DOCSLIB.ORG

Fermions in Two Dimensions and Exactly Solvable Models

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fermions in Two Dimensions and Exactly Solvable Models Doctoral Thesis Fermions in two dimensions and exactly solvable models Jonas de Woul Mathematical Physics, Department of Theoretical Physics School of Engineering Sciences Royal Institute of Technology, SE-106 91 Stockholm, Sweden Stockholm, Sweden 2011 Typeset in LATEX Akademisk avhandling f¨or avl¨aggande av Teknologie Doktorsexamen (TeknD) inom ¨amnesomr˚adet Teoretisk Fysik. Scientific thesis for the degree of Doctor of Philosophy (PhD) in the subject area of Theoretical Physics. ISBN 978-91-7501-174-5 TRITA-FYS-2011:56 ISSN 0280-316X ISRN KTH/FYS/--11:56--SE c Jonas de Woul, November 2011 Printed in Sweden by Universitetsservice US AB, Stockholm November 2011 Abstract This Ph.D. thesis in mathematical physics concerns systems of interacting fermions with strong correlations. For these systems the physical properties can only be described in terms of the collective behavior of the fermions. Moreover, they are often characterized by a close competition between fermion localization versus de- localization, which can result in complex and exotic physical phenomena. Strongly correlated fermion systems are usually modelled by many-body Hamil- tonians for which the kinetic- and interaction energy have the same order of mag- nitude. This makes them challenging to study as the application of conventional computational methods, like mean field- or perturbation theory, often gives un- reliable results. Of particular interest are Hubbard-type models, which provide minimal descriptions of strongly correlated fermions. The research of this thesis focuses on such models defined on two-dimensional square lattices. One motivation for this is the so-called high-Tc problem of the cuprate superconductors. A main hypothesis is that there exists an underlying Fermi surface with nearly flat parts, i.e. regions where the surface is straight. It is shown that a particular continuum limit of the lattice system leads to an effective model amenable to com- putations. This limit is partial in that it only involves fermion degrees of freedom near the flat parts. The result is an effective quantum field theory that is analyzed using constructive bosonization methods. Various exactly solvable models of inter- acting fermions in two spatial dimensions are also derived and studied. Key words: Bosonization, Exactly solvable models, Hubbard model, Mean field theory, Quantum field theory, Strongly correlated systems. iii iv Preface This thesis is the result of my research in the group of Mathematical Physics at the Department of Theoretical Physics, Royal Institute of Technology (KTH) during the period 2007-2011. The thesis is divided into two parts: the first provides an introduction and some complementary material to the scientific papers; the second part consists of the papers in the order listed below. Appended papers I J. de Woul and E. Langmann Partially gapped fermions in 2D Journal of Statistical Physics 139, 1033 (2010): II J. de Woul and E. Langmann Exact solution of a 2D interacting fermion model arXiv:1011.1401 [math-ph] (submitted) III J. de Woul and E. Langmann Gauge invariance, commutator anomalies, and Meissner effect in 2+1 dimensions arXiv:1107.0891 [cond-mat.str-el] (submitted) IV J. de Woul and E. Langmann Partial continuum limit of the 2D Hubbard model (To be submitted) Published papers not included in the thesis V J. de Woul, J. Hoppe and D. Lundholm Partial Hamiltonian reduction of relativistic extended objects in light-cone gauge Journal of High Energy Physics 1101:031 (2011) VI J. de Woul, J. Hoppe, D. Lundholm and M. Sundin A dynamical symmetry for supermembranes Journal of High Energy Physics 1103:134 (2011) v vi Preface My contribution to the appended papers I All analytical computations were done independently by both authors. I did all numerical computations, with a few checked independently by my co-author. The paper was written in close collaboration. II All calculations were done independently by both authors except those in Section 2.3, which were done by my co-author. All parts of the paper were written in close collaboration except Sections 2.3 and 4.4, which were written by my co-author. III All calculations were done independently by both authors. The paper was written in close collaboration. IV The project plan was developed in collaboration. I did all calculations and wrote the appended paper. Revisions will be done in collaboration. Acknowledgments First of all, I want to acknowledge The G¨oran Gustafsson Foundation for providing the scholarship that has funded my doctoral studies in theoretical physics. Doing research for a living is a privilege, and it gets even better with such great co-workers as I’ve had during my time at the department. Above all, I want to thank my supervisor Professor Edwin Langmann. The patience, support and thoughtfulness of Edwin’s mentoring has truly shaped the kind of professional I am today. Beyond that, he has been an invaluable resource for support and advice on matters far beyond science; I proudly call him friend. I also want to thank my other collaborators: to Professor Jens Hoppe for broadening my research interests well beyond lattice fermions, and Martin S. and Douglas whose skills in mathematics I can only dream of. Thanks to all my current and former colleagues in the mathematical physics group: former graduate students Martin H. and Pedram who I shared an office with for many years and who helped me out a lot when I first started, and new student Farrokh who will hopefully find this thesis of some use in his future doctoral studies; to fellow group members Jouko, Teresia, Michael, and all the diploma students and guests who I’ve shared an office with over the years (I don’t dare start naming you all in fear of forgetting someone). Thanks also to the people in the particle physics group for always including me “as one of their own”: to Professor Tommy Ohlsson who has always been very considerate to me; to former members Mattias, Tomas, Thomas, Michal and He; to fellow students Henrik, Johannes and Sofia for always making work a fun place to be. I would not have been able to finish this thesis without the love, support and encouragement from family and friends. My deepest thanks in particular to my sister Sara and brother Mattias, Margaretha, Minna, and Ronny. Contents Abstract.................................... iii Preface v Contents vii I Backgroundandcomplementarymaterial 3 1 Introduction 5 2 Cuprate superconductors 9 2.1 The high Tc problem.......................... 9 2.2 The cuprate x-T phasediagram.................... 12 2.3 Three-band description of the Cu-O layers . 15 3 The Hubbard model 19 3.1 TruncationofamultibandHamiltonian . 20 3.2 Definitionofthemodel......................... 22 3.3 Someexactresults ........................... 24 3.4 Hartree-Focktheory .......................... 27 3.5 Continuumlimitof1Dlatticeelectrons . 31 4 Boson-fermion correspondence 41 4.1 Freerelativisticfermions. 41 4.2 Representations of some -dimensionalLiealgebras . 48 ∞ 4.3 BosonizationandFermionization . 51 4.4 Applications to exactly solvable QFT models . 54 4.5 Bosonizationinhigherdimensions . 60 vii viii Contents 5 Introduction to the papers 63 5.1 Acontinuumlimitof2Dlatticefermions. 63 5.2 Theexactlysolvablenodalmodel . 72 5.3 Couplingnodalfermionstogaugefields . 72 5.4 Generalizingtospinfullfermions . 73 6 Conclusions 75 Bibliography 77 II Scientific papers 95 To my parents Part I Background and complementary material 3 4 Chapter 1 Introduction The number of mobile electrons in a 1 cm3 crystal of copper is about 1023, i.e. a number that can be written as a one followed by twenty-three zeros. To put this abstract number in some perspective, this is roughly the same as the number of sand grains in the Sahara desert if we dig ten meters into the ground!1 Now imagine all these electrons roaming around inside the crystal, bouncing off the atoms and colliding with each other. How could we ever make sense of such an overwhelmingly complex system? One might think the solution to this problem is not that difficult, given that the present theory of everyday matter was written down more than 80 years ago (in the early days of quantum mechanics). Simply take the mathematical equations that describe the electrons and solve them, possibly on a computer. Well, the problem is that, even with the most state-of-the-art supercomputers available to us, we cannot solve these equations for more than a few sand grains on a finger tip. This is where the art of mathematical modelling comes in. If we forego the ambition of trying to solve “everything at once”, we can write down simplified models that describe the properties of some very particular type of system: metals, insulators, magnets, semiconductors, superconductors, etc. What constitutes a good model is of course delicate; it must be simple enough so that we can actually solve the corresponding mathematical equations, while at the same time not too simple as to lose the essential physics. Finding the best compromise to this lies at the very core of theoretical physics. Adopting this philosophy has been highly fruitful in the study of many-electron systems. A nice example is Landau’s Fermi liquid theory of weakly interacting quasiparticles [1–3], applicable for example to conduction electrons in ordinary metals at sufficiently low temperatures. We know that electrons interact strongly 1This analogy is based on the following: The number of conduction electrons in Cu equals the number of ions, the mass density of Cu is 8.96 g cm−3, Cu weighs 63.5 g/mol, each grain is approximately 1 mm3, and the area of the Sahara desert is about 107 km2. 5 6 Chapter 1. Introduction through mutual Coulomb repulsions. Yet, surprisingly, the experimental proper- ties of many metals are largely consistent with those of non-interacting particles. The essential point in Fermi liquid theory is that these particles are not the bare electrons, but so-called quasiparticles that are “dressed up” by interactions.
Load more

Recommended publications
  • First-Principles Calculations and Model Hamiltonian Approaches to Electronic and Optical Properties of Defects, Interfaces and Nanostructures
    First-principles calculations and model Hamiltonian approaches to electronic and optical properties of defects, interfaces and nanostructures by Sangkook Choi A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Grauduate Division of the University of California, Berkeley Committee in charge: Professor Steven G. Louie, Chair Professor John Clarke Professor Mark Asta Fall 2013 First-principles calculations and model Hamiltonian approaches to electronic and optical properties of defects, interfaces and nanostructures Copyright 2013 by Sangkook Choi Abstract First principles calculations and model Hamiltonian approaches to electronic and optical properties of defects, interfaces and nanostructures By Sangkook Choi Doctor of Philosophy in Physics University of California, Berkeley Professor Steven G. Louie, Chair The dynamics of electrons governed by the Coulomb interaction determines a large portion of the observed phenomena of condensed matter. Thus, the understanding of electronic structure has played a key role in predicting the electronic and optical properties of materials. In this dissertation, I present some important applications of electronic structure theories for the theoretical calculation of these properties. In the first chapter, I review the basics necessary for two complementary electronic structure theories: model Hamiltonian approaches and first principles calculation. In the subsequent chapters, I further discuss the applications of these approaches to nanostructures (chapter II), interfaces (chapter III), and defects (chapter IV). The abstract of each section is as follows. ● Section II-1 The sensitive structural dependence of the optical properties of single-walled carbon nanotubes, which are dominated by excitons and tunable by changing diameter and chirality, makes them excellent candidates for optical devices.
    [Show full text]
  • Bosonization and Mirror Symmetry Arxiv:1608.05077V1 [Hep-Th]
    Bosonization and Mirror Symmetry Shamit Kachru1, Michael Mulligan2,1, Gonzalo Torroba3, and Huajia Wang4 1Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA 2Department of Physics and Astronomy, University of California, Riverside, CA 92511, USA 3Centro At´omico Bariloche and CONICET, R8402AGP Bariloche, ARG 4Department of Physics, University of Illinois, Urbana-Champaign, IL 61801, USA Abstract We study bosonization in 2+1 dimensions using mirror symmetry, a duality that relates pairs of supersymmetric theories. Upon breaking supersymmetry in a controlled way, we dynamically obtain the bosonization duality that equates the theory of a free Dirac fermion to QED3 with a single scalar boson. This duality may be used to demonstrate the bosonization duality relating an O(2)-symmetric Wilson-Fisher fixed point to QED3 with a single Dirac fermion, Peskin-Dasgupta-Halperin duality, and the recently conjectured duality relating the theory of a free Dirac fermion to fermionic QED3 with a single flavor. Chern-Simons and BF couplings for both dynamical and background gauge fields play a central role in our approach. In the course of our study, we describe a \chiral" mirror pair that may be viewed as the minimal supersymmetric generalization of the two bosonization dualities. arXiv:1608.05077v1 [hep-th] 17 Aug 2016 Contents 1 Introduction 1 2 Mirror symmetry and its deformations 4 2.1 Superfields and lagrangians . .5 2.2 = 4 mirror symmetry . .7 N 2.3 Deformations by U(1)A and U(1)R backgrounds . 10 2.4 General mirror duality . 11 3 Chiral mirror symmetry 12 3.1 Chiral theory A .
    [Show full text]
  • 8.7 Hubbard Model
    OUP CORRECTED PROOF – FINAL, 12/6/2019, SPi 356 Models of Strongly Interacting Quantum Matter which is the lower edge of a continuum of excitations whose upper edge is bounded by ω(q) πJ cos (q/2). (8.544) = The continuum of excitations develops because spinons are always created in pairs, and therefore the momentum of the two spinons can be distributed in a continuum of different ways. Neutron scattering experiments on quasi-one-dimensional materials like KCuF3 have corroborated the picture outlined here (see, e.g. Tennant et al.,1995). In dimensions higher than one, separating a !ipped spin into a pair of kinks, or a magnon into a pair of spinons, costs energy, which thus con"nes spinons in dimensions d 2. ≥The next section we constructs the Hubbard model from "rst principles and then show how the Heisenberg model can be obtained from the Hubbard model for half- "lling and in the limit of strong on-site repulsion. 8.7 Hubbard model The Hubbard model presents one of the simplest ways to obtain an understanding of the mechanisms through which interactions between electrons in a solid can give rise to insulating versus conducting, magnetic, and even novel superconducting behaviour. The preceding sections of this chapter more or less neglected these interaction or correlation effects between the electrons in a solid, or treated them summarily in a mean-"eld or quasiparticle approach (cf. sections 8.2 to 8.5). While the Hubbard model was "rst discussed in quantum chemistry in the early 1950s (Pariser and Parr, 1953; Pople, 1953), it was introduced in its modern form and used to investigate condensed matter problems in the 1960s independently by Gutzwiller (1963), Hubbard (1963), and Kanamori (1963).
    [Show full text]
  • The Limits of the Hubbard Model
    The Limits of the Hubbard Model David Grabovsky May 24, 2019 1 CONTENTS CONTENTS Contents 1 Introduction and Abstract 4 I The Hubbard Hamiltonian 5 2 Preliminaries 6 3 Derivation of the Hubbard Model 9 3.1 Obtaining the Hamiltonian . .9 3.2 Discussion: Energy Scales . 10 4 Variants and Conventions 12 5 Symmetries of the Hubbard Model 14 5.1 Discrete Symmetries . 14 5.2 Gauge Symmetries . 14 5.3 Particle-Hole Symmetry . 15 II Interaction Limits 17 6 The Fermi-Hubbard Model 18 6.1 Single Site: Strong Interactions . 18 6.2 Tight Binding: Weak Interactions . 20 7 The Bose-Hubbard Model 22 7.1 Single Site: Strong Interactions . 22 7.2 Tight Binding: Weak Interactions . 24 III Lattice Limits 25 8 Two Sites: Fermi-Hubbard 26 8.1 One Electron . 26 8.2 Two Electrons . 27 8.3 Three Electrons . 28 8.4 Complete Solution . 29 9 Two Sites: Bose-Hubbard 33 9.1 One and Two Bosons . 33 9.2 Many Bosons . 35 9.3 High-Temperature Limit . 36 10 Conclusions and Outlook 38 2 CONTENTS CONTENTS IV Appendices 40 A The Hubbard Model from Many-Body Theory 41 B Some Algebraic Results 42 C Fermi-Hubbard Limits 43 C.1 No Hopping . 43 C.2 No Interactions . 44 D Bose-Hubbard Limits 44 D.1 No Hopping . 44 D.2 No Interactions . 46 E Two-Site Thermodynamics 47 E.1 Fermi-Hubbard Model . 47 E.2 Bose-Hubbard Model . 49 F Tridiagonal Matrices and Recurrences 50 3 1 INTRODUCTION AND ABSTRACT 1 Introduction and Abstract The Hubbard model was originally written down in the 1960's as an attempt to describe the behavior of electrons in a solid [1].
    [Show full text]
  • Physics 215C: Particles and Fields Spring 2019
    Physics 215C: Particles and Fields Spring 2019 Lecturer: McGreevy These lecture notes live here. Please email corrections to mcgreevy at physics dot ucsd dot edu. Last updated: 2021/04/20, 14:28:40 1 Contents 0.1 Introductory remarks for the third quarter................4 0.2 Sources and acknowledgement.......................7 0.3 Conventions.................................8 1 Anomalies9 2 Effective field theory 20 2.1 A parable on integrating out degrees of freedom............. 20 2.2 Introduction to effective field theory.................... 25 2.3 The color of the sky............................. 30 2.4 Fermi theory of Weak Interactions..................... 32 2.5 Loops in EFT................................ 33 2.6 The Standard Model as an EFT...................... 39 2.7 Superconductors.............................. 42 2.8 Effective field theory of Fermi surfaces.................. 47 3 Geometric and topological terms in field theory actions 58 3.1 Coherent state path integrals for bosons................. 58 3.2 Coherent state path integral for fermions................. 66 3.3 Path integrals for spin systems....................... 73 3.4 Topological terms from integrating out fermions............. 86 3.5 Pions..................................... 89 4 Field theory of spin systems 99 4.1 Transverse-Field Ising Model........................ 99 4.2 Ferromagnets and antiferromagnets..................... 131 4.3 The beta function for 2d non-linear sigma models............ 136 4.4 CP1 representation and large-N ...................... 138 5 Duality 148 5.1 XY transition from superfluid to Mott insulator, and T-duality..... 148 6 Conformal field theory 158 6.1 The stress tensor and conformal invariance (abstract CFT)....... 160 6.2 Radial quantization............................. 166 6.3 Back to general dimensions......................... 172 7 Duality, part 2 179 7.1 (2+1)-d XY is dual to (2+1)d electrodynamics.............
    [Show full text]
  • Arxiv:2007.01582V2 [Quant-Ph] 15 Jul 2020
    Preparing symmetry broken ground states with variational quantum algorithms Nicolas Vogt,∗ Sebastian Zanker, Jan-Michael Reiner, and Michael Marthaler HQS Quantum Simulations GmbH Haid-und-Neu-Straße 7 76131 Karlsruhe, Germany Thomas Eckl and Anika Marusczyk Robert Bosch GmbH Robert-Bosch-Campus 1 71272 Renningen, Germany (Dated: July 16, 2020) One of the most promising applications for near term quantum computers is the simulation of physical quantum systems, particularly many-electron systems in chemistry and condensed matter physics. In solid state physics, finding the correct symmetry broken ground state of an interacting electron system is one of the central challenges. The Variational Hamiltonian Ansatz (VHA), a variational hybrid quantum-classical algorithm especially suited for finding the ground state of a solid state system, will in general not prepare a broken symmetry state unless the initial state is chosen to exhibit the correct symmetry. In this work, we discuss three variations of the VHA designed to find the correct broken symmetry states close to a transition point between different orders. As a test case we use the two-dimensional Hubbard model where we break the symmetry explicitly by means of external fields coupling to the Hamiltonian and calculate the response to these fields. For the calculation we simulate a gate-based quantum computer and also consider the effects of dephasing noise on the algorithms. We find that two of the three algorithms are in good agreement with the exact solution for the considered parameter range. The third algorithm agrees with the exact solution only for a part of the parameter regime, but is more robust with respect to dephasing compared to the other two algorithms.
    [Show full text]
  • Mott Transition in the Π-Flux SU(4) Hubbard Model on a Square Lattice
    PHYSICAL REVIEW B 97, 195122 (2018) Mott transition in the π-flux SU(4) Hubbard model on a square lattice Zhichao Zhou,1,2 Congjun Wu,2 and Yu Wang1,* 1School of Physics and Technology, Wuhan University, Wuhan 430072, China 2Department of Physics, University of California, San Diego, California 92093, USA (Received 23 March 2018; published 14 May 2018) With increasing repulsive interaction, we show that a Mott transition occurs from the semimetal to the valence bond solid, accompanied by the Z4 discrete symmetry breaking. Our simulations demonstrate the existence of a second-order phase transition, which confirms the Ginzburg-Landau analysis. The phase transition point and the critical exponent η are also estimated. To account for the effect of a π flux on the ordering in the strong-coupling regime, we analytically derive by the perturbation theory the ring-exchange term, which is the leading-order term that can reflect the difference between the π-flux and zero-flux SU(4) Hubbard models. DOI: 10.1103/PhysRevB.97.195122 I. INTRODUCTION dimer orders exist in a projector QMC (PQMC) simulation [30]. With the rapid development of ultracold atom experiments, Considering the significance of density fluctuations in a the synthetic gauge field can be implemented in optical lattice realistic fermionic system, the SU(2N) Hubbard model is a systems [1–4]. Recently, the “Hofstadter butterfly” model prototype model for studying the interplay between density Hamiltonian has also been achieved with ultracold atoms of and spin degrees of freedom. Previous PQMC studies of the 87Rb [5,6]. When ultracold atoms are considered as carriers half-filled SU(2) Hubbard model with a π flux have demon- in optical lattices, they can carry large hyperfine spins.
    [Show full text]
  • Arxiv:1912.06007V3 [Quant-Ph] 30 Nov 2020 Iσ Jσ Google of a Quantum Computation Outperforming a Classical Hi,Ji,Σ I Supercomputer Contained 430 Two-Qubit Gates [8]
    Strategies for solving the Fermi-Hubbard model on near-term quantum computers Chris Cade,1, ∗ Lana Mineh,1, 2, 3 Ashley Montanaro,1, 2 and Stasja Stanisic1 1Phasecraft Ltd. 2School of Mathematics, University of Bristol 3Quantum Engineering Centre for Doctoral Training, University of Bristol (Dated: December 1, 2020) The Fermi-Hubbard model is of fundamental importance in condensed-matter physics, yet is extremely chal- lenging to solve numerically. Finding the ground state of the Hubbard model using variational methods has been predicted to be one of the first applications of near-term quantum computers. Here we carry out a detailed analysis and optimisation of the complexity of variational quantum algorithms for finding the ground state of the Hubbard model, including costs associated with mapping to a real-world hardware platform. The depth com- plexities we find are substantially lower than previous work. We performed extensive numerical experiments for systems with up to 12 sites. The results suggest that the variational ansatze¨ we used – an efficient variant of the Hamiltonian Variational ansatz and a novel generalisation thereof – will be able to find the ground state of the Hubbard model with high fidelity in relatively low quantum circuit depth. Our experiments include the effect of realistic measurements and depolarising noise. If our numerical results on small lattice sizes are representative of the somewhat larger lattices accessible to near-term quantum hardware, they suggest that optimising over quantum circuits with a gate depth less than a thousand could be sufficient to solve instances of the Hubbard model beyond the capacity of classical exact diagonalisation.
    [Show full text]
  • View Publication
    Solving strongly correlated electron models on a quantum computer Dave Wecker,1 Matthew B. Hastings,2, 1 Nathan Wiebe,1 Bryan K. Clark,2, 3 Chetan Nayak,2 and Matthias Troyer4 1Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA 2Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA 3UIUC 4Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland One of the main applications of future quantum computers will be the simulation of quantum mod- els. While the evolution of a quantum state under a Hamiltonian is straightforward (if sometimes expensive), using quantum computers to determine the ground state phase diagram of a quantum model and the properties of its phases is more involved. Using the Hubbard model as a prototypical example, we here show all the steps necessary to determine its phase diagram and ground state properties on a quantum computer. In particular, we discuss strategies for efficiently determining and preparing the ground state of the Hubbard model starting from various mean-field states with broken symmetry. We present an efficient procedure to prepare arbitrary Slater determinants as initial states and present the complete set of quantum circuits needed to evolve from these to the ground state of the Hubbard model. We show that, using efficient nesting of the various terms each time step in the evolution can be performed with just O(N) gates and O(log N) circuit depth. We give explicit circuits to measure arbitrary local observables and static and dynamic correlation functions, both in the time and frequency domain. We further present efficient non-destructive ap- proaches to measurement that avoid the need to re-prepare the ground state after each measurement and that quadratically reduce the measurement error.
    [Show full text]
  • Arxiv:2101.06032V3 [Quant-Ph] 26 May 2021 Quantum Ground States
    The phases of the disordered Bose–Hubbard model with attractive interactions Olli Mansikkam¨aki,Sami Laine, and Matti Silveri Nano and Molecular Systems Research Unit, University of Oulu, P.O. Box 3000, FI-90014 University of Oulu, Finland (Dated: May 27, 2021) We study the quantum ground state phases of the one-dimensional disordered Bose–Hubbard model with attractive interactions, realized by a chain of superconducting transmon qubits or cold atoms. We map the phase diagram using perturbation theory and exact diagonalization. Compared to the repulsive Bose–Hubbard model, the quantum ground state behavior is dramatically different. At strong disorder of the on-site energies, all the bosons localize into the vicinity of a single site, contrary to the Bose glass behavior of the repulsive model. At weak disorder, depending on hopping, the ground state is either superfluid or a W state, which is a multi-site and multi-particle entangled superposition of states where all the bosons occupy a single site. We show that the robustness of the W phase against disorder diminishes as the total number of bosons increases. Introduction The Bose–Hubbard model is a paradig- Bose–Hubbard model is immediately applicable also for matic model of quantum matter and quantum phase tran- cold atoms in optical lattices, where the interaction can sitions, with applications ranging from magnetism to be tuned from repulsive to attractive via the Feshbach disordered superfluid helium [1–4]. It is canonically char- resonance [4, 39]. acterized by a repulsive boson-boson interaction disfavor- In this letter, we use exact diagonalization and perturba- ing local multi-occupancy, together with boson hopping tion theory to construct the ground state phase diagram of which models excitation kinetics.
    [Show full text]
  • Thermodynamics of the 3D Hubbard Model on Approaching the Nйel
    week ending PRL 106, 030401 (2011) PHYSICAL REVIEW LETTERS 21 JANUARY 2011 Thermodynamics of the 3D Hubbard Model on Approaching the Ne´el Transition Sebastian Fuchs,1 Emanuel Gull,2 Lode Pollet,3 Evgeni Burovski,4,5 Evgeny Kozik,3 Thomas Pruschke,1 and Matthias Troyer3 1Institut fu¨r Theoretische Physik, Georg-August-Universita¨tGo¨ttingen, 37077 Go¨ttingen, Germany 2Department of Physics, Columbia University, New York, New York 10027, USA 3Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland 4LPTMS, CNRS and Universite´ Paris-Sud, UMR8626, Baˆtiment 100, 91405 Orsay, France 5Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom (Received 24 September 2010; published 18 January 2011) We study the thermodynamic properties of the 3D Hubbard model for temperatures down to the Ne´el temperature by using cluster dynamical mean-field theory. In particular, we calculate the energy, entropy, density, double occupancy, and nearest-neighbor spin correlations as a function of chemical potential, temperature, and repulsion strength. To make contact with cold-gas experiments, we also compute properties of the system subject to an external trap in the local density approximation. We find that an entropy per particle S=N 0:65ð6Þ at U=t ¼ 8 is sufficient to achieve a Ne´el state in the center of the trap, substantially higher than the entropy required in a homogeneous system. Precursors to antiferromagnetism can clearly be observed in nearest-neighbor spin correlators. DOI: 10.1103/PhysRevLett.106.030401 PACS numbers: 05.30.Fk, 03.75.Ss, 71.10.Fd The Hubbard model remains one of the cornerstone the entropy, energy, density, double occupancy, and spin models in condensed matter physics, capturing the essence correlations—for interactions U up to the bandwidth 12t of strongly correlated electron physics relevant to high- on approach to the Ne´el temperature TN by performing temperature superconductors [1] and correlation-driven controlled large-scale cluster dynamical mean-field calcu- insulators [2].
    [Show full text]
  • Thermodynamics of the Metal-Insulator Transition in the Extended Hubbard Model
    SciPost Phys. 6, 067 (2019) Thermodynamics of the metal-insulator transition in the extended Hubbard model Malte Schüler1,2?, Erik G. C. P.van Loon1,2, Mikhail I. Katsnelson3 and Tim O. Wehling1,2 1 Institut für Theoretische Physik, Universität Bremen, Otto-Hahn-Allee 1, 28359 Bremen, Germany 2 Bremen Center for Computational Materials Science, Universität Bremen, Am Fallturm 1a, 28359 Bremen, Germany 3 Radboud University, Institute for Molecules and Materials, Heyendaalseweg 135, NL-6525 AJ Nijmegen, The Netherlands ? [email protected] Abstract In contrast to the Hubbard model, the extended Hubbard model, which additionally ac- counts for non-local interactions, lacks systemic studies of thermodynamic properties especially across the metal-insulator transition. Using a variational principle, we per- form such a systematic study and describe how non-local interactions screen local cor- relations differently in the Fermi-liquid and in the insulator. The thermodynamics reveal that non-local interactions are at least in parts responsible for first-order metal-insulator transitions in real materials. Copyright M. Schüler et al. Received 09-04-2019 This work is licensed under the Creative Commons Accepted 06-06-2019 Check for Attribution 4.0 International License. Published 14-06-2019 updates Published by the SciPost Foundation. doi:10.21468/SciPostPhys.6.6.067 Contents 1 Introduction1 2 Methods 3 2.1 Extended Hubbard model3 2.2 Variational principle3 3 Results 4 3.1 Double occupation and Entropy4 3.2 Mechanism of screening local correlations.6 3.3 Critical non-local interaction8 3.4 Comparison to experimentally observed first-order transitions8 4 Conclusion 9 A Calculation of the free energy and entropy 10 1 SciPost Phys.
    [Show full text]