Instituto de Física Gleb Wataghin
Bruno Ricardi de Abreu
Quenches numéricos de desordem
no modelo Bose-Hubbard
Numerical quenches of disorder
in the Bose-Hubbard model
CAMPINAS 2018
Numerical quenches of disorder
in the Bose-Hubbard model
Quenches numéricos de desordem
no modelo Bose-Hubbard
Tese apresentada ao Instituto de Física Gleb Wa-taghin da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obten-ção do título de Doutor em Ciências.
Thesis presented to the Institute of Physics Gleb Wataghin of the University of Campinas in par-tial fulfillment of the requirements for the degree of Doctor in Sciences.
Orientador: Silvio Antonio Sachetto Vitiello
Este exemplar corresponde à versão final da tese defendida pelo aluno Bruno Ricardi de Abreu, e orientada pelo Prof. Dr. Silvio An-tonio Sachetto Vitiello.
Campinas 2018
ORCID: ttps://orcid.org/0000-0002-9067-779X
Ficha catalográfica
Universidade Estadual de Campinas Biblioteca do Instituto de Física Gleb Wataghin Lucimeire de Oliveira Silva da Rocha - CRB 8/9174
Abreu, Bruno Ricardi de,
Ab86n AbrNumerical quenches of disorder in the Bose-Hubbard model / Bruno Ricardi de Abreu. – Campinas, SP : [s.n.], 2018.
AbrOrientador: SIlvio Antonio Sachetto Vitiello.
AbrTese (doutorado) – Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin.
Abr1. Superfluidez. 2. Bose-Hubbard, Modelo de. 3. Monte Carlo quântico, Método de. 4. Sistemas desordenados. 5. Átomos ultrafrios. I. Vitiello, Silvio Antonio Sachetto, 1950-. II. Universidade Estadual de Campinas. Instituto de Física Gleb Wataghin. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: Quenches numéricos de desordem no modelo Bose-Hubbard Palavras-chave em inglês:
Superfluidity
Bose-Hubbard model
Quantum Monte Carlo method Disordered systems
Ultracold atoms
Área de concentração: Física Titulação: Doutor em Ciências Banca examinadora:
Silvio Antonio Sachetto Vitiello Ricardo Luis Doretto
Marcos Cesar de Oliveira Raimundo Rocha dos Santos José Abel Hoyos Neto
Data de defesa: 08-08-2018
Programa de Pós-Graduação: Física
RICARDI DE ABREU RA: 80858 APRESENTADA E APROVADA AO INSTITUTO DE
FÍSICA “GLEB WATAGHIN”, DA UNIVERSIDADE ESTADUAL DE CAMPINAS, EM 08/08/2018.
COMISSÃO JULGADORA:
- Prof. Dr. Silvio Antonio Sachetto Vitiello - (Orientador) - IFGW/UNICAMP - Prof. Dr. Ricardo Luis Doretto - IFGW/UNICAMP
- Prof. Dr. Marcos Cesar de Oliveira - IFGW/UNICAMP
- Prof. Dr. Raimundo Rocha dos Santos - INSTITUTO DE FÍSICA - UFRJ - Prof. Dr. José Abel Hoyos Neto - INSTITUTO DE FÍSICA/SÃO CARLOS
A Ata de Defesa, assinada pelos membros da Comissão Examinadora, consta no processo de vida acadêmica do aluno.
CAMPINAS 2018
My entire career as a physics student and, in particular, the work that is presented in this dissertation could not be possibly made without the unconditional love and support that I have always received from my family. I feel like just thanking them on these lines does not even fairly compensate for the most elemental source of motivation that they represent to me. Even so, if not by other means, I express here my gratefulness for having them in my life. It is an extraordinarily comforting pleasure to be sure that they will always be by side no matter the possible different courses that my life would take as a consequence of my choices.
Just as important as them for the construction of my career as a physicist, the development of this work, and for my formation as a citizen and human being during my time as a student at Unicamp is my long-term advisor Silvio Vitiello. Along our journey he has consistently been aware of my feelings, tempering my thoughts when they were too fast and confusing, hastening my ideas when I was moving too slow and wisely advising me in a number of situations of life with distinguished discernment and sagacity. I am deeply thankful for his unrestricted patience and perseverance during this period.
I am also very gratified for people that work at IFGW/Unicamp and made this project possible. From faculty, with highly skilled professors that taught me physics on the finest level, to staff that provided fundamental support such as access to scientific books and articles through the library (BIF), scheduling of classrooms for presentations whenever needed and so many other things, including hot, good coffee and snacks. I am completely sure that they made academic life here a lot easier for me. I must also recognize that this research used the computing resources and assistance of the John David Rogers Computing Center (CCJDR) in IFGW, whose staff has been extremelly supportive as well.
Part of this work was made in the United States, more specifically at the University of Illinois at Urbana-Champaing (UIUC), Institute for Condensed Matter Theory (ICMT), where I have been a visiting scholar with Professor David Ceperley. He has demonstrated that, much more than the extraordinary scientist that his career grants, he is an excellent human being. He helped me by being kind and prestative, keeping my hopes alive during what was, beyond any doubts, the hardest piece of my life so far. I am deeply grateful for his support, which of course extended to academic and research life. During this period I also met Ushnish Ray, an excepetionally talented scientist that has guided me through the subtleties of the subjects and methods that were used throughout this work. I am thankful for his patience, consideration and collaboration.
Last, but not least, I shall say that I have been exceedingly lucky in finding new friends and keeping old ones during these years, both in Brasil and in the United States. Their friendship is
whenever my hard thoughts were overcoming my feelings.
I am thankful for financial support from the Conselho Nacional de Desenvolvimento Científico e Tecnológico – CNPq under grants No. 141242/2014-0 and No. 232682/2014-3 that concern both regular doctorate scholarship and the Science Without Borders program.
Neste trabalho as propriedades das fases superfluida (SF) e vidro de Bose (BG) do modelo Bose-Hubbard desordenado em três dimensões são investigadas usando simulações de Monte Carlo quântico. O diagrama de fases é construído utilizando desordem Gaussiana nas energias de ocu-pação, e dois tipos adicionais de distribuição, exponencial e uniforme, são estudados com respeito às suas influências quantitativas e qualitativas no estabelecimento do super-fluxo que caracteriza o estado superfluido. A estatística de observáveis do sistema pertinente a distribuições de pro-babilidade sobre o ensemble de desordem são estudadas para diversos valores de interação entre átomos e tamanhos da rede, onde fortes efeitos de tamanho são observados. Estes efeitos estão relacionados ao mecanismo que dirige a transição SF-BG e corroboram o entendimento do caráter percolativo da transição. Apesar disso, ambos os parâmetros de ordem, a fração de superfluido e a compressibilidade, permanecem auto-promediantes por toda fase superfluida. Nos arredores do contorno SF-BG, efeitos de tamanho são dominantes mais ainda sugerem que a auto-promediação persiste. Estes resultados são relevantes para experimentos com gases atômicos ultrafrios onde um procedimento sistemático de mediação sobre realizações de desordem não é tipicamente possível, e também para cálculos numéricos que precisam necessariamente considerar efeitos de tamanho quando o sistema apresenta pequenas quantidades de superfluido.
Palavras-chave: superfluidez, Monte Carlo quântico, modelo de Bose-Hubbard, desordem,
vi-dro de Bose, percolação, auto-promediação, modelo de Bose-Hubbard desordenado, gases atômicos ultrafrios
In this work the properties of the superfluid (SF) and Bose-glass (BG) phases in the three-dimensional disordered Bose-Hubbard model are investigated using Quantum Monte-Carlo simu-lations. The phase diagram is generated using Gaussian disorder on the on-site potential, and two additional types of distributions, namely exponential and uniform, are studied regarding both their qualitative and quantitative influence on the establishment of the superflow that characterizes the superfluid state. Statistics pertaining to probability distributions of observables over the disorder ensemble are studied for a range of interaction strengths and system sizes, where strong finite-size effects are observed. These effects are related to the mechanism that drives the SF-BG transition and corroborates the understanding of the percolation character of the transition. Despite this, both order parameters, the superfluid fraction and compressibility, remain self-averaging through-out the superfluid phase. Close to the superfluid-Bose-glass phase boundary, finite-size effects dominate but still suggest that self-averaging holds. These results are pertinent to experiments with ultracold atomic gases where a systematic disorder averaging procedure is typically not pos-sible, and also to numerical calculations that must necessarily address finite-size effects when the system exhibits small amounts of superfluid.
Keywords: superfluidity, Quantum Monte Carlo, Bose-Hubbard model, disorder, Bose-glass,
percolation, self-averaging, disordered Bose-Hubbard model, Stochastic Series Expansion, ultracold atomic gases
1.1 Basis functions from the nearly free particle to the atomic limit . . . 28
1.2 Bloch waves constructed for a one dimensional lattice . . . 29
1.3 Coarse-graining picture in real and momentum space . . . 31
1.4 Typical potential of soft core particles. . . 32
1.5 Cartoon of Bose-Hubbard model terms . . . 33
1.6 Scheme of the superfluid and Mott-insulating phases of the Bose-Hubbard model . . 37
1.7 Superfluid fraction as the response to boundaries motion . . . 43
1.8 Phase diagram of the single-band Bose-Hubbard model . . . 46
1.9 Finite-temperature phase diagram of the BHM . . . 47
2.1 Division of a system into “correlation volumes” . . . 50
2.2 Distribution of local critical parameters in a disordered system . . . 50
2.3 Renormalization flux scheme for a clean critical point . . . 54
2.4 Renormalization flux scheme for class 1 . . . 55
2.5 Renormalization flux scheme for class 2a . . . 57
2.6 Renormalization flux scheme for class 2b . . . 58
2.7 Renormalization flux scheme for class 2c . . . 59
2.8 Quantum-to-classical mapping of quenched disorder . . . 61
2.9 Local values of an observable within a sample of the disorder ensemble . . . 63
2.10 Illustration of the Theorem of inclusions . . . 65
3.1 Illustration of addition of disorder to an experimental optical lattice . . . 67
3.2 Distribution of Bose-Hubbard terms for speckle disorder . . . 68
3.3 Different types of diagonal-disorder distributions . . . 72
3.4 Renormalization flux scheme for the DBHM . . . 73
3.5 Illustration of the DBHM phase diagram . . . 75
3.6 Phase diagram of the DBHM obtained by SMFT . . . 76
3.7 Phase diagram of the DBHM obtained by LMF . . . 77
3.8 Phase diagrams of the DBHM at unit filling . . . 78
3.9 Phase diagram of the DBHM at half-filling . . . 79
3.10 Crossover between the low-𝜅 and high-𝜅 BG . . . 80
3.11 Onset of superfluidity in terms of chemical potential . . . 81
3.12 Percolation picture of the SF-BG transition . . . 83
5.1 Illustration of the configuration space created in Handscomb’s method . . . 109
5.2 Configuration space after truncation of the series . . . 111
5.3 Bond decomposition for the BHM . . . 112
5.4 World Line picture and scattering vertices . . . 114
5.5 Example of bonds for 3 lattice sites with PBC . . . 116
5.6 Example of diagonal update . . . 117
5.7 Illustration of the insertion of a worm for loop update . . . 118
5.8 Possible movements of the worm on a scattering vertex . . . 120
5.9 Example for the estimate of equal-time correlation functions . . . 126
6.1 Thermodynamic quantities calculated via ED and SSE . . . 130
6.2 Dependence of Bose-Hubbard terms on the lattice-depth . . . 132
6.3 Physical properties and grand-canonical phase diagrams for 𝑠 = 10 and 14. . . . 133
6.4 Physical properties and grand-canonical phase diagrams for 𝑠 = 18. . . . 134
6.5 Illustration of the LDA procedure . . . 135
6.6 Radial distributions of physical quantities for 𝑠 = 10. . . 136
6.7 Radial distributions of physical quantities for 𝑠 = 18. . . 137
6.8 Phase diagram for 𝜌 = 0.5 . . . 138
6.9 Phase diagram for 𝜌 = 0.75. . . 139
6.10 Phase diagram for 𝜌 = 1.25. . . 139
6.11 Phase diagram for 𝜌 = 1.0 . . . 140
7.1 Example of the evolution of averages with the number of samples . . . 143
7.2 Example of the evolution of variances with the number of samples . . . 144
7.3 Example of the evolution of probability distributions with the number of samples. . 144
7.4 Sample fluctuations of order parameters over the phase diagram . . . 145
7.5 Relative momenta of 𝜌𝑠 along Δ/𝑈 = 0.5 . . . 147
7.6 Relative momenta of 𝜅 along Δ = 0.5 . . . 147
7.7 Probability distributions and quantile-quantile plots of 𝜌𝑠 for Δ/𝑈 = 0.5 . . . 148
7.8 Probability distributions and quantile-quantile plots of 𝜅 for Δ/𝑈 = 0.5 . . . 149
7.9 Distribution of the samples standard deviation . . . 150
7.10 Probability distributions of order parameters before and after re-scaling energy shifts151 7.11 Probability distributions obtained from shuffling the random potential . . . 152
8.1 Three-dimensional maps of local properties for different samples . . . 156
8.2 Local properties integrated over one spatial direction . . . 157
8.3 Relation between the wave function and the random potential . . . 158
8.4 Maps for a sample with average superfluid fraction . . . 160
8.5 Comparison of 𝜌𝑠 for different distributions of the random potential . . . 161
9.4 Scaling of 𝒫𝐿 for 𝑈/𝑡 = 62.0, Δ/𝑈 = 0.5 . . . 168
9.5 Scaling of 𝒫𝐿 for 𝑈/𝑡 = 72.0, Δ/𝑈 = 0.5 . . . 169
9.6 Scaling of 𝜌𝑠 for 𝑈/𝑡 = 72.0, Δ/𝑈 = 0.5. . . 170
9.7 Comparison of the integrated random potential for 𝐿 = 12 . . . 171
9.8 Scaling of 𝒟 for 𝑈/𝑡 = 62.0, Δ/𝑈 = 0.5 . . . 172
9.9 Scaling of 𝒟 for 𝑈/𝑡 = 72.0, Δ/𝑈 = 0.5 . . . 172
A.1 Illustration of the standard deviation of a distribution . . . 194
A.2 Illustration of skewed distributions . . . 195
A.3 Illustration of distributions with different kurtoses . . . 196
Introduction
15
I
Theory
22
1 The Bose-Hubbard model 23
1.1 The art of coarse-graining . . . 23
1.1.1 The ferromagnet paradigm . . . 23
1.2 Coarse-graining of an excessively microscopic model: derivation of the Bose-Hubbard Hamiltonian . . . 24
1.2.1 A very general Hamiltonian . . . 24
1.2.2 Addition of a lattice: Bloch waves . . . 25
1.2.3 Choice of an appropriate basis set: Wannier functions . . . 28
1.2.4 Further simplifications: energy bands . . . 29
1.2.5 The single-band standard Bose-Hubbard Hamiltonian . . . 30
1.3 Physical properties of the model . . . 33
1.3.1 Insights from the double-well potential . . . 33
1.3.2 Superfluid and Mott-insulating phases . . . 35
1.3.3 Energy spectrum and excitations . . . 37
1.3.4 Order parameters and phase diagram . . . 42
1.3.5 Finite temperature effects . . . 46
2 General effects of disorder on continuous phase transitions 48 2.1 Harris’ criterion . . . 49
2.2 Chayes’ criterion . . . 53
2.3 Fate of critical points under addition of disorder . . . 54
2.4 Self-averaging of observables . . . 58
2.5 Theorem of inclusions. . . 64
3 The disordered Bose-Hubbard model 66 3.1 Addition of disorder to the Bose-Hubbard Hamiltonian . . . 66
3.1.1 Correlation between disorder distributions in Hamiltonian terms . . . 67
3.1.2 Off-diagonal disorder Bose-Hubbard Hamiltonian . . . 69
3.1.3 Diagonal disorder Bose-Hubbard Hamiltonian . . . 69
3.2.2 Character of the Griffiths singularities . . . 72
3.2.3 Exigency of an intervening phase . . . 74
3.3 Phase diagrams . . . 74
3.3.1 Commensurate and incommensurate fillings . . . 75
3.3.2 Reentrant behavior of the superfluid phase . . . 76
3.4 The Bose-glass . . . 78
3.4.1 Onset of superfluidity and the percolation picture . . . 81
3.4.2 Order parameters . . . 82
3.5 Finite-temperature effects . . . 84
II
Methods
85
4 Generic numerical methods 86 4.1 Exact diagonalization . . . 874.1.1 Selection of a suitable basis set . . . 87
4.1.2 Direct numerical diagonalization. . . 88
4.1.3 Example for the DBHM . . . 89
4.2 Monte Carlo . . . 90
4.2.1 A picturesque random experiment . . . 90
4.2.2 Estimators . . . 92
4.3 Sampling techniques . . . 96
4.3.1 Transformation of random variables . . . 96
4.3.2 Acceptance-rejection . . . 100
4.3.3 Metropolis algorithm . . . 102
5 Stochastic Series Expansion 105 5.1 Handscomb’s method . . . 105 5.2 Extended sampling . . . 109 5.3 Diagonal update. . . 115 5.4 Loop update . . . 116 5.5 Observables . . . 121 5.5.1 Z-sector . . . 121 5.5.2 G-sector . . . 124 5.6 CSSER. . . 127
III
Applications
128
6 Preliminary results 129 6.1 Comparison between QMC and exact diagonalization . . . 1296.2.3 Fixed filling maps . . . 135
7 Aspects of the disorder ensemble 141 7.1 Definition of disorder-statistical quantities . . . 142
7.2 Disorder equilibration. . . 143
7.3 Size of fluctuations over the phase diagram . . . 144
7.4 Differences in probability distributions . . . 146
7.5 Origins of non-Gaussian behavior . . . 149
8 Features of the random potential 154 8.1 Local properties in different samples . . . 155
8.2 Effects of different disorder distributions . . . 159
9 Finite-size scaling of quantities 162 9.1 Disorder averages and fluctuations. . . 163
9.2 Probability distributions . . . 165
9.3 Relative variances and the self-average query . . . 171
10 Concluding remarks 174
References
176
Appendices
188
A Statistics toolbox 189
B Central Limit Theorem 198
C Example of experimental setup 200
Introduction
As my initial assertion I shall prompt the reader of this dissertation that I have deliberately opted, in writing this document, for the path of completeness and protraction rather than the path of conciseness, even though I have been advised not to do so. My incitation for this choice comes from the fact that, in several opportunities while studying and researching subjects related to topics that I will discuss here, I have struggled to find references that present methodologies, motivations, and conclusions – elements of the actual process of making science – in a more detailed manner, which I believe is a common dilemma in the realm of scientific papers and articles. Most of my understanding of the physics of bosonic cold atoms systems has only been achieved after reading dissertations that discuss the subject in a more basic level, even though the ultimate knowledge is condensed into articles from several authors. I then accordingly want my work to be presented in a fashion that is similar to those that were so important to me. Furthermore, I strongly believe that subtleties that were traversed by me along my journey to obtain my doctorate are worth being documented because they can help others to avoid obstacles that I have found, which hopefully can facilitate for the progress of this field. For the reader that is interested in a more concise documentation on the subject, I suggest referring to my paper [1] and also to those that I have cited within this dissertation. As a matter of fact, I have also opted for a quite detailed bibliography, citing not only those works that have directly formed and contributed to my knowledge but also the original ones.
The work that is going to be presented here is entirely a result of my efforts to make a relevant contribution to condensed matter physics. This enterprise started with my advisor Professor Silvio Vitiello at IFGW/Unicamp in 2014, and gained a more concrete aspect when I visited Professor David Ceperley at the Institute for Condensed Matter Theory (ICMT) from the University of Illinois at Urbana-Champaign (UIUC), during the 2015/16 school year. A large portion of the results were obtained after I came back to Brazil, specifically in the year of 2017. I should also say that my collaborator Ushnish Ray played an extremely relevant role in obtaining these results. In fact, if it was not for formality, as he is not a professor yet, I would call him an advisor as well. I believe that the conditions for performing scientific research, particularly in condensed matter physics, are great in Campinas and excellent in Urbana-Champaign, therefore I strongly suggest both places for those that would be interested in pursuing academic life.
In this introduction, I will present my motivations for studying the subject of this thesis and also discuss why and how this work can be relevant to the scientific community. I will then explain how the dissertation is organized and give some details on the subject of each chapter.
Motivation
Phases of matter are inherently a subject of interest of the human mind. It has been an object of study since the early days of civilization, playing religious, philosophical and scientific roles throughout the ages. Although very far away from what scientists nowadays find a reasonable description of what a phase of matter is, how it can be changed and, from a more practical point of view, how it can be turned into something useful to society, these notions have long before crossed several minds in different contexts, with diverse concerns and various beliefs. For instance, the social importance of alchemists dates not later than the ancient Egyptian civilization. Similar ideas were proliferated in Greek philosophy, chiefly in the context of the so-called fundamental elements. In modern ages, the capability of manipulating precious metals was indeed a source of power and distinction.
Despite the enormous distance in time and scientific knowledge, there is a common piece of interest that persists. Some minds seem to be attracted by the possibility of changing one phase of matter into other. In this sense, condensed-matter physicists can be seen as alchemists of the scientific era. In technical terms, this comprises the understanding of phase transitions and, more generally, critical phenomena. For that, a deep acquaintance to phases of matter on their own is primordial.
During my graduation at IFGW/Unicamp I was lucky to be exposed, relatively early, to a tool that I think is incredibly valuable in learning the scientific method: computational physics. At least in Brazil, it is a common complaint from students that there is a sort of gap in between learning theoretical and experimental physics, which is terrible since empiricism is vital for the development of science. Computational physics is able to partially fill this gap mostly because computers are, nowadays, largely accessible. One can therefore use simulations to promote the embracement of concepts that are presented in classes and books. On a bigger scale, they can be actually used to endorse scientific development by giving new insights to both theory and experiment. For these reasons I have found motivation in conducting my career to the use of computational tools to study condensed matter physics.
Even though classical phases of matter and phase transitions are extremely interesting and there is a lot of open questions regarding their mechanisms, my attention was completely taken by quantum phases of matter when I was firstly introduced to the phenomenon of superfluidity in 2008, the very first year of my Physics career. Since then, it has been my goal to study macroscopic quantum phenomena that are wonderfully enthralling. Fortunately, to make these two resolutions of mine converge, there exists a powerful method to investigate quantum systems using computers: Quantum Monte Carlo. I have therefore dedicated myself to learning and applying such tool to continuum systems, more specifically4He, during my Master’s degree, and to lattice-systems during
my doctorate, which comprises the work that is to be presented here.
In addition, we have witnessed in the last twenty years or so a prodigious development of what is generally called the field of synthetic materials, chiefly boosted with the help of ultracold atomic gases and artificial lattices that can experimentally realize standard models of condensed matter physics with large precision and great control. This allows for a direct test of our understanding of theoretical concepts and ingredients that we should consider in describing the features of a problem, such as the role of interactions between particles, that lie within the heart of many-body,
collective phenomena. Moreover, such realizations turn possible to study the interplay between these different ingredients, which gives physicist the possibility to distinguish their importance in describing the observed properties of the system, a task that is extremely complex when considering strongly correlated, non-perturbative and/or disordered systems. Regarding applications, these systems potentially will lead to the development of new materials that can exploit the physics of the quantum world in a macroscopic, every-day scale. The most prominent examples are perhaps superfluids and superconductors that can offer a dramatic change of our common perspective of energy and heat transport, for instance.
Concurrently to the development of these experimental techniques that allow one to manipulate the order of 105 atoms, there has been substantial enhancement of the available computational
power, specially with the aid of supercomputers. This increasing of resources now allows us to simulate systems that are actually under the same conditions as the experimental ones, regard-ing control parameters such as temperature, volume or number of particles, interaction strength, disorder and lattice geometries, for instance. It is then possible to directly benchmark theory and experiment, which greatly improves our ability to test theoretical ideas and also propose new experimental instances. In spite of that, the resources are finite and there is a huge amount of problems that society faces today that would have interesting and relevant consequences if they could be addressed. We therefore must use such resources wisely. As a picturesque example, we should not use supercomputers to simulate systems under experimental conditions if results from a smaller system that could be obtained from a laptop are already sufficiently precise!
The work that is presented in this dissertation was intended to make contact with such ideas. I have dedicated my efforts to study systems formed by bosonic atoms in a lattice at very low tem-peratures, which are suitably described by the so-called Bose-Hubbard model. These systems, that are nowadays largely reproduced in laboratories around the world, typically exhibit a superfluid and an insulating phase resulting from the competition between the two primordial ingredients of the model: interaction between atoms and diffusion of atoms throughout the lattice. The specific problem that I chose to address was then what are the effects of the addition of disorder to the lattice, which is then a new ingredient, from both qualitative and quantitative perspectives. At the time I made this decision I was extremely captivated by the possibility of experimentally realizing my own calculations! Even though this is still possible, as I will try to make clear, during my studies on this system, my main goal regarding research has become to help both experimentalists and theorists not to waste their time and resources. Hopefully I will convince the reader that I was indeed able to partially fulfill my ambitions.
Purpose of this work
The qualitative consequences of the addition of quenched disorder to the Bose-Hubbard model have been established in 1988-89 with the seminal works by Fisher et al. [2,3]. Perhaps the most significant effect that was then predicted is the existence of the Bose-glass phase, that exhibits peculiar features that I shall discuss along this dissertation. Since then, this model has received a lot of attention and its properties have been explored using renormalization group (RG) approaches, numerical techniques, as well as experiment [3–13].
The static nature of the quenched type of disorder has the important consequence of demanding averaging the free energy of the system over the different static realizations, making it much more technically challenging than its counterpart, the so-called annealed disorder [14]. In the later case, disorder can be handled by averaging the partition function over the disorder degrees of freedom that are thus in thermal equilibrium with the remaining degrees of freedom. Conversely, in order to properly address a certain physical property of the quenched-disorder system that is encapsulated in an observable 𝑋, one therefore must consider the statistics of 𝑋 – averages, variances, and ultimately probability distributions – over a number of different disorder instances arising from realizations of the random, disordered potential to which the system is subjected. In other words, for a certain set of control parameters, these different disorder instances constitute what we shall call a disorder ensemble. The purpose of this work can be condensed in systematically studying the consequences of exploiting this ensemble.
Even though the disordered Bose-Hubbard model has been largely discussed in the scientific literature from both experimental and theoretical perspectives, I have not found a work that addresses the question of how significant the effects of considering different disorder instances can be when considering the superfluid properties of the model. At the same time, I have not found a discussion of what are the effects of considering different types of disorder distributions from which the random potentials are taken. Furthermore, even though the question of self-averaging of observables in disordered systems is vital, I have not found it addressed in the literature. These questions, that are going to be clarified along this dissertation, make this work original.
The practical relevance of studying these problems is to quantify for both experiments and numerical calculations to what extent one must consider the averaging of physical properties over the disorder ensemble and, additionally, quantify the difference arising from considering more idealistic type of disorder distributions, such as the uniform case, and distributions that are closely related to what is found in experiments, such as the speckle-field one. In order to accomplish for that, I have used Quantum Monte Carlo, specifically the Stochastic Series Expansion method, to simulate the disordered Bose-Hubbard model for a range of physical parameters that covers the superfluid phase and part of the Bose-glass phase, obtaining the order parameters and other thermodynamic properties of the system over the disorder ensemble.
Organization of the text
This dissertation is divided in three parts that were meant to be independent contingent on the interests of the reader. PartI– Theory, contains the theoretical framework that I believe is relevant to discuss the physics of the Disordered Bose-Hubbard Model and is organized in three chapters. Part II – Methods, encloses the numerical methods and techniques that I have used to simulate the model and obtain its physical properties, being composed of two chapters. Finally, Part III – Applications, presents and discusses the results in the light of the theoretical perspective, which were obtained from the methods previously discussed. It contains five chapters. I must beforehand tell the reader that this is where my original contributions to the scientific community are placed. In addition to that, I have included at the end of the document a section with Appendices containing details that, even though not vital in explaining the results and discussing the physics of the system,
I strongly believe are necessary and relevant if one is interested in reproducing the calculations that I have performed.
Before describing in more details the contents of each chapter, I shall notify the reader that, even though I have tried to present the subjects of Parts I and II in a closed, complete form, this is certainly a very difficult task. I therefore recommend consulting with the references that I have cited along the text in case the reader requires further details or clearer explanations. I shall also make a couple of technical remarks. Most of the figures that I have included are in colors and, in spite of my efforts to make them significant even when printed in black and white, I am convinced that printing the whole document in gray scale will represent a certain degree of prejudice in the quality of the dissertation. Furthermore, given the fact that the text is written in LATEX with the
hyperlink package, I recommend reading the text from a digital device rather than from a hard copy.
Overview
Chapter 1 – The Bose-Hubbard model, broadly discusses a few features of condensed matter physics that are relevant to solidify the importance of a model that describes a certain physical system. This is done through the common, text-book paradigm of the ferromagnetic phase tran-sition. I then present a derivation of the Bose-Hubbard Hamiltonian by sequentially introducing the constituting elements of bosonic cold atoms systems: a periodic potential (lattice), interac-tions, diffusion and so on. I also give explicit details on the approximations that are considered to obtain the standard, single-band Bose-Hubbard model (BHM). Notice that, in this chapter, I do not consider disorder in the system. After deriving the Hamiltonian, I then discuss the phys-ical properties of the system using insights from an analytphys-ical solution of a double-well potential and present phase diagrams, also deriving expressions of the energy spectra in both strongly and weakly correlated regimes.
Chapter 2– General effects of disorder on continuous phase transitions, as the very name can tell, discusses the effects of the addition of quenched disorder to a clean model that exhibits a second-order phase transition. Even though the BHM fits in this category, the discussion extends to other systems as well. I present two widely known criteria – Harris’ and Chayes’ – that, under very general assumptions, establish the relevance of the disorder operator to the problem. I then discuss the different possibilities that one could expect regarding the critical properties of the system in the light of renormalization group arguments. Sequentially, I discuss the self-averaging question that is crucial to this dissertation and, finally, the Theorem of Inclusions that, even though quite general, was initially derived in the context of the BHM.
Chapter 3 – The disordered Bose-Hubbard model (DBHM), that closes Part I, collects the concepts from the two previous chapters in an attempt to predict the features of the DBHM. I discuss the different ways that one can actually add disorder to the BHM, giving more details for the case that is central on this dissertation: quenched, diagonal disorder. I also present the three types of disorder distributions that are going to be used in Part III. The relevance of disorder to the system is then discussed, with details on the new intervening phase, the Bose-glass, in a fashion that is similar to the initial chapter: excitations, phase diagrams and other aspects.
The percolation character of the superfluid/Bose-glass transition, a feature that is going to be important for the analysis of the results that I have obtained, is particularly discussed as well.
Chapter 4– Generic numerical methods, that opens Part II, presents relevant numerical tech-niques that were used to obtain the results of this work. Specifically, I discuss direct numerical diagonalization, necessary to obtain physical properties of the model and also to benchmark re-sults from more abstract methods such as Quantum Monte Carlo, which is also discussed on a very basic level. I then discuss the question of numerically sampling distributions from a more direct framework, namely transformation of variables, to the far-reaching, powerful Metropolis algorithm. Chapter 5– Stochastic Series Expansion (SSE), closing PartII, presents the particular method that was used to simulate the DBHM. I start with a derivation of Handscomb’s method, which is the historical root of SSE, and then generalize it to obtain SSE according to what as originally done by the creator of the method, A. Sandvik. I then discuss the sampling procedures with explicit examples of the diagonal update process and the loop-update one that is realized via the directed-loop algorithm. Finally, I present how relevant observables are calculated within the framework and language of SSE.
Chapter 6– Preliminary results, that opens the PartIII, where my original contributions start to be presented, discusses basic features of the DBHM, such as the phase diagrams obtained for the Gaussian type of disorder. I also include a comparison between results obtained from Quantum Monte Carlo and exact diagonalization of the DBHM. This chapter is the only one on the entire dissertation that addresses the question of trapped, non-homogeneous systems that are found in experiments. In particular, using the Local Density Approximation, I present estimates of the features of atomic clouds, such as the shell-structure of phases, that is corroborated in experimental systems. I then discuss different types of maps that one can consider when establishing the phase diagram of the DBHM.
Chapter 7 – Aspects of the disorder ensemble, discusses the features of exploring the random potential to a larger degree by considering simulations for several samples with different disorder realizations. I explicitly address the question of equilibrium in the disorder sense, and subse-quently analyze statistical features of the order parameters of the DBHM, with special attention to the fluctuations over the disorder ensemble and the shape of their probability distributions. I then present the relation between the deviation from Gaussian behavior of the superfluid order parameter to the percolation mechanism of the superluid/Bose-glass transition.
Chapter 8– Features of the random potential, presents a more detailed discussion of the pecu-liarities of the different realizations of the random potential in determining the physical properties of the system. In particular, I show that the wave function of this bosonic system, which is strictly related to the establishment of a superfluid, is directly connected to the formation of puddles where the random potential is negative, which corroborates the understanding of the percolation mech-anism and the consequent deviation from Gaussian behavior of the superfluid order parameter of the system. Additionally, I present results for different disorder distributions – Gaussian, box and exponential – where quantitative differences are observed and related to the particular shape of the distributions. I argue that these differences are consequence of the energetic balance between the interaction energy of the atoms and the occupation energy of lattice sites coming from the random potential.
properties and features that were presented in the previous three chapters. It is shown that, as the size of the lattice is increased, different realizations of the random potential start to look more similar from the perspective of their physical properties. In other words, the order parameters of the system are self-averaging quantities, which is explicitly shown by the scaling of statistical quantities of the disorder ensemble to the thermodynamic limit. Here, we have performed simulations with lattice sizes that are comparable to what is found in experiments.
Chapter 10– Concluding remarks, summarizes and contextualizes the results obtained in Part III of this dissertation. I also discuss some prospects on topics that I think would be interesting to be studied in the future.
Appendix A defines all the statistical quantities that were used in this dissertation, while AppendixB discusses the Central Limit Theorem and AppendixC presents an example of exper-imental setup for ultracold atoms in disordered optical lattices.
Part I
Theory
Chapter 1
The Bose-Hubbard model
One remarkable feature of condensed matter physics is that it can encapsulate a huge amount of physical knowledge into what we call models. To construct these models, the art of coarse-graining is far-reaching, being quite popular in the field. In this chapter I will present a suitable derivation of the Bose-Hubbard Model (BHM) that simulates soft-core bosons in an optical lattice, discussing the meaning of the physical parameters and the scope of the model. I will also discuss pertinent physical properties and present the phase diagram for this system.
1.1
The art of coarse-graining
I will start this discussion with the paradigmatic example of a ferromagnet. From a layman point of view this may not be the best option since the very bear concept of magnetization carries on a significant amount of physical knowledge, but for the purposes of what condensed-matter physics really wants to clarify, it is consensually the most pictorial and intuitive case. Moreover, I am assuming that a layman that puts his eyes on this thesis is not that much of a layman. The following discussion is strongly based on Ref. [15], in similarity to what is found in Refs. [16, 17].
1.1.1
The ferromagnet paradigm
A ferromagnet is a certain material – a crystal, for instance – that exhibits a liquid, finite magnetization in its natural state. This means that even when it is left alone, without any kind of fields, external interferences or, in general, in the absence of any perturbations, the material is magnetized. If you come along with a compass and put it close to a ferromagnet, the needle will do some crazy movements. The nature of the constituents of such material, in particular the way they interact, is responsible for the macroscopic emergent behavior that is the magnetization. They fabulously arrange themselves in order to produce this unique, important physical property. Furthermore, we also know that if we heat up a piece of ferromagnet too much, it loses this property. It becomes a paramagnet. There is no finite magnetization unless we provide an external magnetic field. By changing the temperature, it is then possible to transform one phase into the other. The same material, with exactly the same constituents, has different physical properties according to a certain control parameter.
In order to describe the emergency of the spontaneous magnetization in a crystal, we need to specify the constituting parts that form the material. Perhaps the more fundamental picture that we can draw, because I do not intend to go into particle physics depths, is that the composing elements are atomic nuclei and electrons interacting via the Coulomb force. I will call this the approach number 1. It is very fundamental, microscopic. If we can derive the magnetization of a ferromagnet from there, we would definitely have a successful theory. On the other hand, this route is poorly practical and actually it is not smart. We know more than this. For instance, from solid state physics, we can think of electrons in a prescribed crystal lattice with an effective interaction (approach 2). Parameters specifying the interactions, as well as the band structure, crystal fields and so on are widely known. This route is more suitable because the formation of crystal lattices and inner atomic shells are remote from the spontaneous magnetization that we want to describe.
There is a little bit more that we can say that will further simplify our approaches. The source of magnetization is undoubtedly the electronic spins of incomplete electronic shells (d and f shells in Fe, Ni and Co, for instance). We also know that the exchange effect, that combines Coulomb interaction and Pauli exclusion, tends to align spins in an effective short-range interaction, since it lowers the energy of the system. With that in mind, a proper route would be considering classical spins, one in each unit cell of a crystal lattice, with specified spin-spin interaction that would come from parameters adjusted to simulate what approach 2 would imply. This is approach 3. The quantum nature and electronic motion are ignored. However, the physics of interest lies in how a large number of spins behave together, namely, the liquid magnetization. Being a little bit crude on the unit cell scale cannot matter much.
The approaches 1, 2 and 3 constitute models for a ferromagnet. They are representations in terms of parameters that comprise interactions between the elemental constituents of a system that we want to describe. The transition from a more microscopic level of description to a less refined one is what we call the coarse-graining procedure. In each step, we need to know a lot of physics to put as much information as we can into parameters that will mimic the physics that lies within lower scales that we suspect are likely to not play a central role in the observed macroscopic physical properties of the system.
1.2
Coarse-graining of an excessively microscopic model:
derivation of the Bose-Hubbard Hamiltonian
Models are usually specified by their Hamiltonian, which encloses the form with which the components of the model interact and, therefore, governs the dynamics and thermodynamics of the described system. I will start a series of simplifying hypotheses that will transform a very complicated Hamiltonian into the pragmatical Bose-Hubbard model.
1.2.1
A very general Hamiltonian
The use of field operators is reasonable to start modeling a system from the most general perspective because it requires the minimum amount of physical knowledge about the system,
which can be seen by its definition: ^
𝜓†(⃗𝑥) = ∑︁ 𝜈
⟨⃗𝑥|𝜈⟩ ^𝑎†𝜈, (1.2.1)
where |𝜈⟩ is a single-particle eigenstate and ^𝑎†𝜈 is an operator that creates a particle in such state. Thus, ^𝜓†(⃗𝑥) creates a particle at position ⃗𝑥 in any possible single-particle state. We do not know
anything about the quantum state of this particle except for its position.
It is plausible to presume that the particles have kinetic degree of freedom and that they interact in pairs via some potential 𝑈 (⃗𝑥1, ⃗𝑥2). In the scope of this thesis, three-body and
higher-order interactions will not play any important role. I will also consider that the particles can be subjected to a external field 𝑉 (⃗𝑥) that is a one-body potential. The Hamiltonian operator that
describes these particles, in second quantization form, can then be written as
^ 𝐻 = ∫︁ 𝑑𝑑𝑥 ^𝜓†(⃗𝑥) [︃ −~ 2 2𝑚 ⃗ ∇2+ 𝑉 (⃗𝑥) ]︃ ^ 𝜓(⃗𝑥) + 1 2 ∫︁ 𝑑𝑑𝑥1 ∫︁ 𝑑𝑑𝑥2𝜓^†(⃗𝑥1) ^𝜓†(⃗𝑥2)𝑈 (⃗𝑥1, ⃗𝑥2) ^𝜓(⃗𝑥2) ^𝜓(⃗𝑥1). (1.2.2) From the above discussion and taking into account the integrations in configuration space, this Hamiltonian describes particles at any point of space, occupying any possible single-particle states. It could describe any system whose particles pairwise interact, so it is really very general. The first term is composed of one-body operators, so it essentially constitutes many problems at the single-particle level that are likely to be soluble using textbook-like tools of quantum mechanics since the wave-functions would be separable. The second term introduces interactions, characterizing the many-body problem. It clearly makes the Hamiltonian non-diagonal, and most of the times diagonalizing such general form is just unfeasible. Actually, it is hard to even get any physical insights from this form. We need to be more specific in what we want to describe in order to obtain a more treatable form.
1.2.2
Addition of a lattice: Bloch waves
The Bose-Hubbard model is a lattice model. In many problems of condensed matter physics, the existence of a lattice is a consequence of the mechanism of spontaneous symmetry breaking that underlies phase transitions. For instance, a liquid freezes into a solid as we reduce its tem-perature. The solid is rigid, and its macroscopic physical properties differ from the liquid because its constituents exhibit a periodical arrangement throughout the entire space that the system oc-cupies. This periodic arrangement, which is called a lattice, arises from the interactions between the particles and is responsible for the stiffness, or rigidity, of the solid. Particles in a liquid do not possess an organized spatial structure. In contrast to the solid, the knowledge of the position of a certain particle does not bring information about the position of all other particles. In other words, the density of particles does not exhibit long-range correlations. The symmetry that is behind such phase transition is that of spatial translation and rotation.
Although a question of its own interest and importance, a lot of times the formation of a lattice does not concern the description of a physical problem. In the field of synthetic materials, this is
very often the case because the lattice itself is prescribed, and does not necessarily arise from the dynamics of the components of the system. Conversely, it determines the dynamics of the system and, in that being so, it can be seen as a fundamental ingredient of the model describing such material. In this case, the lattice is part of the model. This is precisely the case of cold-atoms systems in optical lattices, which are the main experimental realization of the model that is central to this thesis.
A practical manner to include a lattice in the Hamiltonian that we have written in the last section is by considering the fact that this structure can, as a matter of fact, be encapsulated by a periodic one-body potential 𝑉 (⃗𝑥) to which the particles are subjected. By periodic we mean that
𝑉 (⃗𝑥) = 𝑉 (⃗𝑥 + ⃗𝑎) (1.2.3)
where ⃗𝑎 is any linear combination of the translation vectors of the lattice. This periodicity results in a band structure, where the eigenstates of the Hamiltonian can be grouped in bands. Consider a single particle in one spatial dimension such that the periodicity condition can be written as
𝑉 (𝑥) = 𝑉 (𝑥 + 𝑚𝑎) where 𝑎 is the lattice parameter and 𝑚 ∈ 𝒵. A natural route to deal with
periodic functions is to cast their Fourier series expansion,
𝑉 (𝑥) = +∞ ∑︁ 𝑛=−∞ ˜ 𝑉 (𝑞𝑛)𝑒𝑖𝑞𝑛𝑥, (1.2.4)
where 𝑞𝑛 = 𝑛𝜋/𝑎 are the reciprocal vectors. Using this form, it is interesting to notice the action
of the single particle Hamiltonian 𝐻(𝑥) = −~2 2𝑚
𝑑2
𝑑𝑥2+ 𝑉 (𝑥) in a plane wave function 𝑓𝑘(𝑥) =
1 √ 𝐿𝑒 𝑖𝑘𝑥: 𝐻(𝑥)𝑓𝑘(𝑥) = 1 √ 𝐿 [︃ − ~ 2 2𝑚 𝑑2 𝑑𝑥2 + +∞ ∑︁ 𝑛=−∞ ˜ 𝑉 (𝑞𝑛)𝑒𝑖𝑞𝑛𝑥 ]︃ 𝑒𝑖𝑘𝑥 = √1 𝐿 [︃ ~2𝑘2 2𝑚 𝑒 𝑖𝑘𝑥+ +∞ ∑︁ 𝑛=−∞ ˜ 𝑉 (𝑞𝑛)𝑒𝑖(𝑞𝑛+𝑘)𝑥 ]︃ (1.2.5) = [︃ ~2𝑘2 2𝑚 𝑓𝑘(𝑥) + +∞ ∑︁ 𝑛=−∞ ˜ 𝑉 (𝑞𝑛)𝑓𝑘+𝑞𝑛(𝑥) ]︃
so that the resulting function belongs to the subspace 𝒮𝑘 ≡ {𝑓𝑘, 𝑓𝑘+𝑞1, 𝑓𝑘−𝑞1, 𝑓𝑘+𝑞2, 𝑓𝑘−𝑞2, ...}, and
it is clear that the action of 𝐻(𝑥) in any member of such subspace is a closed operation. This means that these functions span such subspace so that a general solution for the eigen-functions of the Hamiltonian 𝐻(𝑥) can be written as
𝜓𝑘(𝑥) = ∑︁ 𝑛 ˜ 𝑢𝑛(𝑘) 1 √ 𝐿𝑒 𝑖(𝑞𝑛+𝑘)𝑥 ≡ 𝑒𝑖𝑘𝑥𝑢 𝑘(𝑥), (1.2.6) where 𝑢𝑘(𝑥) = 1 √ 𝐿 ∑︁ 𝑛 ˜ 𝑢𝑛(𝑘)𝑒𝑖𝑞𝑛𝑥 (1.2.7)
is called a Bloch wave. Notice that it represents a plane wave modulated by a periodic function. In addition to that, two subspaces 𝒮𝑘 and 𝒮𝑘′ are equal if, and only if, 𝑘′ = 𝑘 + 𝑛(2𝜋/𝑎), therefore
the set of 𝑘 values in the range −𝜋/𝑎 < 𝑘 < +𝜋/𝑎 are unique labels to the corresponding set of subspaces {𝒮𝑘}. This set is the so-called first Brillouin zone.
The time-independent Schrödinger equation for 𝜓𝑘(𝑥) then reads
−~ 2 2𝑚 𝑑2 𝑑𝑥2𝜓𝑘(𝑥) + 𝑉 (𝑥)𝜓𝑘(𝑥) = 𝐸𝑘𝜓𝑘(𝑥), (1.2.8) so that −~ 2 2𝑚 ∑︁ 𝑛 ˜ 𝑢𝑛(𝑘) 1 √ 𝐿 𝑑2 𝑑𝑥2𝑒 𝑖(𝑞𝑛+𝑘)𝑥+ 𝑉 (𝑥)∑︁ 𝑛 ˜ 𝑢𝑛(𝑘) 1 √ 𝐿𝑒 𝑖(𝑞𝑛+𝑘)𝑥= 𝐸 𝑘 ∑︁ 𝑛 ˜ 𝑢𝑛(𝑘) 1 √ 𝐿𝑒 𝑖(𝑞𝑛+𝑘)𝑥 ∑︁ 𝑛 ~2(𝑘 + 𝑞𝑛)2 2𝑚 𝑢˜𝑛(𝑘) 1 √ 𝐿𝑒 𝑖(𝑞𝑛+𝑘)𝑥+∑︁ 𝑛 ˜ 𝑢𝑛(𝑘)𝑉 (𝑥) 1 √ 𝐿𝑒 𝑖(𝑞𝑛+𝑘)𝑥 = 𝐸 𝑘 ∑︁ 𝑛 ˜ 𝑢𝑛(𝑘) 1 √ 𝐿𝑒 𝑖(𝑞𝑛+𝑘)𝑥, (1.2.9)
and taking the inner product with √1
𝐿𝑒 𝑖(𝑘+𝑞𝑚)𝑥 we obtain [︃ ~2(𝑘 + 𝑞𝑛)2 2𝑚 𝛿𝑚𝑛+ ˜𝑉 (𝑞𝑚− 𝑞𝑛) ]︃ ˜ 𝑢𝑛(𝑘) = 𝐸𝑘𝑢˜𝑛(𝑘). (1.2.10)
This is a matrix equation that can be diagonalized to obtain the required solutions {𝜓𝑘𝑛(𝑥), 𝐸𝑘𝑛}, where 𝑛 denotes a band. As stated above, the periodic potential brings in this structure of eigenpairs {𝜓𝑖, 𝐸𝑖}, where 𝑖 = 𝑖(𝑘, 𝑛) indexes the band 𝑛 and the crystal momentum 𝑘, that
characterize the solution of the Schrödinger equation for the system.
We can then rewrite the single-particle field operator using these basis functions in terms of momentum as ^ 𝜓(⃗𝑥) =∑︁ 𝑛 ∑︁ ⃗ 𝑘 𝜓⃗𝑘𝑛(⃗𝑥)^𝑏𝑛,⃗𝑘, (1.2.11)
where now ^𝑏𝑛,⃗𝑘 annihilates a particle with momentum ⃗𝑘 in the energy band 𝑛. The Hamiltonian (1.2.2) then becomes ^ 𝐻 =∑︁ 𝑛,⃗𝑘 𝐸𝑛,⃗𝑘^𝑏† 𝑛,⃗𝑘^𝑏𝑛,⃗𝑘+ ∑︁ 𝑛1,𝑛2,𝑛3,𝑛4 ⃗𝑘1,⃗𝑘2,⃗𝑘3,⃗𝑘4 𝑈⃗𝑘1,⃗𝑘2,⃗𝑘3,⃗𝑘4 𝑛1,𝑛2,𝑛3,𝑛4 ^𝑏† 𝑛1,⃗𝑘1 ^ 𝑏† 𝑛2,⃗𝑘2 ^𝑏 𝑛3,⃗𝑘3 ^ 𝑏𝑛 4,⃗𝑘4. (1.2.12)
Although we have now included the lattice potential to the problem, this form of Hamiltonian is no less complicated than the previous one, therefore we have not gained much from coarse-graining perspectives. For such purposes, the choice of basis functions is very important since it can potentially help analyzing the structure of the continuous Hamiltonian. To obtain the last equation (1.2.12) we have used an expansion in Bloch waves that, even though carry the information of the lattice, are still completely delocalized functions and therefore approach the nearly free particle regime. More localized functions, usually obtained from the atomic limit, are suitable for deeper lattices. This situation is described in Fig. 1.1.
Figure 1.1: Basis functions for different physical situations ranging from the free particle regime (left) to the atomic limit (right). As the lattice depth is plugged in, the band structure emerges, becoming energy levels in the atomic limit, where the band description is no longer needed. Figure from Ref. [18].
1.2.3
Choice of an appropriate basis set: Wannier functions
We now know that any linear combination of Bloch waves is a solution to the problem of particles in an underlying periodic potential. This gives the freedom to choose the relative phase between these states in whatever fashion we may want. A more localized wave-function can be constructed as following:
𝑤𝑗𝑛(⃗𝑥) =∑︁ ⃗𝑘
𝑒𝑖⃗𝑘·⃗𝑥𝑗𝜓𝑛
⃗𝑘(⃗𝑥), (1.2.13)
where ⃗𝑥𝑗 is any point in space that we may want to localize the function around. A convenient
choice is the location of maximal weight of such state, which should coincide with the minima of the lattice potential. This gives a maximally localized state. For localization to be granted, energy bands cannot overlap, as it will be in the cases we are going to deal with. These so-called Wannier functions have the very useful feature of being orthogonal to each other. A comparison between such states and the former Bloch waves is shown in Fig. 1.2. The extended character of Bloch waves, compared to the localization of Wannier states, is evident.
Using this set of functions to write field operators in terms of lattice sites, ^ 𝜓(⃗𝑥) =∑︁ 𝑗 ∑︁ 𝑛 𝑤𝑗𝑛(⃗𝑥)^𝑏𝑛,𝑗, (1.2.14)
we obtain the following form for the Hamiltonian (1.2.2): ^ 𝐻 = −∑︁ 𝑛,𝑚 ∑︁ 𝑖̸=𝑗 𝑡𝑛𝑚𝑖𝑗 ^𝑏†𝑛,𝑖^𝑏𝑚,𝑗+ ∑︁ 𝑛𝑚 ∑︁ 𝑖 𝜖𝑛𝑚𝑖 ^𝑏†𝑛,𝑖^𝑏𝑚,𝑖+ 1 2 ∑︁ 𝑖𝑗𝑘𝑙 ∑︁ 𝑛1𝑛2𝑛3𝑛4 𝑈𝑛1𝑛2𝑛3𝑛4 𝑖𝑗𝑘𝑙 ^𝑏 † 𝑛1,𝑖 ^𝑏† 𝑛2,𝑗 ^𝑏𝑛 3,𝑘^𝑏𝑛4,𝑙 (1.2.15) where 𝑡𝑛𝑚𝑖𝑗 ≡ − ∫︁ ¯ 𝑤𝑛𝑖(⃗𝑥) [︃ − ~ 2 2𝑚∇ 2+ 𝑉 (⃗𝑥) ]︃ 𝑤𝑗𝑚(⃗𝑥)𝑑⃗𝑥, (1.2.16)
Figure 1.2: Examples of Bloch waves and the resulting Wannier state for a 𝐿 = 20 lattice. Top: Bloch waves for two different wave vectors within the first Brillouin zone. Note that they extend all over the system. Bottom: resulting Wannier state centered around the tenth lattice site. It rapidly vanishes away from its center. A deeper lattice would exhibit an even more localized state.
𝜖𝑛𝑚𝑖 ≡ ∫︁ ¯ 𝑤𝑛𝑖(⃗𝑥) [︃ −~ 2 2𝑚∇ 2+ 𝑉 (⃗𝑥) ]︃ 𝑤𝑖𝑚(⃗𝑥)𝑑⃗𝑥, (1.2.17) and 𝑈𝑛1𝑛2𝑛3𝑛4 𝑖𝑗𝑘𝑙 ≡ ∫︁ ∫︁ ¯ 𝑤𝑛1 𝑖 (⃗𝑥1) ¯𝑤𝑛𝑗2(⃗𝑥1)𝑈 (⃗𝑥1, ⃗𝑥2)𝑤𝑘𝑛3(⃗𝑥2)𝑤𝑙𝑛4(⃗𝑥2)𝑑⃗𝑥1𝑑⃗𝑥2. (1.2.18)
The bar over the functions ( ¯𝑤) denote their complex conjugates. Note that off-diagonal terms in
the single-particle sector arise from the fact that the Wannier states are not eigenstates of that one-body Hamiltonian.
1.2.4
Further simplifications: energy bands
At first glance, equation (1.2.15) may seem more complicated than the previous field Hamilto-nian since we ended up creating yet another non-diagonal term. However, as it can be seen from the summation over 𝑖𝑗, which denote the minima of the lattice potential, we are starting to face a more lattice-like form of Hamiltonian, which is our goal. It is very important to notice that the
creation operators that we are handling now create particles in states localized at the bottom of the lattice wells and at a certain energy band.
Considerations about these energy bands can greatly simplify the problem. For instance, in the case of fermions, it is often possible to eliminate all bands except for the valence and conduction ones. For hard-core bosons a similar analysis hold because their interactions somehow mimics Pauli’s exclusion principle. For the cases that are important to this thesis, namely soft-core bosons, one single band is capable to accommodate all the particles that will compose the dynamics of the system. Transitions to other bands can be achieved by thermal excitation. Thus, provided that the temperature is low enough, we can safely disregard all energy bands 𝑛 > 0. In more precise terms, this is valid when
𝑘𝐵𝑇 ≪ ⎯ ⎸ ⎸ ⎷ ~2 𝑚 𝜕2𝑉 (⃗𝑥) 𝜕2⃗𝑥 ⃒ ⃒ ⃒ ⃒ ⃒⃗𝑥 𝑖 , (1.2.19)
where 𝑇 is the temperature and 𝑘𝐵 the Boltzmann constant. This comes from the fact that, close
to the minima of the potential (⃗𝑥𝑖) we can harmonically approximate it, so that the right hand
side of the last equation is a measure of the separation between the lowest energy band (𝑛 = 0) and the closest one (𝑛 = 1). Furthermore, this single-band form of Hamiltonian can be written in a hierarchy of terms corresponding to summations over sites, over nearest-neighbors sites ⟨...⟩, over second-nearest neighbors sites ⟨⟨...⟩⟩ and so on,
^ 𝐻 = ⎡ ⎣ ∑︁ 𝑖 𝜖𝑖^𝑏†𝑖^𝑏𝑖− ∑︁ ⟨𝑖𝑗⟩ 𝑡𝑖𝑗^𝑏†𝑖^𝑏𝑗 − ∑︁ ⟨⟨𝑖𝑗⟩⟩ 𝑡𝑖𝑗^𝑏†𝑖^𝑏𝑗 + ... ⎤ ⎦+ 1 2 ⎡ ⎣ ∑︁ 𝑖 𝑈𝑖^𝑛𝑖(^𝑛𝑖 − 1) + ∑︁ ⟨𝑖𝑗⟩ 𝑈𝑖𝑗𝑛^𝑖𝑛^𝑗 + ... ⎤ ⎦, (1.2.20) where ^𝑛𝑖 = ^𝑏 †
𝑖^𝑏𝑖 is the number operator.
For the lattice character of such Hamiltonian to be even more prominent, it is essential to realize that disregarding higher order energy bands is equivalent to ignoring the dynamics that takes place at length scales smaller than the lattice constant, as shown in Fig. 1.3. This procedure lies within the core of coarse-graining techniques. Integrating out high-momenta modes is also a fundamental tool in the context of the Renormalization Group (RG) that has found enormous success in describing the mechanisms behind phase transitions [19–21].
1.2.5
The single-band standard Bose-Hubbard Hamiltonian
Up to this point, we have simplified a very general Hamiltonian to a lattice form that, in principle, could be more treatable. Although quite useful, these simplifications were made upon very mild and broad considerations, namely the introduction of a periodic one-body potential and the coarse-graining over short-length scales that are not relevant to the problem. Further simplifications will need more specific assumptions. In order to obtain the standard form of the Bose-Hubbard Hamiltonian, we will consider three of them.
The first one concerns the range of the interactions between the bosons that we want to de-scribe. In cold atoms systems, soft-core bosons are paradigmatic. Their interacting potentials can come in different shapes, but must possess the following attributes: being short-range and
Figure 1.3: Coarse graining scheme for ignoring high order energy bands. To the left, real position space length scales 𝑎 (lattice constant) and 𝑙. To the right, corresponding momentum-space scales. As we can see, taking into account only the first Brillouin zone (indicated by red) is equivalent to accounting only for the dynamics that takes place in length scales larger than a lattice constant in real position space. Larger momentum-scales are essentially integrated out. Figure from Ref. [18].
predominantly repulsive. Note that, according to equation (1.2.18), the interaction term is a re-sult of the interacting potential integrated either over different Wannier states or over the same ones. In being short-ranged, we expect that the former case cannot contribute much (recall that the Wannier states themselves are localized!), whereas the later one is relevant. This results in positive interaction energy terms regardless how far away from each other the occupied lattice sites are. The interaction usually is composed of a repulsive part that origins from the Coulomb forces between the electronic clouds of the atoms, and an attractive part that arises from the dipole-dipole interaction. Potentials very often have the well-known Lennard-Jones shape, as in Fig. 1.4, which illustrates both attributes. This amounts to the fact that we can ignore interaction terms in the Hamiltonian (1.2.20) that account for different sites, keeping only the local term:
^ 𝐻 = ⎡ ⎣ ∑︁ 𝑖 𝜖𝑖^𝑏 † 𝑖^𝑏𝑖− ∑︁ ⟨𝑖𝑗⟩ 𝑡𝑖𝑗^𝑏 † 𝑖^𝑏𝑗 − ∑︁ ⟨⟨𝑖𝑗⟩⟩ 𝑡𝑖𝑗^𝑏 † 𝑖^𝑏𝑗 + ... ⎤ ⎦+ 1 2 ∑︁ 𝑖 𝑈𝑖𝑛^𝑖(^𝑛𝑖− 1). (1.2.21)
The second assumption regards the diffusion of particles throughout the lattice or, in other words, the behavior of the terms defined in (1.2.16). They are commonly called hopping terms, which can be understood by the fact that they account for the energy of atoms hopping, or jumping, to different lattice sites. From their definitions, they are tunneling amplitudes between quantum states localized on different sites, or more precisely, they are matrix elements of the underlying potential plus the kinetic term between different Wannier states. Intuitively, the deeper the lattice, the more difficult for a particle to tunnel to a different site. Also, tunneling to larger distances
Figure 1.4: Illustration of a typical soft-core bosonic interacting potential. Important features are strong repulsion and short-rangeness (see text). Figure from the web, source not specified.
has to be harder than tunneling to a close site. Actually, from a text-book-like calculation with a particle facing a repulsive box potential, it can be seen that these dependencies are exponential. As we have a lattice model, we do not ever want the lattice to be too shallow, otherwise it would not make any sense on having a lattice at all. Therefore, hopping terms to sites beyond closest-neighbors are, to a very good approximation, negligible. Our Hamiltonian then becomes:
^ 𝐻 =∑︁ 𝑖 𝜖𝑖^𝑏 † 𝑖^𝑏𝑖− ∑︁ ⟨𝑖𝑗⟩ 𝑡𝑖𝑗^𝑏 † 𝑖^𝑏𝑗 + 1 2 ∑︁ 𝑖 𝑈𝑖𝑛^𝑖(^𝑛𝑖− 1). (1.2.22)
The third and last consideration is, perhaps, the most obvious one: we have no reason (yet!) to distinguish between lattice sites. Just as the particles that we have in the problem, lattice sites are identical. In other words, our system is homogeneous and isotropic, so all spatial directions are equivalent. It is also a clean system, in the sense that there are no impurities or defects. Maybe at some point it will make sense to relax these conditions, but surely not right at the start. This measures up to having no lattice dependence in any of the terms defined by equations (1.2.16) to (1.2.18): ^ 𝐻 =∑︁ 𝑖 𝜖^𝑏†𝑖^𝑏𝑖− ∑︁ ⟨𝑖𝑗⟩ 𝑡^𝑏†𝑖^𝑏𝑗+ 1 2 ∑︁ 𝑖 𝑈 ^𝑛𝑖(^𝑛𝑖− 1). (1.2.23)
The first term is no more than an overall energy that depends on the total number of particles
𝑁 . If 𝑁 is fixed, it is a constant and it is then irrelevant. However, in the more general case
of the grand-canonical ensemble, where the system is in contact with a particle-reservoir, it can become important since it is possible to attribute a chemical potential 𝜇 that will then control the average number of atoms in the system. We then finally arrive at the Hamiltonian for the so-called single-band standard Bose-Hubbard model:
^ 𝐻 = −𝑡∑︁ ⟨𝑖𝑗⟩ ^ 𝑏†𝑖^𝑏𝑗+ 𝑈 2 ∑︁ 𝑖 ^ 𝑛𝑖(^𝑛𝑖− 1) − 𝜇 ∑︁ 𝑖 ^ 𝑛𝑖. (1.2.24)
Figure 1.5: Cartoon of the interaction term 𝑈 and hopping term 𝑡 of the Bose-Hubbard model. Although atoms appear in two colors in the figure, they are indistinguishable for our purposes. Figure from the OLAQUI website.
1.3
Physical properties of the model
A lot of information about the behavior of a system described by a certain model can be obtained by analyzing possible solutions of the Hamiltonian in regimes where they are somehow easily achieved. Most of the times, for interacting systems, they are either the weakly interacting regime or the strongly interacting regime. For instance, solutions can be obtained by perturbatively treating the appropriate terms in the Hamiltonian. This procedure leads to a better understanding of the physics of the model and also can bring insights on what to expect in non-perturbative regimes. But before that, we can learn a lot by considering the simplest situation that the Bose-Hubbard model could possibly describe: a one-dimensional double-well potential.
1.3.1
Insights from the double-well potential
We consider here a system composed of two lattice sites with a certain number of particles 𝑁 such that the Bose-Hubbard Hamiltonian reads
^ 𝐻 = −𝑡(︁^𝑏†1^𝑏2+ ^𝑏 † 2^𝑏1 )︁ + 𝑈 2𝑛^1(^𝑛1− 1) + 𝑈 2𝑛^2(^𝑛2− 1). (1.3.1) In the non-interacting limit, 𝑈 = 0, ^𝐻 can be diagonalized by defining the operators
^𝑏 + = 1 √ 2 (︁ ^𝑏 1+ ^𝑏2 )︁ (1.3.2) ^𝑏− = √1 2 (︁ ^ 𝑏1− ^𝑏2 )︁ (1.3.3)
