Hasting, can be employed here. One just needs to start with some configurationfor the f-electrons, propose a new one and then use the variation in the free energy to determine theprobability of accepting the new configuration. So basically we are able to sample f-electron configura-tions that respect the equilibrium distribution. For each iteration the algorithm needs to diagonalize thesingle particle Hamiltonian h(nf ), which takes a time that scales like O(V 3).

In this picture we can also think of the f-electrons as interacting with each other via the c-electrons,so F (nf ) can be seen as the energy of an effective model that encodes those interactions.

3.2 Thermodynamic and f-electron Properties

For completeness, we introduce, in Appendix A, the thermodynamic quantities needed to conduct thetreatment of the MC data, specifically: the specific heat, the f-electron charge susceptibility and theBinder cumulant. These quantities are used to perform the finite size scaling (FSS) analysis (alsodetailed in Appendix A) required to estimate critical temperatures and critical exponents. In this sectionwe provide expressions for those quantities in the context of the FK model.

3.2.1 Order Parameter and Its Susceptibility

The symmetry group of an Hamiltonian consists on the set of transformations under which it is invariant.At low temperature, the state of the system tends to break some of the symmetries of the underlyingHamiltonian, being instead invariant under a smaller symmetry group. States that transform onto eachother by applying a transformation in the larger group have the same energy. If the system is in equilib-rium, those states have an equal probability of occurring and, over time, the system will spend an equalamount of time in each of them (ergodic hypothesis). At low temperature however, the system may getstuck in a subspace of states and the ergodicity is broken. The fact that some states are no longer

14

available to the system means that expectation values of thermodynamic observables do not have thesame value that they would otherwise have. One can take advantage of this to define quantities thatare zero in the symmetric phase and are different from zero in the symmetry broken phase (also calledordered phase). Such quantities are called order parameters and are useful to identify ordered phasesin a phase diagram. This discussion is only valid after taking the thermodynamic limit. If this is not donethen for a long enough time the system can escape the subspaces where it may be locked and so theergodicity is not broken.

For the Ising model with a ferromagnetic interaction (presented in Appendix A) the magnetizationwas a suitable order parameter. For the FK model, the order parameter depends on the type of orderpresent, which in turn depends on features such as the type of lattice and the filling. Here we give theorder parameter for the square lattice at half filling as an example. All the quantities that depend on theorder parameter can easily be generalized to describe order on the triangular lattice.

On the square lattice at half filling the staggered f-electron occupation can be used as an orderparameter

φ =1

V

∑r

ei(π,π)·r(2nf,r − 1) =1

V

∑r

(−1)rx+ry (2nf,r − 1) . (3.7)

In this case the ordered phase has a checkerboard pattern and for both ground states the order param-eter can take the values φ = ±1. We can also define the susceptibility of the order parameter in theusual way

χφ = βV(〈φ2〉 − 〈|φ|〉2

), (3.8)

which is related to the charge susceptibility at the momentum Q = (π, π). The Q vector is the nestingvector that connects states at the Fermi surface and makes the system susceptible to instabilities thatgap the spectrum all over the Fermi surface and order the system (just like the Peierls instability). Onecan also use it to define the Binder cumulants

U2 = 1− 〈φ2〉

3〈|φ|〉,

U4 = 1− 〈φ4〉

3〈φ2〉, (3.9)

for which the crossing point for different system sizes provides a very weakly biased estimate for thecritical temperature, as is shown more carefully in Appendix A.

3.2.2 Specific Heat

We now derive the specific heat for the FK model. Let us first write the partition function as

Z =∑{nf}

e−βF (nf ) =∑{nf}

P (nf ) , (3.10)

whereP (nf ) = eβµfNf det

[1 + e−βh(nf )

]= eβµfNf

∏ν

[1 + e−βεν

]. (3.11)

The internal energy per volume is given by

U =1

V〈H〉 = − 1

V

∂ logZ

∂β= − 1

V Z

∂Z

∂β= − 1

V Z

∑{nf}

∂P (nf )

∂β, (3.12)

15

where the derivative of P (nf ) is

∂P (nf )

∂β=

(µfNf −

∑ν

ενe−βεν

1 + e−βεν

)eµfNf det

[1 + e−βh(nf )

]=

(µfNf − Tr

[h(nf )

1 + eβh(nf )

])P (nf ) = −E(nf )P (nf ) . (3.13)

So, finally we can write the internal energy as

U =1

V Z

∑{nf}

E(nf )P (nf ) , (3.14)

where the energy of a configuration is defined as

E(nf ) = Tr

[h(nf )

1 + eβh(nf )

]− µfNf . (3.15)

The specific heat is defined as

CV =β2

V

∂2 logZ

∂β2=β2

V

∂

∂β

− 1

Z

∑{nf}

E(nf )P (nf )

= − β2

V Z

∑{nf}

[∂E(nf )

∂βP (nf ) + E(nf )

∂P (nf )

∂β

]+

β2

V Z2

∂Z

∂β

∑{nf}

E(nf )P (nf ) . (3.16)

Now, all that is left is the computation of the derivative of the energy of a configuration with respect to β:

∂E(nf )

∂β= −

∑ν

ε2ν(1 + eβεν )2

eβεν = −∑ν

ε2ν1 + e2βεν + 2eβεν

eβεν

= −∑ν

ε2νeβεν + e−βεν + 2

= −1

2

∑ν

ε2ν [1 + cosh (βεν)]−1

= −1

2Tr[h2(nf )[1 + cosh (βh(nf ))]−1

]= −δ2E(nf ) . (3.17)

Then, we finally have

CV =β2

V

1

Z

∑{nf}

P (nf )[E2(nf ) + δ2E(nf )

]−

1

Z

∑{nf}

E(nf )P (nf )

2

=β2

V

[〈E2 + δ2E〉 − 〈E〉2

]. (3.18)

3.2.3 Other Quantities

A FSS analysis can be used in order to estimate critical exponents and critical temperatures (see Ap-pendix A). For example, the order parameter susceptibility χφ can be used to extract the exponent γ.On the other hand, the derivatives of the logarithm of the order parameter and the derivatives of theBinder cumulants can be used to extract the exponent ν. Expressions for these quantities, provided inAppendix A for the Ising model, are given here for the FK model.

Here we provide d log〈|φ|〉dβ as an example. A similar procedure can be used for the remaining quanti-

16

ties. We start with the expectation value of the order parameter

〈|φ|〉 =1

Z

∑{nf}

|φ|P (nf ) . (3.19)

Varying this quantity with respect to β yields

d log〈|φ|〉dβ

=1

1Z

∑{nf} |φ|P (nf )

1

Z

∑{nf}

|φ|∂P (nf )

∂β− 1

Z2

∂Z

∂β

∑{nf}

|φ|P (nf )

=

11Z

∑{nf} |φ|P (nf )

− 1

Z

∑{nf}

|φ|E(nf )P (nf ) +1

Z2

∑{nf}

|φ|P (nf )

∑{nf}

E(nf )P (nf )

= 〈E〉 − 〈|φ|E〉

〈|φ|〉. (3.20)

Similarly,

d log〈φ2〉dβ

= 〈E〉 − 〈φ2E〉〈φ2〉

. (3.21)

The derivatives of the Binder cumulants with respect to β are given by

dU2(β)

dβ=

1

3〈|φ|〉2

[〈φ2〉〈E〉 − 2

〈φ2〉〈|φ|E〉〈|φ|〉

+ 〈φ2E〉]

= (1− U2)

[〈E〉 − 2

〈|φ|E〉〈|φ|〉

+〈φ2E〉〈φ2〉

],

dU4(β)

dβ= (1− U4)

[〈E〉 − 2

〈φ2E〉〈φ2〉

+〈φ4E〉〈φ4〉

]. (3.22)

3.3 c-electron’s Properties

Up until now we have not addressed quantities that are specific to the c-electrons. However these mobileelectrons are the ones determining the transport properties of the system. By looking at the spectrumof c-electrons one can make assertions about whether or not the system behaves as an insulator or aconductor, for example by noticing if the density of states has a gap around the Fermi level or not.

As stated in the previous chapter, the c-electrons interaction with the f-electrons can make a systembehave as insulator, even when simple band theory would predict it to be a conductor. There is aninterplay between the kinetic part of the Hamiltonian and the Coulomb repulsion that leads to resultsthat would not be possible with just one of those terms [16]. For large U it is intuitive that there shouldbe a gap in the spectrum of the c-electrons. The repulsion is so strong that each species of electronwill stay away from each other and there is basically an electron of either type at each lattice point. Thepresence of the gap prevents c-electrons from being easily added to the system, which would come atgreat energy cost due to the large U .

Another important thing to characterize is if the states available to the c-electrons are localized ornot, specifically the ones near the Fermi surface. If the f-electrons are disordered then the c-electronsbasically see a random potential across the lattice from which they scatter. The scattered waves interferein such a way that the resulting c-electron wave functions are localized around lattice sites. This is whatis called Anderson localization, and even though there might not be a gap at the Fermi surface, sincethe states are localized, they cannot carry any current and the system behaves as an insulator. To inferthe presence of such a phenomenon one can use a quantity called the inverse participation ratio. Both

17

the density of states and the inverse participation ratio of the c-electron wave functions and their MCaverages are defined in the following.

3.3.1 Density of States

The density of states for a particular configuration of the f-electrons is defined as

DOS(nf , ω) =∑ν

δ(ω − εν(nf )) . (3.23)

We can make a thermal average over the configurations of the f-electrons to get an expected value forthis quantity

DOS(ω) =1

Z

∑{nf}

P (nf )DOS(nf , ω) . (3.24)

This means that we can compute it through the normal Monte Carlo method. We just need to keep theeigenvalues of the configurations sampled. We cannot, however, use (3.23) directly. Instead we need toreplace the Dirac delta functions with Lorentzians with some arbitrary width

fν(nf , ω) =1

π

γ

[ω − εν(nf )]2 + γ2, (3.25)

where γ is a width that decreases when the system size increases, so that in the thermodynamic limitwe recover the Dirac delta functions.

There is yet another way to compute this quantity. By making a histogram over some energy range,one can divide it in bins and then count the number of times the eigenvalues for the configurationssampled fall in each bin. Then one divides that number by the sample size and the volume. This methodprovides an advantage over the previous one since it does not require for one to set an arbitrary widthfor the Lorentzians. This would be specially problematic when computing DOS(ω = 0), which basicallyis what is used to determine if there is a gap or not and which could, due to a bad choice of the width,give faulty values. For this reason, the second method is the one used.

3.3.2 Inverse Participation Ratio

The inverse participation ratio of a single particle state |ψ〉 is given by

IPR|ψ〉 =∑r

| 〈r|ψ〉 |4 . (3.26)

This is a quantity that provides insight into the degree of localization of a normalized state |ψ〉. This canbe easily seen for limiting cases: if the state is completely localized to a single position then only oneof the projections 〈r|ψ〉 is going to be different than zero and we get IPR|ψ〉 = 1; if, on the other hand,the state is equally delocalized over all possible positions, like a free particle in some momentum state,then | 〈r|ψ〉 |2 = 1

V , in which case IPR|ψ〉 = 1V . A simple generalization of these limiting cases shows

that if the IPR scales with V , the state is delocalized and if it does not scale at all it is localized. Notethat the IPR is allowed to scale with any power of the volume between −1 and 0, which corresponds tointermediary degrees of localization, but this in practice only happens for very specific systems.

For each f-electron configuration one can introduce the IPR for a given energy as:

IPR(nf , ω) =

∑ν IPR|ν〉δ(ω − εν(nf ))∑

ν δ(ω − εν(nf )). (3.27)

18

Once again we can make a thermal average:

IPR(ω) =1

Z

∑{nf}

P (nf )IPR(nf , ω) . (3.28)

It is also possible to use the previous histogram to compute the IPR. Provided one already has, foreach bin, the number of times an eigenvalue fell inside, one can then take the corresponding IPR|ψ〉’sand average them inside each bin. This provides with an estimate for IPR(ωi), where ωi labels therespective bin.

Both the IPR and the DOS were used in Ref. [27] to detect insulator phases of the FK model on thesquare lattice at half filling. Their consideration was crucial to identify a Mott and Anderson insulatingphases, as well as the respective phase transition lines.

3.4 Tools to Locate Candidates for Liquid Phases

The liquid phases, among other things are characterized by not breaking any symmetry at low tempera-ture. In this section we mention two methods that can be used to look for ordered phases. If they fail thatdoes not necessarily mean that we have a liquid phase, but at least we can propose it as a candidate.Those candidates can then later be studied using other tools that may allow to make some strongerstatements.

3.4.1 Charge Susceptibility

The charge susceptibility is defined as:

χf (q) =1

V

∑r,r′

ei(r−r′)q〈nf,rnf,r′〉 . (3.29)

This quantity is found to have peaks for the nesting vector. For this vector this is related to quantitythat we defined as the susceptibility of the order parameter. If one does not know for which momentumthe system order, one can compute this quantity numerically for a window of the momenta to see if thereis a peak that signals some CDW phase.

3.4.2 Principal Component Analysis (PCA)

Principal Component Analysis (PCA) is a method that applies an orthogonal transformation to go froma data set of observations with possibly correlated variables to a new set with variables (called principalcomponents) that are linearly uncorrelated [33]. After the transformation the first principal componenthas the largest possible variance, the second has the largest possible variance with the constraint ofbeing orthogonal to the first and so on. In the end the principal components form an orthogonal basis.

This method can be used for dimensional reduction when visualizing multivariate data. By lookingonly at the first few components, one sees the data in the directions where it varies the most, which arethe most relevant when it comes to discriminating data. This is used to try to find patterns in the data.

In practice PCA works as follows: One starts with a data sample of size N from a multivariate randomvariable x = (x1, . . . , xi, . . . , xM ), where M is the number of variables that compose it; The mean ofeach variable is subtracted from the data (center the data) x(k)

i = x(k)i − µi, where i = 1, . . . ,M labels

the variable, k = 1, . . . , N labels the observation and µi is the mean value of the variable xi; Then thecovariance matrix of the data is computed along with its eigenvalues and eigenvectors (normalized); Theeigenvalues are basically the variances of the corresponding principal components. The eigenvectors

19

can now be used to transform the centered data yi = x ·vi, where vi is the eigenvector that correspondsto the i’th principal component and yi is the projection of the data along its direction.

In Ref. [34] PCA is applied for the 2D ferromagnetic Ising model to distinguish configurations fromthe ordered and disordered phases. In this context M = V is the number of lattice sites, xi are the spinvariables which can be ±1 and N is the number of configurations sampled via Monte Carlo for severaldifferent temperatures. After the method is applied the configurations are shown projected in the spacespanned by the first two principal components y1 and y2. It was observed that the high temperatureconfigurations formed a cluster around the origin, whereas the low temperature configurations formedtwo clusters for symmetrical and finite values in y1 axis. This meant that the method was able to identifythe two ordered phase configurations (every spin up or down) from just looking at configurations fromdifferent temperatures and without any explicit information about the Hamiltonian or the lattice geometry.Not only that, but y1 turned out to be the magnetization, so PCA was also able of finding the orderparameter.

This method shall be used below in the context of the FK model on the triangular lattice to providehints at the presence of possible ordered phases.

20

Chapter 4

Results

This chapter is devoted to the presentation and discussion of Monte Carlo simulations of the FK modelon a triangular lattice for two different filling fractions, corresponding to a geometrical non-frustrated andto a frustrated situation.

The non-frustrated case is obtained by setting the fillings: ρf = 1/3 and ρc = 2/3 by an appropriatechoice of the chemical potentials computed by Newton’s method. In the reminder of this thesis, this setupis called the 1/3 filling scenario. The results obtained for this case, for large and moderate values of Tand U , are compatible with the ones obtained for the square lattice at half-filling, with some importantdifferences explained below.

The geometrically frustrated case is obtained setting both species of electrons to half-filling: ρf = 1/2

and ρc = 1/2. We call this the half-filling scenario. At low temperatures, the results are expectedto fundamentally differ from those of the square lattice and the likelihood of finding liquid phases isenhanced.

It is important to understand geometrical frustration in the context of the FK model. In general,frustration happens when there are competing constraints that cannot be simultaneously satisfied. Away to understand it is through equation (2.23), which represents an effective model for the f-electronsobtained via an expansion in U−1 for the ground state energy. Here, the c-electrons have been integratedout, yielding an effective interaction between the f-electrons that, for finite U is long ranged. Thus, theeffective spin model, besides Ising-like nearest neighbours interactions, also possesses interactionswith higher order neighbors, as well as higher order terms in the field, such as plaquette terms. Forincreasing values of U longer range interactions are suppressed and the leading terms correspond toan antiferromagnetic Ising model on the triangular lattice with a non-vanishing magnetic field term thatfixes the overall magnetisation (i.e. the filling, in the particle picture).

For 1/3 filling, a ground state configuration of the Ising term in eq.(2.23) can be obtained by keepingthe f-electrons as far away from each other as possible. This leads to the full occupancy of one of the 3sub-lattices that compose the triangular lattice, as shown in Fig.4.3. In these low energy configurationsevery triangular plaquette has exactly one f-electron, which means that two of the bonds lower the overallenergy and one raises it. Note that, if there were triangles with two f-electrons instead, there would stillbe two bonds that lower the energy and one that raises it, but in addition completely empty triangleswould also exist with three bonds that raise the energy. This simple argument for 1/3 filling leads tothe conclusion that the configuration presented, along with two additional ones obtained by translation,minimize the energy. Note that only the Ising term was considered so this alignment is only valid forlarge U . For the half-filled square lattice the large-U ordered phase has been proven to be preservedfor lower values of U as well. For the triangular lattice no such prove exists so, in principle, other phasescould exist for lower values of U .

21

Subdominant terms in the Hamiltonian tend to take a more noticeable effect when the Ising interactionleads to a macroscopic degeneracy of the ground state. In this case higher 1/U terms may contribute tolift the degeneracy of the ground-state manifold. This scenario arises for half-filling, where the numberof ground states of the Ising term increases proportionally to the volume of the system. Here, the bestone can do to minimize the Ising term, is to have half the triangles with two f-electrons and the other halfwith just one, being that both those types cost the same energy. Configuration made of these buildingblocks are indistinguishable by the Ising term. This huge ground-state degeneracy prevents the onset oforder and leads to a liquid-like state. It is in this situation that the sub-leading terms can play a crucialrole by choosing among those configurations. At the very least, they will partially lift the degeneracy ofthe macroscopical degenerate ground-state manifold. But, it is also possible that they induce orderedstates or more exotic phases.

The results presented below were obtained by simulating the model with the Metropolis-Hastingalgorithm with sequential updates, while using periodic boundary conditions. A review of the algorithmis presented in Appendix A where it is benchmarked for the Ising model on a square lattice. The hoppingstrength is set to t = 1 and used as a unit of energy and the interaction strength U is always taken to bepositive.

4.1 1/3 Filling

The main result of this section is presented in Fig.4.1, that depics the phase diagram for the FK modelat 1/3 filling in the T − U plane.

Figure 4.1: Phase diagram for the Falicov-Kimball model on the triangular lattice with 1/3 filling in theT − U plane, composed of five different regions: a charge density wave (CDW) ordered phase at lowtemperature and U ≥ 7. For high temperature there is a Mott insulator (MI) for large values of U ,an Anderson insulator (AI) for intermediate ones and a metallic-like (WL) phase for low U . For lowtemperature and U < 7 the results are still inconclusive.

With the exception of the a region low temperature and U < 7, where our results are inconclusive,this phase diagram is very similar to the one obtained in Ref. [27] for the square lattice at half-filling,which as was already said was to be expected.

For U ≥ 7 and low temperature there is a charge density wave (CDW) phase. This ordered phasecorresponds to the configurations like the ones shown in Fig.4.3, the other two low energy states beingobtained by translation. Similarly to the checkerboard phase for the square lattice at half-filling, this

22

ordered phase results from a perfect nesting of the Fermi surface that leaves the system susceptible toinstabilities that order both species of electron with a certain period.

For high temperature there are three disordered phases that behave differently with respect to theproperties of the c-electrons. For low U there is a weakly localized phase with metallic-like behavior andfor large U a Mott insulator is observed. For intermediate values there is an Anderson insulator. Aswas stated in the previous chapter the density of states (DOS) and the inverse participation ratio (IPR)were used to identify these phases. The Mott insulator is identified by a gap in the DOS around theFermi surface. The weakly localized phase is identified by an IPR that scales with the volume for statesaround the Fermi surface, indicating they are delocalized and that the system is not an insulator. For theAnderson phase, even though there is no gap around the Fermi surface, those states are localized andtherefore the system behaves as an insulator.

For low temperature and U < 7 results are still inconclusive. Some hypothesis for this fact andproposals for further investigation of this region are presented in the conclusion section.

To complement this picture we also show the specific heat as a function of the temperature for U = 7

and different system sizes in Fig.4.2a.

(a) (b)

Figure 4.2: Specific heat as a function of the temperature for (a) U = 7 and (b) U = 20 for differentsystem sizes.

The specific heat can be related to variations in the system’s entropy CV = T ∂S∂T . A local maximum

of the CV means that suddenly a large number of states have become available to the system. For lowtemperature there is a peak that has a scaling with the volume and is related to the phase transitioninto the ordered phase. The second peak does not have a scaling with the volume and we associate itwith a crossover from a regime where the system only has access to the subspace of states for whichthe c-electrons and f-electrons are at different lattice sites to a regime where the energy is such that thesystem has access to doubly occupied states.

As was proved in Chapter 2, for U > 2zt there is always a gap in the c-electron’s spectrum so long asthere is at least a lattice site with a f-electron and a lattice site that is vacant. The number of c-electronstates from the low energy band is equal to V −Nf and the number of states in the upper band is Nf .This means that for U > 2zt there is always a Mott insulator and that the c-electrons and f-electronsalways occupy different sites (as was also shown). However for very high temperature the c-electronsstart to be able to occupy states from the upper band and the system starts to have states with bothspecies of electron on the same lattice site. We assign this crossover to the high temperature bump ofthe specific heat. A detailed study of the double occupancy is still needed to confirm our hypothesis: ifthis picture holds, for all U > 2zt (in this case U > 12) there should be a temperature of the order of

23

the gap for which the system starts to have doubly occupied states. In Fig.4.2b we show the specificheat as a function of temperature for U = 20 and different system sizes. This value of U is large enoughto open of a gap for all temperatures. The bump is shifted to the right for larger values of U as shouldbe expected, since for a larger U the gap is wider and the temperatures need to be higher for thermalactivation to occur.

For smaller values of U for which a gap might not be present at high temperatures this picture isslightly changed. For example, for U = 7, while in the Mott insulator phase, the system is not orderedbut it is restricted to states where the c-electrons occupy lattice sites without f-electrons. After going intothe Anderson phase the gap vanishes and the system is allowed to be in states where the c-electronsand the f-electrons occupy the same lattice sites. The temperature for which the gap vanishes andthe system goes from the Mott to the Anderson phase in Fig.4.1 is close to the temperature where thesecond bump starts in Fig.4.2a. Therefore in this phase the high temperature maximum in the specificheat coincided with the Anderson-Mott insulator transition. We note that this transitions is a propertyof the electrons at the Fermi energy only and therefore is not a thermodynamic transition. That is thereason why no divergence of the specific heat is observed.

Note that in Fig.4.2a the scaling of the CV with the volume is not perfect. There are some largersystems with a smaller peak than smaller systems. This is due to the fact that there are lattice sizes forwhich the period of the ordered phase is not commensurable with. Only for L’s that are multiples of 3

do we have an ordered phase without defects. These non-commensurable system sizes were includedfor comparison reasons with the half-filling case, because there we do not know what is the orderedphase (or if there is even one) and therefore we do not know the correct system sizes for which to runsimulations. There are also some features around T = 0.1 that shall be discussed later in the half-fillingsection.

In Ref. [27] it is proposed that in the square lattice at half-filling the weakly localized phase is just afinite size effect and that in the thermodynamic limit the system should be an Anderson insulator instead.That claim is based on the fact that for very high temperature all the f-electron configurations should beequally probable and one is left with a random disorder model where Anderson localization is to beexpected. For low U , as the temperature is lowered this phase should still be present and therefore themetallic-like phase should be a finite size artifact. This argument should still be valid for the presentphase diagram.

In the next sections we discuss one-by-one how the different phases present in the phase diagramwere identified and how the transition lines were determined.

4.1.1 Chemical Potentials

As presented in Chapter 2, the FK model for bipartite lattices possesses a point (µc, µf ) = (U/2, U/2)

for which the model is particle-hole invariant and for which the filings are fixed to ρc = ρf = 1/2. Forthe triangular lattice there is no such symmetry points and as a consequence of that we had to devise amethod to fix the desired filling fractions for both species of electrons. The way this problem was solvedwas by using a Newton-like method for solving a system of non-linear equations. For each iterationof the method, the values of the chemical potentials for which the system has the desired fillings areapproached. This method is explained in detail in Appendix B.

The chemical potentials had to be determined for all the combinations of U , T and L that we runour simulations for. Luckily, chemical potentials determined for small system sizes, in certain conditions,could be used for larger systems and still provide the right fillings. Because of this we applied the methodfor the smallest lattice size possible that still provided chemical potentials that fixed the correct fillingsfor larger system sizes. For large U one can compute chemical potentials for a system of 6× 6 and they

24

remain valid for all larger system sizes used. As U is lowered one has to start using increasingly largersystems when computing the chemical potentials.

When we had chemical potentials for a certain U and T we could use them as the initial guess ofthe algorithm to determine the chemical potentials for slightly different values of the parameters. Thishowever at some point would require us to make an initial guess without any idea of what the valuescould be. In Ref. [35] the ground state energy is expanded in a power series for U−1. This expansion isthen used to determine for T → 0 a phase diagram in the µf −µc space which detailed different possibleground states with different fillings. For a certain window of the chemical potentials there was the 1/3

filling ground state present in Fig.4.3. Values of the chemical potentials inside that window were usedas the very first initial guesses for the Newton method.

4.1.2 Z3 Order Parameter

Figure 4.3: Ground state configuration of the f-electrons at 1/3 filling for the CDW phase.

The CDW phase presented in the phase diagram of Fig.4.1 has the ground state configuration shownin Fig.4.3, which corresponds to the f-electrons occupying one of the three sublattices that compose thetriangular lattice.

The ground state configurations persist for finite temperatures until a critical value is reached thatdestroys them. In order to determine the critical temperature of this order-disorder phase transition onehas to apply a finite size scaling (FSS) method. Such method, however, requires an order parameter.An appropriate order parameter for this ordered phase is given by

φ =3

V

∑r

ei2π3 (rx−ry)nf,r , (4.1)

where r is the lattice label, with rx being its x and ry its y coordinate on the lattice, such that r = ryL+rx.As was stated, there are 3 possible configurations for the ordered phase. There is the one in Fig.4.3and the others are obtained by shifting every position by one and two steps to the side. For each ofthem, φ has a different value φ = 1, ei

2π3 , ei

4π3 . This is easy to verify: the exponent in (4.1) can only be

θ = 0, i 2π3 , i

4π3 + 2πin, with n being an integer; a rotation by 2πin represents the same complex number.

So depending on which ordered configuration we have, only one of this 3 complex numbers is goingto be summed. So in the end we get the already stated results for φ. If however we have a perfectlydisordered configuration then we will have in the sum an equal amount of these 3 complex numbers andso in the end we get φ = 0. Therefore, 〈|φ|〉 = 0 corresponds to a disordered phase and 〈|φ|〉 = 1 to anordered one. We can thus conclude that this quantity is a good order parameter and we can use it todefine the susceptibility χφ = βV

(〈|φ|2〉 − 〈|φ|〉2

), the Binder cumulant U4 = 1−〈|φ|4〉/3〈|φ|2〉 and so on

for the remaining quantities mentioned in the previous chapter. This quantities help us to characterizethe order-disorder transition in the next section.

25

4.1.3 Order-disorder Transition

The transition line between the ordered phase and the disordered one was obtained by determiningthe crossing points of the Binder cumulants for different lattice sizes. We can see some of them inFig.4.4. This results were obtained with the Metropolis-Hasting algorithm with sequential updates. Aftera complete sweep over the lattice is done, the measurements of E and |φ| are performed. This wasrepeated 100000 times with the first 10000 data points being discarded to minimize the risk of usingnon-equilibrated data. In Fig.4.4 we also present some the integrated correlation times (τ|φ|,int) for |φ|so that we are sure that we have enough uncorrelated data points. The effective sample size, i.e. thenumber of uncorrelated points is Neff = N

2τO,int. The values for the correlation time were at worst

around τ|φ|,int ≈ 20, which corresponds to Neff ≈ 2200. This is usually considered good enoughstatistics [36]. We also see that the 10000 points discarded initially are always significantly more than20τ , which empirically is considered more than adequate to ensure equilibration [36].

(a) U = 30 (b) U = 30

(c) U = 40 (d) U = 40

Figure 4.4: (a),(c) Binder cumulants as a function of the temperature for several system sizes L and twodifferent interaction strengths U . (b),(d) Integrated correlation time τ|φ|,int for several system sizes L andtwo different interaction strengths U .

Critical Exponents

For U = 30 we also ran longer simulations in order to obtain the critical exponents of the transition.Those were executed in a computer cluster that allowed us to run 40 Markov chains in parallel. Weacquired N = 1000000 points, with each Markov chain discarding 3500 configurations before storing25000 of them. Here we do an analysis very similar to the one present in Appendix A. A reader notfamiliar with Monte Carlo methods or with FSS is advised to read that appendix first, where a detailedreview of those methods is presented along with an example of their application to the Ising model.

In Fig.4.5a we present the susceptibility of the order parameter with respect to T for different system

26

sizes. Similar plots were obtained for the derivatives of the Binder cumulant and for the derivatives ofthe logarithm of the order parameter. Fig.4.5b presents the order parameter correlation time defined in(A.32).

(a) (b)

Figure 4.5: (a) Plot of χ(β) for different system sizes. (b) Integrated correlation time τ|φ|,int shown fordifferent system sizes.

The same scaling analysis as in Appendix A is performed here for the FK model. Fig.4.6 shows thefits of the scaling equations in (A.45) to the maxima of the quantities shown in the legend of the graph,using a log− log scale. These fits are used to determine the critical exponents ν and γ, with the resultspresented in Tab.4.1.

Figure 4.6: Plot of the maxima of d log〈|φ|p〉dβ with p = 1, 2 and dU2

dβ . The lines are the fits of equations(A.45) (with a log− log scale) to the maxima and are used to extract the critical exponents ν. This scalewas chosen so that the fits could be linear. The same thing for χφ(β) is also done here in order to extractthe critical exponent γ.

d log〈|φ|〉/dβ d log〈|φ|2〉/dβ dU2/dβ weighted average χ

ν 0.8003(114) 0.7962(128) 0.8347(221) 0.8031(114) γ 1.4748(329)

Table 4.1: Values of ν determined using different quantities. The average value weighted by the errorsis also provided. The value of γ determined using the maxima of χ is shown as well.

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We also determine the critical temperature using the same methods as in Appendix A. We can seethe results in Fig.4.7 and in Tab.4.2.

(a) (b)

Figure 4.7: (a) Plot of the Binder cumulant U4 for different system sizes. The inset shows a close up ofthe crossing point at the critical temperature. (b) Plot of the pseudo-transition points βmax for the samequantities as in Fig.4.6. Equation A.46 is fitted to the data, where we take ν to be the value estimatedbefore.

χ d log〈|φ|〉/dβ d log〈|φ|2〉/dβ dU2/dβ weighted average Binderβc 0.01778(3) 0.01756(2) 0.01761(7) 0.01770(4) 0.01766(2) 0.0175

Table 4.2: Values of βc determined using different quantities. The average value weighted by the errorsis also provided along with the value determined via the crossing of the Binder cumulants.

The values for the critical temperature obtained for this long run do not deviate too much from theones obtained before for the shorter runs, which validates the critical temperatures estimated in Fig.4.1even though they use less statistics.

As the ground states are related by a Z3 symmetry, similar to what happens in the Potts model forq = 3, we expected that the critical exponents obtained for this transition were compatible with the onesfrom the Potts model for 2D. The values for the critical exponents present in the literature are ν = 5/6

and γ = 13/9 [37]. The results obtained are compatible with this exact values, being that the exact valueof γ is contained inside the errorbars and the value of ν inside twice the errorbars. In the future largersystem sizes should be used to strengthen this result.

4.1.4 Mott and Anderson Insulator Phases and Respective Transition Lines

Now we would like to analyze the disordered phases. For that we used the density of states and theinverse participation ratio defined in Chapter 3. In Fig.4.8 we plot the DOS(ω) for different U and L. Wesee that for low U there is a finite density of states at the Fermi level (ω = 0). As U increases the DOSstarts to get depleted at a region around ω = 0, and by U = 6 an energy gap has formed. Therefore thesystem in this phase behaves as an insulator.

We now look at the IPR, which we plot in Fig.4.8 for different U and L. This quantity scales withthe volume if the states of the c-electrons are completely delocalized and remains constant if they arecompletely localized. For low U we see that there is a scaling with the volume, which combined withthe fact that there is no gap in the DOS leads us to conclude that we have a metal-like phase. Forintermediate U there is no gap yet in the DOS around ω = 0 but the IPR is constant there. This means

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that even though there is no gap to keep more electrons from going into the system, the electron statesare localized and that the system behaves like an Anderson insulator.

(a) DOS for U = 1 (b) IPR for U = 1

(c) DOS for U = 3 (d) IPR for U = 3

(e) DOS for U = 6 (f) IPR for U = 6

Figure 4.8: DOS(ω) and IPR(ω) computed via a histogram technique. All of the graphs are for T = 0.03.

In Fig.4.9a we plot DOS(ω = 0) against U for different temperatures. This allows us to know for eachtemperature where the gap opens. This is what lets us know in the phase diagram in Fig.4.1 wherethe transition line into the Mott insulator phase occurs. The DOS(ω = 0) was computed by opening awindow around ω = 0 and counting the electron states inside.

In Fig.4.9b we also plot IPR(ω = 0) against U , for several T and L. We see that for low U , theIPR(ω = 0) curves for the same T are separated according to L. As the U increases this curves start tocome closer until they fall onto each other. The point where this happens is the point where we go fromthe metal-like phase to the Anderson insulator phase. This is how that transition line was estimated.

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(a) (b)

Figure 4.9: (a) DOS(ω = 0). (b) IPR(ω = 0). These quantities are used to determine the transition linefrom Anderson to Mott insulator and from weakly localized to Anderson insulator respectively.

4.1.5 Inconclusive Region

For low temperature and U < 7 there is a region in the phase diagram of Fig.4.1 for which the resultsare not yet fully understood. Following the established tendency for larger U , it was expected that thesame ordered phase would also be present here for low enough temperature. In addition to this, for thesquare lattice at half filling the ordered phase persisted for all U and so the same was expected for thepresent non-frustrated scenario.

As it was already said, in Ref. [35] an expansion in U−1 for the zero temperature energy was obtained,which allowed to determine a phase diagram in the chemical potentials space, detailing the differentpossible ground state configurations with different fillings. For the terms of the expansion considered inthat paper, there was a window of chemical potentials for which the ground state is the one in Fig.4.3.For U smaller than a certain value that window closes. This does not necessarily mean that there isdefinitely no ground state for low U and 1/3 filling, it just means that there is not one for the number ofexpansion terms considered in the paper. If further terms are added, then further windows are addedas well to the phase diagram, but they will be increasingly smaller. What this means is that it may bepossible that there is a window of chemical potentials for U < 7 for which there is a ground state at 1/3

filling. But that window may be so small that it requires a severe degree of fine-tuning for one to be ableto determine the right chemical potentials using the Newton method. If this picture is indeed true, thenthe actual phase diagram should have an ordered phase where ours has the unknown region, but dueto numerical constraints it may be very hard to get it.

The configurations obtained in this region appear to correspond to the ordered phase for ρ = 1/4,which is presented in Fig.4.10, but with some defects that ensure the filing to be 1/3. An hypothesesfor this is that the filling functions ρf/c(µc, µf , U, T, L) have stationary points in regions of the parameterspace where there are ordered phases. This means that in those regions the filling is the one associatedwith that ordered phase and that the filling function is not very sensible to small perturbations of theparameters. It might be that for U < 7 there is no stationary point that can gives 1/3 filling and theclosest one is the one for 1/4 filling, which could explain why that is almost the configuration that weobserve.

It is also possible that there is a different ordered phase in this region but its period is not commen-surable with the system sizes tested for, which lead us to look at configurations that are in fact orderedbut that have defects.

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Figure 4.10: Ground state configuration of the f-electrons at 1/4 filling.

4.1.6 Charge Susceptibility and PCA

As was said in Chapter 3 the PCA method is capable of determining the linear combination of the f-electron occupations nf,i that is better capable of distinguishing between configurations sampled for arange of temperatures that goes from the ordered phase to the disordered one. In other words, it is amethod that is capable of determining the most efficient order parameter that corresponds to a linearcombination of the degrees of freedom, which in this case are the f-electrons occupations.

In Chapter 3 it was also said that there is a quantity called charge susceptibility χf (q) that in momen-tum space has peaks for the nesting vectors of CDW phases.

Both this methods can be used to verify the presence of ordered phases. We use them in this sectionbecause we want to see how different are the results when we go from the non-frustrated case to thefrustrated scenario. If the methods fail to detect an ordered phase that does not necessarily mean thatthere is a disordered one, but they can at least help to shed some light to the problem and may be usedto propose possible candidates of liquid phases, which can be further investigated in the future.

In Fig.4.11a we show the result of the PCA method for L = 12. There are three surrounding clustersfor configurations that belong to low temperatures. These are associated with the three possible config-urations for the ordered phase. The central cluster is composed from high temperature configurations,which indicates it corresponds to the disordered phase. In Fig.4.11b we also show the result for L = 10,for which it more or less looks like that structure for the previous case it is trying to from. We did this tosee what would happen if a system size that is not commensurable with the period of the ordered phasewas used instead. This provides important insight for the half-filling case. For that scenario the orderedphase, if it exists, is unknown and therefore it is important to know if this method may help us to locate iteven if we use system sizes not commensurable with its period.

(a) (b)

Figure 4.11: Projection of the configurations in the subspace of the first two principal components y1 andy2. The color represents the temperature of a given configuration. The temperature range used includesthe critical temperature of the phase transition. (a) L = 12 and (b) L = 10 both for U = 7.

31

In Fig.4.12a we plot the charge susceptibility for T = 0.014, U = 30 and L = 12. This showstwo peaks which correspond to the nesting vector for which the CDW phase orders Q = 2π

3 (1,−1). InFig.4.12b we also plot the same thing for L = 10 which once again is not commensurable with the periodof the order. The image shows peaks around the same moment, but it has an additional structure. Thismight hint at the fact that there is an ordered phase but the system size is not the right one to see it.This quantity may be useful in the half-filling regime to find an unknown ordered phase.

(a) (b)

Figure 4.12: Plot of the projection of the configurations for different temperatures in the subspace of thetwo principal components determined by the PCA method for (a) L = 12 and (b) L = 10.

4.2 Half-Filling

In this section the results for the half-filling scenario are presented. This case is frustrated and the lowtemperature picture is expected to be significantly different from the 1/3 filling case. In Fig.4.13 thephase diagram for the FK model at half-filling in the T − U plane is shown.

Figure 4.13: Phase diagram for the Falicov-Kimball model on the triangular lattice with half-filling in theT − U plane, composed of four different regions: For high temperature there is a Mott insulator (MI) forlarge values of U , an Anderson insulator (AI) for intermediate ones and a metallic-like (WL) phase forlow U . For low temperature it is hard to get a definite picture of the phases present.

For high temperature the results are very similar to the ones obtained for 1/3 filling in the triangularlattice and the square lattice at half filling. For high temperature and high U there is a Mott insulator, for

32

intermediate U there is an Anderson insulator and at low U there is the metallic-like phase. This similarityis to be expected and in fact we propose that this picture should be very weakly dependent on latticetypes or filling fractions. For high U there will always be a gap in the system and if the temperature is highenough for the system to be out of a possible ordered phase, then the only thing left for the system to beis a Mott insulator. When the gap closes for lower U and for high enough temperatures the c-electronswill always see a random potential generated by the f-electrons, which leads to Anderson localization.

For low temperature the results are generally not very conclusive. On one hand this is problematicbecause it does not allow us to make many solid conclusions. On the other hand, the very fact that thelow temperature regime for the half-filling case is hard to analyze is exactly what we set out to find. Thefact that the tools employed to characterize the previous case do not work so well here means that whatis going on is significantly different and more complex.

Just like before we try to complement the picture by looking at the specific heat. Since we do notknow what the ordered phase is or if there is even one, we cannot define an order parameter which wouldhelp with analyzing the transition. Because of this the only way to locate a possible phase transition orcrossover is via the specific heat.

In Fig.4.14 we show the specific heat for different values of U and different system sizes. For U = 5

there is the presence of three bumps in the specific heat. The one for large temperature was alreadyreported for 1/3 filling and corresponds to the fact that the system starts to have doubly occupied states.The other two possess some scaling with the volume which might signal some kind of phase transition.However since most likely the system sizes used are not commensurable with the period of the order (ifthere is one) the scaling is not perfect. They may also just correspond to crossovers and just mean thatthe distribution of the states the system occupies somehow changes after going through them. We notethat the bump for lowest temperature starts to get pushed for even lower temperatures as U increasesand eventually disappears. Besides this nothing very certain was found about it. However for the middlebump there is something that can be said.

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(a) U = 5 (b) U = 6

(c) U = 7 (d) U = 10

Figure 4.14: Specific heat as a function of temperature for different values of U and different systemsizes.

-150 -100 -50 00

2000

4000

6000

8000

ΣBsB

#

-150 -100 -50 0 50 100 1500

200

400

600

800

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1200

ΣPsp

#

0 20 40 60 80 100 120 1400

500

1000

1500

N f

#

L=12; U=10;

-50 0 500

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ΣBsB

#

0 20 40 60 800

1000

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#

-50 0 500

500

1000

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#

L=9; U=5;

T=0.015

T=0.3T=0.05

T=0.2T=0.015

Figure 4.15: Histograms for the number of f-electrons, the energy distribution of the Ising term and forthe triangular plaquettes term. These were computed using the configurations sampled via Monte Carlofor different values of U and T . The temperatures chosen were the ones before and after the bumps inspecific heat.

In Fig.4.15 we show histograms of the antiferromagnetic Ising contributions to the energy along withthe contribution from the triangular plaquettes. These are terms from the effective model expansionpresent in (2.23) and the histograms were computed with the configurations sampled via Monte Carlo.

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There are histograms for U = 9: for a temperature lower than the left bump; another for a temperaturebetween the first and the second and another between the second and the third. There is also his-tograms for U = 10 for a temperature before and after the second bump. What is observed is that thedistribution of the energy contributions from the Ising term changes significantly when crossing the sec-ond bump. This can be seen specially well for U = 10. It suggests that with the crossing some constraintis lifted that allows for the population of a greater variety of configurations, which explains the bump inthe specific heat. This conclusion also suggests that maybe the lowest temperature bump comes froma similar effect for a different term of the expansion, and since the succeeding terms depend on largerpowers of U−1, this would explain why the lowest bump eventually vanishes as U increases. Howeverthe histograms for the plaquettes did not allow to make any more conclusions supporting this line ofthought.

4.2.1 Mott and Anderson Insulator Phases and Respective Transition Lines

(a) DOS for U = 2 (b) IPR for U = 2

(c) DOS for U = 3 (d) IPR for U = 3

(e) DOS for U = 6 (f) IPR for U = 6

Figure 4.16: DOS(ω) and IPR(ω) computed via a histogram technique. All of the graphs are for T =0.03.

Just like before, we look at the DOS and the IPR to identify the different high temperature phases alongwith its transition lines. In Fig.4.16 we have the plots for the DOS(ω) and IPR(ω) for different U and L.Once again for low U the density of states around the Fermi surface is finite and the inverse participation

35

ration scales with the volume there. This leads to the conclusion that for the half-filling case for low U

and high temperature the metallic-like phase is still present. For high values of U there is an energy gapat the Fermi surface and therefore we still have a Mott insulator. For intermediate U the DOS is still finitebut the IPR tells us the c-electron states at the Fermi surface are localized, so the system behaves asan Anderson insulator.

Once again the DOS(ω = 0) and IPR(ω = 0) were used to determine the transition lines betweenthe phases mentioned. In Fig.4.17 we plot those quantities as a function of U for different temperaturesto determine where the transitions occur. When the gap opens for DOS(ω = 0), this signals the systemwent from the Anderson insulator to the Mott one. When the IPR(ω = 0) curves for the same temper-ature but different volume meet, then the IPR at the Fermi surface no longer scales with the volumewhich the indicates the system no longer behaves in a metallic-like fashion, but is now an Andersoninsulator.

(a) (b)

Figure 4.17: (a) DOS(ω = 0). (b) IPR(ω = 0). These quantities are used to determine the transitionline from Anderson to Mott insulator and from weakly localized to Anderson insulator respectively.

4.2.2 PCA

In this section we use the PCA method in hope that it can cast some light for the low temperaturesregimes. For U = 5 there were three bumps in the specific heat. We have plausible justifications forboth of the high temperature bumps but none for the lowest temperature one. We apply this methodwith the configurations that come from temperatures in a range that includes this first bump. If the bumpcorresponds to a phase transition, then the method should provide some insight into a possible orderedphase. In Fig.4.18 is shown the result of the application of the method.

Figure 4.18: Projection of the configurations in the subspace of the first two principal components y1

and y2 for U = 5. The color represents the temperature of a given configuration. The temperature rangeused includes the first bump in the specific heat.

36

We can see that the high temperature configurations form a ball in the middle and the low tem-perature ones are spread in non uniform way. Not much can be said except that the low temperatureconfigurations are not completely disordered. If they were, then we would just see a ball for configu-rations of all temperatures which would mean that no linear combination of the f-occupations would bepreferable in distinguishing configurations. In such scenario that would mean that there was no orderparameter (at least not one that is defined linearly with the degrees of freedom), which would hint thatprobably the bump corresponds to no phase transition. However what we see is that there seems tobe a restricted set of different types of low temperature configurations. By looking at the configurationsthemselves we were not able to spot anything special except that they tended to form stripes. What thismay mean is that there are "rules" for the possible stripes that the system can display. Even if this istrue, it also does not necessarily mean that this corresponds to a phase transition. There might not beany symmetry breaking and the only thing that happens is that the allowed states have to obey someset of rules that restricts its number (which would explain the bump in the specific heat).

37

38

Chapter 5

Conclusion

As stated in the very beginning, liquid phases are poorly understood phases of matter that possesssome very exotic phenomena. The question that this work set out to answer was whether or not asimple semi-classical model put under geometrical frustration is capable of displaying the properties ofsuch phases of matter. If so, then a new effective and systematic way of studying them would be open.

To answer this question the Falicov-Kimball model was chosen, since this is one of the simplestmodels for correlated electron systems and can be simulated efficiently via Monte Carlo. A program thatimplemented the simulations was developed along with many tools to extract physical observables andtools for numerical analysis.

Results were obtained for two different filling fractions, one frustrated and another not, so the resultscould be contrasted.

For 1/3 filling the results obtained, except for an inconclusive region, were compatible with the onesfor the square lattice at half-filling. There was an ordered phase for low temperature and for high tem-peratures there were weakly localized, Anderson insulator and Mott insulator phases. Inclusively theorder-disorder transition also belongs to the same universality class as a classical model: the Pottsmodel for q = 3, whereas for the square lattice it was the Ising model.

For half-filling, the high temperature picture was the same as before and we proposed that this pictureshould not depend too strongly on lattice types and filling fractions. For low temperature the resultswhere somewhat inconclusive. We were able to justify the presence of two of the three bumps observedfor the specific heat. The remaining one might signal a crossover or a transition to an interesting lowtemperature phase, as was suggested by the non-uniform structure obtained with the PCA. The fact thatthe methods used to analyze the 1/3 filling case failed to lead to more solid conclusions for this frustratedscenario is on itself interesting. The goal was to find regimes in the phase diagram that display unusualphenomena that is hard to characterize. Such regimes could be candidates for liquid phases, but furtherinvestigation is required to be able to make such strong claims.

Future work shall focus on studying the low temperature regimes obtained for the half-filing. Inparticular discovering the origin of the lowest temperature bump in the specific heat might provide someinsight into the nature of the regime present just before the bump. Given that one of the other bumpswas linked directly to the Ising term in the effective model, it would be interesting to simulate the classicalmodels that are obtained by truncating that expansion. It might be possible that by adding specific termsof the expansion one might be able to reproduce what was observed for the FK model. This may providean origin for the unknown bump.

It may also be of value to try and come up with methods to identify specifically liquid phases in thephase diagram, since the current approaches are more focused in looking at what liquid phases are notinstead of what they are.

39

40

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[36] A. D. Sokal, “Monte carlo methods in statistical mechanics: Foundations and new algorithms,” 1996.

[37] F. Y. Wu, “The potts model,” Rev. Mod. Phys., vol. 54, pp. 235–268, Jan 1982.

[38] N. Goldenfeld, Lectures on phase transitions and critical phenomena. Frontiers in Physics, 85,Westview Press, illustrated edition ed., 1992.

[39] A. R. a. H. F. R. S. A. W. e. Ralf Schneider, Amit Raj Sharma, Computational Many-Particle Physics.Lecture Notes in Physics 739, Springer-Verlag Berlin Heidelberg, 1 ed., 2008.

[40] R. Ren, C. J. O’Keeffe, and G. Orkoulas, “Sequential metropolis algorithms for fluid simulations,”International Journal of Thermophysics, vol. 28, pp. 520–535, Apr 2007.

[41] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state cal-culations by fast computing machines,” The Journal of Chemical Physics, vol. 21, no. 6, pp. 1087–1092, 1953.

[42] W. K. Hastings, “Monte carlo sampling methods using markov chains and their applications,”Biometrika, vol. 57, no. 1, pp. 97–109, 1970.

[43] R. Ren and G. Orkoulas, “Acceleration of markov chain monte carlo simulations through sequentialupdating,” The Journal of Chemical Physics, vol. 124, no. 6, p. 064109, 2006.

43

44

Appendix A

Monte Carlo Method

In this chapter we review the standard approach to Monte Carlo simulations. We use the Ising modelon a square lattice to do so. Due to its relative simplicity, there are a number of properties of this modelavailable, either exactly or with high numerical precision. Thus, this model is an ideal benchmark.

The main purpose of this section is for a reader who is not familiar with Monte Carlo Methods to geta basic idea of how it works and to provide a compilation of references. We do it in the hope that thewhole thesis will be self-contained.

In the following, we show in particular how to compute average values of thermodynamic quantities,error estimations, critical temperatures and critical exponents. The same methods were applied to theFK model, with some differences that are covered in Chapter 3.

In general a Monte Carlo Algorithm is an algorithm that makes use of random numbers to computeits result. The output can be incorrect within a certain probability which can be controlled by runing thealgorithm many times. In our particular case the Monte Carlo algorithm allows us to sample states froma sytem obeying the equilibrium distribution and compute average values of thermodynamic quantitiesand their associated statistical errors. Those errors can be lowered by generating more states and willin general be proportional to 1/

√N where N is the sample size. This information should serve as a

warning because this is a slow convergence rate when compared with other numerical methods. This isone of the reasons why one should only resort to Monte Carlo methods when nothing else is better [36].

Before geting into more details about algorithms we first give a review of some necessary conceptsof statistical physics.

We will use the Ising model as an example, but this exposition can easily be generalized to othermodels. The Ising model is described by the following Hamiltonian:

H = −J∑<ij>

σiσj , (A.1)

where the sum is over nearest neighbors. J is the coupling energy which can be positive or negativefavoring either ferromagnetic or anti-ferromagnetic alignment and σi is the spin at lattice site i, whichcan take the values ±1.

A.1 Statistical Mechanics

Statistical Mechanics of systems in equilibrium rests on the idea of the partition function, which con-tains the necessary information about the microscopic details of the system to compute thermodynamic

45

quantities. The partition function is given by

Z =∑{s}

e−βH(s) , (A.2)

where the sum is over all possible states of the system and β ≡ 1/kBT is the inverse of the temperature[1]. H is the Hamiltonian of the model used to describe the system and encodes all the interactionsbetween the various degrees of freedom present. For a specific state s of the system, H(s) is going togive us its energy.

The partition function serves as a normalization constant and allows one to know the probability thata system is in a given state s, also known as Boltzmann weight:

P (s) = e−βH(s)/Z . (A.3)

As such, one can use this probability density to compute the expected value of observables:

〈O〉 =∑{s}

Ose−βH(s)/Z , (A.4)

where Os is the value of the observable for state s.

If we could sample states directly from the Boltzmann distribution then we could approximate thisexpected values as the mean values over the sample

〈O〉 ≈ O =1

N

N∑i=1

Oi , (A.5)

where the sum is over the states sampled and N is the sample size. The problem is that we can’t do thisdirect sampling because doing so would require knowledge of the exact value of the partition function,and that is something we can only very rarely do. There is however an alternative based on the idea ofsetting up a stochastic process that automatically generates states obeying the Boltzmann distribution.The process used is what is called a Markov Chain and we talk about them in the next section. For nowwe keep laying some of the fundamentals about Statistical Physics that are required to understand thenumerical method used in this work.

An important concept in Statistical Mechanics and Condensed Matter Physics is that of phases ofmatter and equally important is the notion of a phase transitions. Materials have a very different behaviorand physical properties depending on a set of external parameters. For specific values of the parametersthat separate qualitative different phases there are sudden changes in those properties. An example ofthis is a material that at high temperature behaves as a paramagnet, that is, it has no net magnetizationwhen not in the presence of an external magnetic field. As the temperature is lowered, at some point,the magnetic moments align in a certain direction and we get a net magnetization.

There are two types of phase transition, the discontinuous (first order) transitions and the continuous(second order) ones. First order transitions are characterized by latent heat, that is, during the transitionthe system absorves or emits heat but its temperature remains constant [38]. At the transition thesystem is in a mixed phase where some portions are in the high temperature phase and others in thelow temperature one. Besides the discontinuity in the energy, other first derivatives of the free energymay also have discontinuities present.

Second order phase transitions on the other hand are driven by a divergent correlation lenght ξ asthe critical temperature Tc is aproached [38]. Inside an ordered phase the correlation function betweentwo degrees of freedom is typically given by G(r) ∼ e−r/ξ where r is the distance between the degrees

46

of freedom (translational invariance was assumed). The correlation length is the typical length for whichfluctuations of the degrees of freedom of the system are correlated with each other. When this quantitydiverges near the transition, what this means is that fluctuations are important on all length scaleswhich in turn implies that the degrees of freedom will have to have longer range correlations. At thetransition the correlation function no longer has an exponential decay and instead behaves as a powerlaw G(r) ∼ r−k, which has intrinsic scale. All of this leads to power law singularities in thermodynamicfunctions.

Near the transition the leading contribution to the correlation length is given by

ξ =

ξ+0 |t|−ν + . . . (T ≥ Tc)

ξ−0 t−ν + . . . (T ≤ Tc)

, (A.6)

where ν is a critical exponent, t = 1 − T/Tc and ξ±0 are the amplitudes on both sides of the transition.The leading terms for the singularities of some of the other quantities are given by

C = C0|t|−α + . . . ,

m = m0tβ + . . . ,

χ = χ0|t|−γ + . . . , (A.7)

where α, β and γ are critical exponents and C0, m0 and χ0 are the amplitudes near the transition fortheir respective functions [1].

Even though the correlation length diverges, when we are simulating a finite system this quantity isgoing to be limited by the linear size of the system ξ → L. This means that in the scaling relation in(A.7), t gets replaced by L−1/ν . This scaling relations, can be written more generally as

C(T, L) = Lα/νfC(x)[1 + . . . ] ,

m(T, L) = Lβ/νfm(x)[1 + . . . ] ,

χ(T, L) = Lγ/νfχ(x)[1 + . . . ] , (A.8)

where f(x) are scaling function of the scaling variable x = tL1/ν [39]. This means that if we plot forexample χ(T, L)/Lγ/ν against x, data for different L and T are going to fall over the same curve.

In the thermodynamic limit divergences of some thermodynamic quantities identify the critical tem-perature. In finite systems what happens is that those divergences are smeared and shifted. This canbe understood since near the transition temperature Tc, where the scaling relations are valid, a certainscaling function has a maximum for some value of the scaling variable xmax. Since x = tL1/ν and themaximum of f(x) is always going to be xmax then this implies the following relation [39]

Tmax = Tc

(1− xmaxL−1/ν

)+ . . . . (A.9)

This shows that the temperature Tmax for which the maximum of this quantities occurs is going to beshifted from its thermodynamic limit value, and gets increasingly closer as L is increased.

A.2 Markov Chain

A Markov Chain is a stocastic process that consists of sucessive transitions between different states.This process can be described by a transition matrix P , such that if i and j are states that belong to thestate space S, then Pij gives us the probability of going from state i to state j [40].

47

Line i of P contains the probabilities of a transition between i and all the other states. So for there toexist conservation of probability, we have to have∑

j

Pij = 1 . (A.10)

If we start with a certain probability distribution of states f(0) = (f1(0), f2(0), . . . ) we have the follow-ing after one transition:

fT (1) = fT (0) · P , (A.11)

where f is a column vectors and fT is its transpose.

After successive applications of the transition matrix, it is possible for some Markov chains to achievea stationary distribution π:

limn→∞

fT (n) = fT (0) · limn→∞

Pn = πT . (A.12)

If this limit exists, then such distribution is left invariant under further iterations of the process

πT · P = πT , (A.13)

or put another way, for each j ∈ S we have [36]∑i

πiPij = πj . (A.14)

A sufficient, but not necessary, condition for a Markov process to be stationary is:

πiPij = πjPji ∀i, j ∈ S . (A.15)

This is called detailed balance condition. We can see that this leads to (A.14) by summing the previousequation over i and then applying (A.10) on the right hand side. Therefore if we can verify that thiscondition is satisfied for a certain process, then we know that the Markov Chain eventually converges toa stationary distribution of states.

A.3 Metropolis-Hasting Algorithm

We present here the Metropolis-Hasting algorithm, proposed in [41] and then generalized in [42]. Thisalgorithm allows one to build a Markov chain that converges to a desired stationary distribution. Theidea behind the algorithm goes as follows:

We start with a state i and then propose a new state j with a certain probability qij . This new stateis then accepted with a probability αij . We can therefore write the transition matrix as:Pij = qijαij , ∀i 6= j

Pii = 1−∑j 6=i Pij

. (A.16)

We can then define the acceptance probability as:

αij = min

[1,qjiqij

πjπi

], (A.17)

which as we will see satisfies the detailed balance condition.

We can assume without loss of generality that qjiπj < qijπi. In this case αij =qjiqij

πjπi

and αji = 1.

48

We can then write:Pij = qijαij = qij

qjiqij

πjπi

= qjiπjπi

= qjiαjiπjπi

= Pjiπjπi

, (A.18)

which is equivalent to detailed balance.

For applications in statistical mechanics, the desired stationary distribution is the Boltzmann distribu-tion, where

πi = e−βEi/Z , (A.19)

with Ei being the energy of state i. We see then that the probability of accepting the new state is thenproportional to:

πjπi

= e−β(Ej−Ei) = e−β∆E . (A.20)

Notice that the partition function does not appear explicitly. This is what allows the method to work sinceone can only rarely compute explicitly the partition function.

Applying the algorithm then is very simple. One starts with a state i and proposes a new state j

which is accepted with probability αij . Then the method is repeated over and over. When convergenceis achieved the states are sampled with respect to the Boltzmann distribution.

For the Ising model the proposal of a new state is achieved by flipping a spin. The choice of thespin can be random or one can make sequencial sweeps across the whole lattice. In the former casedetailed ballance is always satisfied in all updates. For the sequencial update that does not happen, butit can be shown that the Markov chain still converges to the right distribution [43].

A.4 Computing Average Values of Observables

Now that we have a way of sampling states with the Boltzmann distribution, we are able to estimateexpectation values of physical observables:

O =1

N

N∑i=1

Oi , (A.21)

whereOi is the value of the observable for the sampled state i andO is the mean value of the observable.Notice that there is a difference between the mean value O and the expected value 〈O〉. The later is anumber which represents an exact result, while the former is an estimator of the expected value and isin fact a random variable. This variable will oscillate around 〈O〉 and through those fluctuations one canestimate a statistical error for O. Since for a single simulation, we only get one value for the randomvariable O, naively it would seem that one needs to run the simulation many times in order to estimateits variance σ2

O. However, as we will see, the variance of the estimator can be related to the variance of

the individual measurements σ2O, which means that the simulation only needs to be ran once.

We start by writing [39] :

σ2O

=⟨(O − 〈O〉

)2⟩= 〈O2〉 − 〈O〉2

=1

N2

N∑i,j=1

〈OiOj〉 −1

N2

N∑i,j=1

〈Oi〉〈Oj〉 . (A.22)

By separating diagonal and off-diagonal terms we get [39] :

σ2O

=1

N2

N∑i=1

(〈O2

i 〉 − 〈Oi〉2)

+1

N2

N∑i 6=j

(〈OiOj〉 − 〈Oi〉〈Oj〉) . (A.23)

49

The first term is a sum of the variances of the measurements at time i, but in equilibrium those willnot depend on i. This means that we are just summing the variance of the measurements N times.For the second term we can write 〈OiOj〉 − 〈Oi〉〈Oj〉 = 〈(Oi − 〈Oi〉)(Oj − 〈Oj〉)〉, which representsthe correlation between the value measured at time i with the one at j. If we assume the data isuncorrelated (which in general is a doubtful assumption) the second term vanishes and we are then leftwith the following expression for the error of the estimator:

ε2O

= σ2O

=σ2O

N. (A.24)

A.5 Correlations Between Measurements

In the previous section, at some point we assumed that the measurements were completely uncorrelatedin order to get the statistical error of the estimator of the expected value for some observable. But thisvery clearly is not a good assumption. The Markov chain will sample a new state from a previous one,but the new (depending on the algorithm used) will not be too far away from the old in state space. Thismeans that they will both share a lot information and clearly will not be uncorrelated from each other.What this implies, as we will see, is that the error bars determined in the previous section were actuallyunderestimated.

Going back to (A.23) we can use the symmetry between i and j to write the sum of the second termas [39]:

σ2O

=σ2O

N+

2

N2

N∑i=1

N∑j=i+1

(〈OiOj〉 − 〈Oi〉〈Oj〉)

=σ2O

N+

2

N2

N∑k=1

(〈OsOs+k〉 − 〈Os〉〈Os+k〉) (N − k)

=1

N

[σ2O + 2

N∑k=1

(〈OsOs+k〉 − 〈Os〉〈Os+k〉)(

1− k

N

)]. (A.25)

In the second step the terms were rearranged by using the time-translation symmetry which has to holdin thermal equilibrium.

We can then write the error as [39]

ε2O

= σ2O

= 2σ2O

NτO,int , (A.26)

where τO,int is the integrated correlation time and is defined as

τO,int =1

2+

N∑k=1

ρ(k)

(1− k

N

), (A.27)

with ρ(k) given by

ρ(k) =〈OsOs+k〉 − 〈O〉2

σ2O

. (A.28)

ρ(k) is the normalized autocorrelation function and in the limit of large k decays exponentially [39]:

limk→∞

ρ(k) = ae−k/τO,exp , (A.29)

where τO,exp is called the exponential correlation time. We notice that the correction in the parentesis in

50

(A.27) only matters for large k, however for those values ρ(k) is exponentially small. [39] Therefore wecan ignore the correction and simplify the integrated correlation time

τO,int =1

2+

N∑k=1

ρ(k) . (A.30)

By looking at (A.26) we can see that considering the correlation time increases the error bars fromthe uncorrelated case. Another way of looking at this is by introducing Neff = N/2τO,int, which allowsus to write σ2

O= σ2

O/Neff . This means that the time correlations have the effect of reducing the effectivesample size.

A.6 Estimation of Correlation Times

We saw in the previous section that with time correlations the effective sample size is lower than theactual sample. If the correlation time is too long, as it happens near a phase transition, the effectivesample size may become too small for the results obtained via Monte Carlo to be trusted. It is thereforevery important for correlation times to be provided in any trustworthy Monte Carlo study, or else one cannever be sure that the obtained results are statistically meaningful.

We can estimate (A.28) via [36]

ρ(k) =1

N−k∑N−ki=1

(Oi −O

) (Oi+k −O

)σ2O

, (A.31)

where σ2O is the estimator of the variance σ2

O. For large k the variance of ρ(k) starts diverging and if wewere to use (A.30) to estimate the correlation time we would run into problems. What is usually done isto cutoff the sum at some kmax

τO,int(kmax) =1

2+

kmax∑k=1

ρ(k) . (A.32)

kmax can be determined self-consistently by stopping the sum at kmax > 6τO,int(kmax) [36]. In Fig.A.1awe can see the correlation time as a function of kmax for a simulation with the Metropolis-Hasting algo-rithm with sequential update for the Ising model on the square lattice. We see that at some point thevalue converges around τO,int ≈ 5.7.

51

(a) (b)

Figure A.1: (a) Integrated correlation time for the energy and (b) its binning analysis. The results wereobtained for the Ising model on a 16× 16 square lattice simulated with the Metropolis-Hasting algorithmwith sequential updates close to βc = log(1 +

√2)/2.

A.6.1 Binning Analysis

A more convinient way of estimating errors of mean values is through binning analysis. Here we dividethe series of N measurements into NB non-overlapping bins with length l. We can then compute theaverage of the observable inside each bin [39]

O(B)i =

1

l

l∑j=1

O(i−1)l+j , (A.33)

where i = 1, . . . , NB . Thus the number of measurements has been reduced to l. If l � τ then thesemeasurements are almost uncorrelated and so we can compute the variance of the mean value accord-ingly σ2

O= σ2

B/NB . We should also note that the average over the bins is equal to the original meanvalue O(B) = O. We can then estimate the error via [39]

ε2O

= σ2O

= σ2B/NB =

1

NB(NB − 1)

NB∑i=1

(O

(B)i −O(B)

)2

. (A.34)

If we compare this with (A.26) we see that σ2B/NB = 2τO,intσ

2O/N . Noting that l = N/NB we can use

this to estimate the correlation timeτO,int = lσ2

B/2σ2O . (A.35)

This result is in agreement with Fig.A.1b where we plot the right hand side of the previous equationagainst the bin length. We see that this converges to a value of the correlation time that agrees with theone obtained from Fig.A.1a .

A.6.2 Jackknife Analysis

The problem with all of the previous analysis is that computing error estimates of quantities that are notdirectly measured in the simulation, such as the specific heat or the susceptibility, is not trivial at all. Onecan not naively apply any of the previous methods for this quantities that are nonlinear combinations ofthe basic observables. One way to make the error estimates for this quantities is with error propagationbut that is very unyielding. An alternative is the Jackknife analysis.

The Jackknife method consists of forming large blocks that contain almost all the data points except

52

the ones present in a particular bin from the binning method used previously. One can then compute thequantity inside a Jackknife block by removing the quantity evaluated inside the respective bin [39]:

O(J)i =

NO − lO(B)i

N − l, (A.36)

with i = 1, . . . , NB and l the same as in the binning analysis. This means that each Jackknife block isusing N − l of the original data points.

The Jackknife blocks are obviously correlated since the same data is present in NB − 1 of them.This means that the variance over the Jackknife blocks is much lower than the variance in the binningmethod. To adjust for this, one can multiply the Jackknife variance by a factor of (NB − 1)2. We thenhave the following estimate for the error [39]:

ε2O

= σ2O

=NB − 1

NB

NB∑i=1

(O

(J)i −O(J)

)2

. (A.37)

A.7 Cluster Algorithms

As was explained in the previous sections, the data generated via the Markov chain is going to becorrelated. In fact, near a second order phase transition, the correlation time scales as τ ∝ Lz, wherez is called the dynamical critical exponent [1]. This effect is called critical slowing down and is speciallypronounced in local update algorithms.

Critical slowing down can be a serious issue when one wants to generate enough uncorrelated datafor larger systems. For some models one can solve this problem by constructing cluster algorithms,whose exponent z tends to be lower than local update ones. Cluster algorithms propose new configu-rations by changing large groups of degrees of freedom across the system instead of just a single (ora few) local ones. The problem is that these type of algorithms are not as easy to generalize to othermodels as for example the Metropolis-Hasting.

In the case of the FK model there is no cluster algorithm known that would converge to the rightequilibrium distribution. For that reason we will use the Metropolis-Hasting and thus it is not necessaryto get into further details about cluster algorithms.

A.8 Finite Size Scaling

When simulating a physical model, one is always limited by the computational resources available. Thereare memory limitations to be considered as well as the ones related with time. This considerations limit,among other things, the maximal possible size of the system simulated. Larger systems will take morememory and take longer to generate enough uncorrelated data. The problem is that for addressing thethermodynamic behavior of a real world system one needs to take the thermodynamic limit (L → ∞)when doing calculations. This, however, is obviously not feasible to do in a simulation and one has noalternative but to use a finite size system to approximate the real system. The problem is that thereis a danger that the results obtained are nothing but some finite size effect that vanishes when thethermodynamic limit is taken.

A finite size scaling (FSS) analysis is a way to try to go around this problem. It consists of simulating amodel with increasingly larger system size and analyze how quantities scale as the size increases. Thisallows one to check if certain behaviors persist or start to be mitigated when considering a larger systemsize and allows for extrapolations of the behavior at the thermodynamic limit. When studying continuous

53

phase transitions in particular, this allows one to estimate critical values of the transition parameters aswell as critical exponents. This in turn helps one to identify the universality class of a transition.

In the next sections we will provide an overview on how this method is applied to study continuousphase transitions. Finally, we apply it to the Ising model as an example.

A.8.1 Necessary Thermodynamic Quantities

To deploy the FSS method one has to run simulations for several system sizes and several tempera-tures and store in memory the series of measurements of the basic observables. In this case thoseobservables are the energy E and the magnetization M =

∑i σi, which in this model serves as an order

parameter. An order parameter is a quantity that vanishes in the disordered phase and has a finite valueafter the transition into the ordered phase. It is therefore useful to identify the presence of a phase tran-sition. With the series of measurements of this basic observables one can compute more complicatedquantities, such as the specific heat C(β) and the susceptibility χ(β)

C(β) = β2V(〈e2〉 − 〈e〉2

),

χ(β) = βV(〈m2〉 − 〈|m|〉2

), (A.38)

where e = E/V and m = M/V .

Note that in (A.38) the actual definition of susceptibility χ(β) = βV(〈m2〉 − 〈m〉2

)was not used. The

reason for this is that in a finite system the symmetry is not actually broken and for a long enough timethe system will on ocasion flip between m = ±1 configurations. This results in 〈m〉 = 0. If we imaginethe distribution of m we see that it has two symetric peaks and that their respective values are going tobe close to the values m would assume if the symmetry had been spontaniously broken. So using 〈|m|〉instead provides us with a good estimator [39].

It is also going to be useful to compute the magnetic Binder cumulants

U2(β) = 1− 〈m2〉3〈|m|〉2

,

U4(β) = 1− 〈m4〉3〈m2〉2

, (A.39)

as well as their derivatives

dU2(β)

dβ=

V

3〈|m|〉2

[〈m2〉〈e〉 − 2

〈m2〉〈|m|e〉〈|m|〉

+ 〈m2e〉]

= V (1− U2)

[〈e〉 − 2

〈|m|e〉〈|m|〉

+〈m2e〉〈m2〉

],

dU4(β)

dβ= V (1− U4)

[〈e〉 − 2

〈m2e〉〈m2〉

+〈m4e〉〈m4〉

]. (A.40)

We also make use of the derivatives of the magnetization

d log〈|m|〉dβ

= V

(〈|m|e〉〈|m|〉

− 〈e〉)

,

d log〈m2〉dβ

= V

(〈m2e〉〈m2〉

− 〈e〉)

. (A.41)

As was mentioned before, a many of these quantities develop singularities near a second-orderphase transition, which for finite systems are going to be smeared. [39] They scale with volume near the

54

transition temperature in accordance with

C = Lα/νfC(x)[1 + . . . ] ,

〈|m|〉 = Lβ/νfm(x)[1 + . . . ] ,

χ = Lγ/νfχ(x)[1 + . . . ] , (A.42)

where α, β, γ and ν are critical exponents and the dots represent smaller corrections to the mainbehavior. fi(x) is a scaling function and x is the scaling variable

x = (β − βc)L1/ν . (A.43)

The Binder cumulants are specially useful because, apart from the small corrections, they do notscale with the volume near the transition

U2p = fU2p(x)[1 + . . . ] , (A.44)

with p = 1, 2. This means that the curves for diferent system sizes will intercept at the transition [39].This allows one to have an unbiased estimation for the critical temperature.

The remaining quantities mentioned before scale according to

dU2p

dβ= L1/νf ′U2p

(x)[1 + . . . ] ,

d ln〈|m|p〉dβ

= L1/νfdmp(x)[1 + . . . ] , (A.45)

with p = 1, 2 as well [39].The quantities in eqs. (A.38)–(A.41) all have maxima at pseudo-transition points βmaxi(L). Requiring

that the scalling variable be held constant at the phase transition implies that these maxima scale withvolume according to

βmaxi(L) = βc + aiL−1/ν + . . . (A.46)

A.8.2 Method

In the following we describe how to extract the critical exponents from the Monte Carlo data.We notice that the scaling of the quantities in (A.45) only depends on the exponent ν. This means

that one can make use of least-squares fits to extract an estimate for this exponent by using the maximafor these quantities. Those fits can be made linear if one uses a log− log scale instead, which makesthem more reliable.

Using the location of the maxima βmaxi(L) for the different quantities in eqs. (A.38)–(A.41) and usingeqs. (A.46) with the previously estimated ν allows for the estimation of the critical temperature βc viaa linear fit as well. One can then check this estimates against the one using the crossing point of theBinder cumulants.

Once ν is known one can also similarly use (A.42) to estimate the remaining critical exponents.

A.8.3 Example: Ising model on a two-dimensional square lattice

We now apply the method to the Ising model on a square lattice. We take the coupling energy J = 1.In Fig.A.2 we plot the susceptibility for different system sizes. A log scale was chosen for convenience

in visualizing the data. The smooth curves through the data points are based on spline fits so that we canhave a continuous function of β. This means a standard maximization routine can be used to compute

55

the maxima. Similar plots were obtained for d log〈|m|p〉dβ with p = 1, 2 and dU2

dβ Now that we have themaxima for these quantities we can apply the method.

Figure A.2: Plot of χ(β) for different system sizes. A logarithmic scale was used to help with visualization.

We start with the exponent ν. In Fig.A.3 we plot the maxima of the quantities mentioned above ona log− log scale. We then fit equations (A.45) to the data in order to extract ν. The results of the fitsare shown in table A.1, where we present the values of ν determined through each quantity. We alsothen present the average weighted by the errors. Since the data for all this different fits is from thesame simulation, the different values are obviously correlated. Because of this, if we simply use errorpropagation to estimate the error of the weighted average we end up underestimating the error. Soinstead, to heuristically account for the correlations, the smallest error from the original ν values is takento be the final one [39]. We then obtain ν = 1.0041± 0.0004 as the final result which is comparable withthe exact value νexact = 1.

Figure A.3: Plot of the maxima of d log〈|m|p〉dβ with p = 1, 2 and dU2

dβ . The lines are the fits of equations(A.45) (with a log− log scale) to the maxima and are used to extract the critical exponent ν. This scalewas chosen so that the fits could be linear. We also do the same thing for χ(β) here in order to extractthe critical exponent γ.

56

d log〈|m|〉/dβ d log〈m2〉/dβ dU2/dβ weighted averageν 1.0058(14) 1.0038(04) 1.0324(52) 1.0041(04)

Table A.1: Values of ν determined using different quantities. The average value weighted by the errorsis also provided.

We now turn to the critical temperature. There are two ways of computing this. The first is using thecrossing point of the Binder cumulant which we show in Fig.A.4a. The value obtained with this methodis βc = 0.44066 which isn’t too far away from the exact value βc,exact = log

(√2 + 1

)/2 ≈ 0.440687.

(a) (b)

Figure A.4: (a) Plot of the Binder cumulant U4 for different system sizes. The inset shows a close up ofthe crossing point at the critical temperature. (b) Plot of the pseudo-transition points βmax for the samequantities as in Fig.A.3. Equation A.46 is fitted to the data, where we take ν = 1 (the exact value).

The second way of estimating the critical temperature is by fitting equation (A.46) to the maxima loca-tions for the different quantities. This is shown in Fig.A.4b and the corresponding results are presentedin tabel A.2. Since the exact value of ν is known we used it here, however if that was not the case onewould have had to use the value estimated previously. The value of the critical temperature determinedwith this method was βc = 0.44064± 0.00008.

χ d log〈|m|〉/dβ d log〈m2〉/dβ dU2/dβ weighted averageβc 0.44065(13) 0.44155(24) 0.44099(61) 0.44059(15) 0.44077(13)

Table A.2: Values of βc determined using different quantities. The average value weighted by the errorsis also provided.

Finally we determine γ the same way we did ν. We plot the maxima of χ on a log− log scale inFig.A.3 and fit the last equation of (A.42). We show the result of the fit in table A.3. The value obtainedwas γ = 1.7682± 0.0035 which can be compared with the exact one γexact = 1.75.

χ

γ 1.7697(30)

Table A.3: Value of γ determined using the maxima of χ.

In this chapter we reviewed the standard approach to the Monte Carlo method for simulating physicalmodels as well as how to perform error and FSS analysis. This tools are used in the remaining work tostudy the FK model.

57

58

Appendix B

Newton’s Method For Fixing theFillings

As was stated in the main text, the FK model for the square lattice possesses a point (µc, µf ) =

(U/2, U/2) where the model is invariant under particle-hole transformations. In this regime one has〈Nf 〉V = 〈Nc〉

V = 1/2. On the triangular lattice however, there is no such symmetry point, so we had tocome up with another approach to be able to fix the filling of both species of electrons. The way wewere able to do this was using Newton’s method for solving systems of non-linear equations. Let us firstreview the method and then explain how it was used to achieve our purpose.

B.1 The method

Newton’s method is an iterative algorithm used to find closer and closer approximations of the zerosof some function f(x) = 0. One starts with an initial guess for the root and then applies the methodsuccessively until convergence has been reached. The update for the approximation of the solution isgiven by

xn+1 = xn −f(xn)

f ′(xn), (B.1)

where f ′(xn) is the derivative of f(x) evaluated at xn.

The method can be generalized for a system of k equations and k variables

Jf (xn)(xn+1 − xn) = −f(xn) , (B.2)

where Jf (xn) is the Jacobian matrix evaluated at point xn. f(xn) and xn are column vectors withdimension k.

B.2 Application to the filling fixing problem

We now see how the method was used to find the chemical potentials necessary to fix a particular filling.We start by restating some things:

Z =∑{nf}

e−βF (nf ) ; F (nf ) = − 1

β

∑ν

log[1 + e−β(εν−µc)

]− µfNf . (B.3)

59

We now can define the fillings of both species nf , nc with respect to the partition function as

ρf =1

V β

d logZ

dµf=

1

V

∑{nf}Nfe

−βF (nf )

Z=〈Nf 〉V

, (B.4)

andρc =

1

V β

d logZ

dµc=

1

V Z

∑{nf}

∑ν

1

eβ(εν−µc) + 1e−βF (nf ) =

〈∑ν nF (εν)〉V

=〈Nc〉V

, (B.5)

where nF (εν) is the Fermi occupation number of the state with energy εν .

The goal of the method is then to find the chemical potentials µc and µf such that ρc and ρf arewhatever number we want between 0 and 1. If for example we want a filling of 1/3 for the f-electrons and2/3 for the c-electrons then the system of equations that has to be solved is the followingf1(µf , µc) = ρf (µf , µc)− 1/3 = 0

f2(µf , µc) = ρc(µf , µc)− 2/3 = 0. (B.6)

If we want some other filling we just have to change the numbers in this equations and everything thatfollows next applies as well.

We start from (B.2) and apply it to our case: ∂f1∂µf

∂f1∂µc

∂f2∂µf

∂f2∂µc

(µf,n+1

µc,n+1

)=

∂f1∂µf

µf,n + ∂f1∂µc

µc,n − f1(µf,n;µc,n) ≡ a

∂f2∂µf

µf,n + ∂f2∂µc

µc,n − f2(µf,n;µc,n) ≡ b

. (B.7)

We then use Gaussian elimination∂f1∂µf

∂f1∂µc

0 ∂f2∂µc− ∂f1

∂µc

∂f2∂µf∂f1∂µf

≡ c

(µf,n+1

µc,n+1

)=

a

b− a∂f2∂µf∂f1∂µf

, (B.8)

and finally arrive at the solution µc,n+1 =

(b− a

∂f2∂µf∂f1∂µf

)/c

µf,n+1 =a− ∂f1∂µc

µc,n+1

∂f1∂µf

(B.9)

So basically the method to fix the desired fillings works as follows. We start with an initial guess for thechemical potentials. We run the normal Monte Carlo method for a while and then evaluate the averagevalues of the derivatives of f1,2 with respect to the chemical potentials. Then we use (B.9) to update thevalues of the chemical potentials. When the difference between the values of the fillings computed withMonte Carlo and the intended values falls bellow a certain threshold we stop the algorithm.

B.2.1 Computing the derivatives

In this section we show the explicit formulas for the derivatives required to apply the method.

We first see that ∂f1∂µf

=∂ρf∂µf

, ∂f2∂µf

= ∂ρc∂µf

, ∂f1∂µc

=∂ρf∂µc

and ∂f2∂µc

= ∂ρc∂µc

. Now we compute the firstderivative

dρfdµf

=β

V Z

∑{nf}

N2f e−βF (nf ) − β

V Z2

∑{nf}

Nfe−βF (nf )

2

=β

V

(〈N2

f 〉 − 〈Nf 〉2), (B.10)

60

and the second

dρcdµf

=β

V Z

∑{nf}

(∑λ

nF (ελ)

)Nfe

−βF (nf )−

− β

V Z2

∑{nf}

Nfe−βF (nf )

∑{nf}

(∑λ

nF (ελ)

)e−βF (nf )

=β

V(〈NfNc〉 − 〈Nf 〉〈Nc〉) . (B.11)

Now we notice that the second is equal to the third dρcdµf

=dρfdµc

. And finally we write the last one as

dρcdµc

=β

V

(〈N2

c 〉 − 〈Nc〉2 +

⟨∑λ

n2F (ελ)eβ(ελ−µc)

⟩). (B.12)

We now know everything we need to apply this method for fixing the fillings of both species of elec-trons.

The mean values can be calculated by Monte Carlo sampling.

61

62

Appendix C

Spline Functions and Error Estimation

As stated in Appendix A, to perform a FSS analysis it is necessary to determine the location of themaxima of some thermodynamic functions, such as the one in Fig.A.2. The problem is that for that tobe possible those functions should be continuous with temperature, but the Monte Carlo simulations areonly run for specific values of temperature so we only have access to a discrete version of them. Towork around such problem, spline interpolation is used. A spline is a polynomial defined in branchesand here third degree polynomials are used.

For n + 1 points {(xi, yi) : i = 0, 1, . . . , n} there are n third degree polynomials y = qi(x), i =

1, 2, . . . , n, one for each branch. The values of xi represent the temperatures for which the simulationwas run and yi represent the values of a particular thermodynamic function computed via the Jack-knife method. The goal is then to obtain the spline for each thermodynamic function and then use amaximization routine on it to find the corresponding maxima needed for the FSS analysis.

C.1 Determining the Polynomials

In this section the results are just stated, but one can check their derivation in ref().

The polynomials can be written as

qi = (1− t)yi−1 + tyi + t(1− t)[ai(1− t) + bit] , (C.1)

and are valid for xi−1 ≤ x ≤ xi with i = 1, . . . , n. The following definitions are also usedt = x−xi−1

xi−xi−1

ai = ki−1(xi − xi−1)− (yi − yi−1)

bi = −ki(xi − xi−1) + (yi − yi−1)

, (C.2)

where ki are determined via a set of linear equations. n− 1 of those equations are the following

ki−1

xi − xi−1+

(1

xi − xi−1+

1

xi+1 − xi

)2ki +

ki+1

xi+1 − xi= 3

(yi − yi−1

(xi − xi−1)2+

yi+1 − yi(xi+1 − xi)2

), (C.3)

63

for i = 1, . . . , n− 1. The remaining two are given by

2

x1 − x0k0 +

1

x1 − x0k1 = 3

y1 − y0

(x1 − x0)2,

1

xn − xn−1kn−1 +

2

xn − xn−1kn = 3

yn − yn−1

(xn − xn−1)2. (C.4)

(C.3) together with (C.4) give a set of n+ 1 linear equations that uniquely define k0, k1, . . . , kn. One canalso write that system of equations in matrix form, which is convenient further below

Ak = b , (C.5)

where A is a (n+ 1)× (n+ 1) matrix that holds the coefficients of the ki’s on the left side of the previousequations. k is a column vector that holds the ki’s and b is another column vector that holds the righthand side of the previous equations.

By solving the system of linear equations, one uniquely determines the polynomial in each branchand obtains the desired spline.

However there is a problem. The set of points {(xi, yi) : i = 0, 1, . . . , n} used to determine the splinehas an error bar σyi associated with each value yi, which is just the statistical error determined viaJackknife when computing a particular thermodynamic quantity. This means that all the points qi(x) inall the branches have an error that can be traced back to the errors of the yi’s. It is important to knowthat error to be able to estimate an error for the computed maxima.

C.2 Error Estimation

The goal is to write the polynomials qi(x) as a linear combination of the yi’s. This makes it simple tocompute an error for qi(x) by using error propagation. It is convenient to invert equation (C.5)

k = A−1b = Bb , (C.6)

where B is the inverse matrix of A. This equation can also be written in component form

ki =∑j

Bijbj . (C.7)

If (C.2) along with (C.7) is plugged in (C.1) and then bi replaced by their definitions in (C.3) and (C.4)one gets

qi(x) = (xi − xi−1)

n−1∑j=1

3

(xj − xj−1)2

[t(1− t)2(Bi−1,j +Bi−1,j−1)− t2(1− t)(Bi,j +Bi,j−1)

]yj−

− (xi − xi−1)

n−1∑j=1

3

(xj+1 − xj)2

[t(1− t)2(Bi−1,j +Bi−1,j+1)− t2(1− t)(Bi,j +Bi,j+1)

]yj−

− (xi − xi−1)3

(x1 − x0)2

[t(1− t)2(Bi−1,0 +Bi−1,1)− t2(1− t)(Bi,0 +Bi,1)

]y0+

+ (xi − xi−1)3

(xn − xn−1)2

[t(1− t)2(Bi−1,n +Bi−1,n−1)− t2(1− t)(Bi,n +Bi,n−1)

]yn+

+ (xi − xi−1)3

(x1 − x0)2

[t(1− t)2Bi−1,0 − t2(1− t)Bi,0

]y1−

− (xi − xi−1)3

(xn − xn−1)2

[t(1− t)2Bi−1,n − t2(1− t)Bi,n

]yn−1 + c1yi−1 + c2yi . (C.8)

64

The following definitions were also usedc1 = 2t3 − 3t2 + 1

c2 = −2t3 + 3t2. (C.9)

Equation (C.8) can be simplified by just writing

qi(x) =

n∑j=0

Cij(x)yj , (C.10)

where all those coefficients are just collected in Cij(x). Writing the polynomials like this allows for aneasy computation of the error by using quadratic error propagation

σ2qi(x) =

n∑j=0

(∂qi(x)

∂yj

)2

σ2yj =

n∑j=0

C2ij(x)σ2

yj . (C.11)

C.2.1 Error in xmax

Knowing the spline allows for the computation of the maximum via a standard maximization routine thatstarts with a guess for its location and then with each iteration updates the value using the derivative.This routine also determines in which branch the maximum is. By doing the same as before and writingthe position of the maximum xmax as a linear combination of yj ’s, one can estimate its error.

We start by solving for q′i(x) = 0, where i already labels the branch in which the maximum is located.By simplifying the equation one eventually obtains

d0 + d1t+ d2t2 = 0⇔ t =

−d1 ±√d2

1 − 4d0d2

2d2, (C.12)

where we have d0 = yi−yi−1+ai

xi−xi−1

d1 = 2bi−4aixi−xi−1

d2 = 3 ai−bixi−xi−1

, (C.13)

with ai and bi defined as before.

We now do the same as in the previous section and write d0, d1 and d2 as linear combinations of yj ’seach. We obtain

d0 = 3

n−1∑j=1

[Bi−1,j +Bi−1,j−1

(xj − xj−1)2− Bi−1,j +Bi−1,j+1

(xj+1 − xj)2

]yj − 3

Bi−1,0 +Bi−1,1

(x1 − x0)2y0+

+ 3Bi−1,n +Bi−1,n−1

(xn − xn−1)2yn + 3

Bi−1,0

(x1 − x0)2y1 − 3

Bi−1,n

(xn − xn−1)2yn−1 , (C.14)

65

d1 = −3

n−1∑j=1

[4Bi−1,j + 4Bi−1,j−1 + 2Bi,j + 2Bi,j−1

(xj − xj−1)2− 4Bi−1,j + 4Bi−1,j+1 + 2Bi,j + 2Bi,j+1

(xj+1 − xj)2

]yj+

+ 34Bi−1,0 + 4Bi−1,1 + 2Bi,0 + 2Bi,1

(x1 − x0)2y0 − 3

4Bi−1,n + 4Bi−1,n−1 + 2Bi,n + 2Bi,n−1

(xn − xn−1)2yn−

− 34Bi−1,0 + 2Bi,0

(x1 − x0)2y1 + 3

4Bi−1,n + 2Bi,n(xn − xn−1)2

yn−1 + 6yi − yi−1

xi − xi−1(C.15)

and

d2 = 3

n−1∑j=1

[Bi−1,j +Bi−1,j−1 +Bi,j +Bi,j−1

(xj − xj−1)2− Bi−1,j +Bi−1,j+1 +Bi,j +Bi,j+1

(xj+1 − xj)2

]yj−

− 3Bi−1,0 +Bi−1,1 +Bi,0 +Bi,1

(x1 − x0)2y0 + 3

Bi−1,n +Bi−1,n−1 +Bi,n +Bi,n−1

(xn − xn−1)2yn+

+ 3Bi−1,0 +Bi,0

(x1 − x0)2y1 − 3

Bi−1,n +Bi,n(xn − xn−1)2

yn−1 − 2yi − yi−1

xi − xi−1. (C.16)

This allows for the computation of the maximum location tmax according to (C.12). Even thoughthere are two solutions for the equation, only one of them is in the correct branch. One can also do thesame as in (C.11) and compute the errors for σd0 , σd1 and σd2 , which via error propagation allow for thecomputation of the error for tmax

σ2tmax =

(∂t

∂d0

)2

σ2d0 +

(∂t

∂d1

)2

σ2d1 +

(∂t

∂d2

)2

σ2d2 , (C.17)

where ∂t∂d0

= ∓(d21 − 4d0d2)−1/2

∂t∂d1

=−1±d1(d21−4d0d2)−1/2

2d2

∂t∂d2

= ∓d0d2 (d21 − 4d0d2)−1/2 − t

d2

. (C.18)

Using the definition of t we can write xmax = tmax(xi − xi−1) + xi−1, which finally gives an error

σxmax = (xi − xi−1)σtmax . (C.19)

In practice what happens is that a maximization routine is used to find the branch where the maximumis located. Then the values of d0, d1 and d2 are computed which determines tmax via (C.12). This givesthe position of the maximum xmax together with its error. One can then use (C.11) to compute the errorof the y coordinate of the maximum σymax = σqi(xmax).

Note that the maximization routine is merely used to find the branch where the maximum is located.Even though it also provides an estimation for xmax it has an error associated with the step size used inthe routine, that is not accounted for here. So instead we just use it to find the correct branch and thenuse the formulas derived in this section to determine the maximum and its error.

66