DOCSLIB.ORG

Contents More Information

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Contents More Information Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Table of Contents More information Contents From the Preface to the first edition page xiii Preface to the second edition xvii Part I Introduction 1 1 Basic concepts 3 1.1 What is a quantum phase transition? 3 1.2 Nonzero temperature transitions and crossovers 5 1.3 Experimental examples 8 1.4 Theoretical models 9 1.4.1 Quantum Ising model 10 1.4.2 Quantum rotor model 12 1.4.3 Physical realizations of quantum rotors 14 2 Overview 18 2.1 Quantum field theories 21 2.2 What’s different about quantum transitions? 24 Part II A first course 27 3 Classical phase transitions 29 3.1 Mean-field theory 30 3.2 Landau theory 33 3.3 Fluctuations and perturbation theory 34 3.3.1 Gaussian integrals 36 3.3.2 Expansion for susceptibility 39 Exercises 42 4 The renormalization group 45 4.1 Gaussian theory 46 4.2 Momentum shell RG 48 4.3 Field renormalization 53 4.4 Correlation functions 54 Exercises 56 vii © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Table of Contents More information viii Contents 5 The quantum Ising model 58 5.1 Effective Hamiltonian method 58 5.2 Large-g expansion 59 5.2.1 One-particle states 60 5.2.2 Two-particle states 61 5.3 Small-g expansion 64 5.3.1 d = 264 5.3.2 d = 166 5.4 Review 67 5.5 The classical Ising chain 67 5.5.1 The scaling limit 70 5.5.2 Universality 71 5.5.3 Mapping to a quantum model: Ising spin in a transverse field 72 5.6 Mapping of the quantum Ising chain to a classical Ising model 74 Exercises 77 6 The quantum rotor model 79 6.1 Large-g˜ expansion 79 6.2 Small-g˜ expansion 80 6.3 The classical XY chain and an O(2) quantum rotor 82 6.4 The classical Heisenberg chain and an O(3) quantum rotor 88 6.5 Mapping to classical field theories 89 6.6 Spectrum of quantum field theory 90 6.6.1 Paramagnet 91 6.6.2 Quantum critical point 92 6.6.3 Magnetic order 92 Exercises 95 7 Correlations, susceptibilities, and the quantum critical point 96 7.1 Spectral representation 97 7.1.1 Structure factor 98 7.1.2 Linear response 99 7.2 Correlations across the quantum critical point 101 7.2.1 Paramagnet 101 7.2.2 Quantum critical point 103 7.2.3 Magnetic order 104 Exercises 107 8 Broken symmetries 108 8.1 Discrete symmetry and surface tension 108 8.2 Continuous symmetry and the helicity modulus 110 8.2.1 Order parameter correlations 112 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Table of Contents More information ix Contents 8.3 The London equation and the superfluid density 112 8.3.1 The rotor model 115 Exercises 115 9 Boson Hubbard model 117 9.1 Mean-field theory 119 9.2 Coherent state path integral 123 9.2.1 Boson coherent states 125 9.3 Continuum quantum field theories 126 Exercises 130 Part III Nonzero temperatures 133 10 The Ising chain in a transverse field 135 10.1 Exact spectrum 137 10.2 Continuum theory and scaling transformations 140 10.3 Equal-time correlations of the order parameter 146 10.4 Finite temperature crossovers 149 10.4.1 Low T on the magnetically ordered side, >0, T 151 10.4.2 Low T on the quantum paramagnetic side, <0, T || 157 10.4.3 Continuum high T , T || 162 10.4.4 Summary 168 11 Quantum rotor models: large-N limit 171 11.1 Continuum theory and large-N limit 172 11.2 Zero temperature 174 11.2.1 Quantum paramagnet, g > gc 175 11.2.2 Critical point, g = gc 177 11.2.3 Magnetically ordered ground state, g < gc 178 11.3 Nonzero temperatures 181 11.3.1 Low T on the quantum paramagnetic side, g > gc, T + 186 11.3.2 High T , T +,− 186 11.3.3 Low T on the magnetically ordered side, g < gc, T − 187 11.4 Numerical studies 188 12 The d = 1, O(N ≥ 3) rotor models 190 12.1 Scaling analysis at zero temperature 192 12.2 Low-temperature limit of the continuum theory, T + 193 12.3 High-temperature limit of the continuum theory, + T J 199 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Table of Contents More information x Contents 12.3.1 Field-theoretic renormalization group 201 12.3.2 Computation of χu 205 12.3.3 Dynamics 206 12.4 Summary 211 13 The d = 2, O(N ≥ 3) rotor models 213 13.1 Low T on the magnetically ordered side, T ρs 215 13.1.1 Computation of ξc 216 13.1.2 Computation of τϕ 220 13.1.3 Structure of correlations 222 13.2 Dynamics of the quantum paramagnetic and high-T regions 225 13.2.1 Zero temperature 227 13.2.2 Nonzero temperatures 231 13.3 Summary 234 14 Physics close to and above the upper-critical dimension 237 14.1 Zero temperature 239 14.1.1 Tricritical crossovers 239 14.1.2 Field-theoretic renormalization group 240 14.2 Statics at nonzero temperatures 242 14.2.1 d < 3 244 14.2.2 d > 3 248 14.3 Order parameter dynamics in d = 2 250 14.4 Applications and extensions 257 15 Transport in d = 2 260 15.1 Perturbation theory 264 15.1.1 σI 268 15.1.2 σII 269 15.2 Collisionless transport equations 269 15.3 Collision-dominated transport 273 15.3.1 expansion 273 15.3.2 Large-N limit 279 15.4 Physical interpretation 281 15.5 The AdS/CFT correspondence 283 15.5.1 Exact results for quantum critical transport 285 15.5.2 Implications 288 15.6 Applications and extensions 289 Part IV Other models 291 16 Dilute Fermi and Bose gases 293 16.1 The quantum XX model 296 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Table of Contents More information xi Contents 16.2 The dilute spinless Fermi gas 298 16.2.1 Dilute classical gas, kB T |μ|, μ<0 300 16.2.2 Fermi liquid, kB T μ, μ>0 301 16.2.3 High-T limit, kB T |μ| 304 16.3 The dilute Bose gas 305 16.3.1 d < 2 307 16.3.2 d = 3 310 16.3.3 Correlators of ZB in d = 1 314 16.4 The dilute spinful Fermi gas: the Feshbach resonance 320 16.4.1 The Fermi–Bose model 323 16.4.2 Large-N expansion 327 16.5 Applications and extensions 331 17 Phase transitions of Dirac fermions 332 17.1 d-wave superconductivity and Dirac fermions 332 17.2 Time-reversal symmetry breaking 335 17.3 Field theory and RG analysis 338 17.4 Ising-nematic ordering 342 18 Fermi liquids, and their phase transitions 346 18.1 Fermi liquid theory 347 18.1.1 Independence of choice of k0 354 18.2 Ising-nematic ordering 355 18.2.1 Hertz theory 356 18.2.2 Fate of the fermions 358 18.2.3 Non-Fermi liquid criticality in d = 2 360 18.3 Spin density wave order 363 18.3.1 Mean-field theory 364 18.3.2 Continuum theory 365 18.3.3 Hertz theory 367 18.3.4 Fate of the fermions 368 18.3.5 Critical theory in d = 2 369 18.4 Nonzero temperature crossovers 370 18.5 Applications and extensions 374 19 Heisenberg spins: ferromagnets and antiferromagnets 375 19.1 Coherent state path integral 375 19.2 Quantized ferromagnets 380 19.3 Antiferromagnets 385 19.3.1 Collinear antiferromagnetism and the quantum nonlinear sigma model 385 19.3.2 Collinear antiferromagnetism in d = 1 388 19.3.3 Collinear antiferromagnetism in d = 2 390 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51468-2 - Quantum Phase Transitions: Second Edition Subir Sachdev Table of Contents More information xii Contents 19.3.4 Noncollinear antiferromagnetism in d = 2: deconfined spinons and visons 395 19.3.5 Deconfined criticality 401 19.4 Partial polarization and canted states 403 19.4.1 Quantum paramagnet 405 19.4.2 Quantized ferromagnets 406 19.4.3 Canted and Néel states 406 19.4.4 Zero temperature critical properties 408 19.5 Applications and extensions 410 20 Spin chains: bosonization 412 20.1 The XX chain revisited: bosonization 413 20.2 Phases of H12 423 20.2.1 Sine–Gordon model 425 20.2.2 Tomonaga–Luttinger liquid 428 20.2.3 Valence bond solid order 428 20.2.4 Néel order 431 20.2.5 Models with SU(2) (Heisenberg) symmetry 431 20.2.6 Critical properties near phase boundaries 433 20.3 O(2) rotor model in d = 1 435 20.4 Applications and extensions 436 21 Magnetic ordering transitions of disordered systems 437 21.1 Stability of quantum critical points in disordered systems 438 21.2 Griffiths–McCoy singularities 440 21.3 Perturbative field-theoretic analysis 442 21.4 Metallic systems 445 21.5 Quantum Ising models near the percolation transition 447 21.5.1 Percolation theory 447 21.5.2 Classical dilute Ising models 448 21.5.3 Quantum dilute Ising models 449 21.6 The disordered quantum Ising chain 453 21.7 Discussion 460 21.8 Applications and extensions 461 22 Quantum spin glasses 463 22.1 The effective action 464 22.1.1 Metallic systems 469 22.2 Mean-field theory 470 22.3 Applications and extensions 477 References 479 Index 496 © in this web service Cambridge University Press www.cambridge.org.
Load more

Recommended publications
  • Computing Quantum Phase Transitions Preamble
    COMPUTING QUANTUM PHASE TRANSITIONS Thomas Vojta Department of Physics, University of Missouri-Rolla, Rolla, MO 65409 PREAMBLE: MOTIVATION AND HISTORY A phase transition occurs when the thermodynamic properties of a material display a singularity as a function of the external parameters. Imagine, for instance, taking a piece of ice out of the freezer. Initially, its properties change only slowly with increasing temperature. However, at 0°C, a sudden and dramatic change occurs. The thermal motion of the water molecules becomes so strong that it destroys the crystal structure. The ice melts, and a new phase of water forms, the liquid phase. At the phase transition temperature of 0°C the solid (ice) and the liquid phases of water coexist. A finite amount of heat, the so-called latent heat, is required to transform the ice into liquid water. Phase transitions involving latent heat are called first-order transitions. Another well known example of a phase transition is the ferromagnetic transition of iron. At room temperature, iron is ferromagnetic, i.e., it displays a spontaneous magnetization. With rising temperature, the magnetization decreases continuously due to thermal fluctuations of the spins. At the transition temperature (the so-called Curie point) of 770°C, the magnetization vanishes, and iron is paramagnetic at higher temperatures. In contrast to the previous example, there is no phase coexistence at the transition temperature; the ferromagnetic and paramagnetic phases rather become indistinguishable. Consequently, there is no latent heat. This type of phase transition is called continuous (second-order) transition or critical point. Phase transitions play an essential role in shaping the world.
    [Show full text]
  • Quantum Phase Transitions
    P1: GKW/UKS P2: GKW/UKS QC: GKW CB196-FM July 31, 1999 15:48 Quantum Phase Transitions SUBIR SACHDEV Professor of Physics Yale University iii P1: GKW/UKS P2: GKW/UKS QC: GKW CB196-FM July 31, 1999 15:48 PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK http://www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA http: //www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain C Subir Sachdev 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United States of America Typeface in Times Roman 10/12.5 pt. System LATEX2ε [TB] A catalog record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Sachdev, Subir, 1961– Quantum phase transitions / Subir Sachdev. p. cm. Includes bibliographical references and index. ISBN 0-521-58254-7 1. Phase transformations (Statistical physics) 2. Quantum theory. I. Title. QC175. 16.P5S23 2000 530.4014 – dc21 99-12280 CIP ISBN 0 521 58254 7 hardback iv P1: GKW/UKS P2: GKW/UKS QC: GKW CB196-FM July 31, 1999 15:48 Contents Preface page xi Acknowledgments xv Part I: Introduction 1 1 Basic Concepts 3 1.1 What Is a Quantum Phase Transition?
    [Show full text]
  • Emergent Phenomena in Quantum Critical Systems
    University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations Dissertations and Theses July 2018 Emergent Phenomena in Quantum Critical Systems Kun Chen University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/dissertations_2 Part of the Atomic, Molecular and Optical Physics Commons, Condensed Matter Physics Commons, Other Physics Commons, Quantum Physics Commons, and the Statistical, Nonlinear, and Soft Matter Physics Commons Recommended Citation Chen, Kun, "Emergent Phenomena in Quantum Critical Systems" (2018). Doctoral Dissertations. 1224. https://doi.org/10.7275/11928423.0 https://scholarworks.umass.edu/dissertations_2/1224 This Open Access Dissertation is brought to you for free and open access by the Dissertations and Theses at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. EMERGENT PHENOMENA IN QUANTUM CRITICAL SYSTEMS A Dissertation Presented by KUN CHEN Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY May 2018 Department of Physics c Copyright by Kun Chen 2018 All Rights Reserved EMERGENT PHENOMENA IN QUANTUM CRITICAL SYSTEMS A Dissertation Presented by KUN CHEN Approved as to style and content by: Nikolay Prokof'ev, Co-chair Boris Svistunov, Co-chair Jonathan Machta, Member Murugappan Muthukumar, Member Narayanan Menon, Department Chair Department of Physics ACKNOWLEDGMENTS First and foremost I would like to express my sincere gratitude to my supervisors Prof. Nikolay Prokof'ev and Prof. Boris Svistunov for the continuous support of my Ph.D study and related research, for their patience, motivation, intensive discussion, and immense knowl- edge.
    [Show full text]
  • Quantum Phase Transitions of Correlated Electrons in Two Dimensions: Connections to Cuprate Superconductivity
    Quantum Phase Transitions of Correlated Electrons in Two Dimensions: Connections to Cuprate Superconductivity Vamsi Akkineni Department of Physics, University of Illinois at Urbana-Champaign ABSTRACT The physics of the two dimensional electron gas is important to the high-Tc superconductivity in the cuprates where the behavior of electrons in the CuO2 planes determines the properties of the material. Due to the ¯lling of closely spaced atomic orbitals the electrons are strongly interacting at length scales where quantum e®ects are important. The ground state of such a system could have very di®erent order as a function of some parameter, with these di®erent phases connected by quantum phase transitions. Since the nature of the ground state is a crucial determinant of the behavior at ¯nite temperature, the study of these phases and their excitations are important to high-Tc superconductiv- ity. This paper starts with an introduction to quantum phase transitions including the important quantum-classical mapping described for the quantum rotor model. The coupled ladder an- tiferromagnet model is introduced and the quantum phases and excitations of this model are detailed. Finally, the magnetic tran- sitions in the cuprate superconductors are described and shown to be of the same universal class as those of the coupled ladder antiferromagnet. 1 1 Introduction 1.1 Quantum Phase Transitions Consider a system whose degrees of freedom reside on the sites of an in¯- nite lattice, and which is described by a microscopic Hamiltonian containing two non-commuting operators. The Hamiltonian also involves a continuously variable parameter g that represents the essential tension between the com- peting ordering tendencies of the two non-commuting operators.
    [Show full text]
  • The Tensor-Entanglement Renormalization Group of the 2D Quantum Rotor Model
    The tensor-entanglement renormalization group of the 2D quantum rotor model Rolando La Placa Department of Physics, Harvard University, Cambridge, MA, 02138 (Dated: May 16, 2014) We review the tensor-renormalization group (TRG)[1], and the tensor-entanglement renormaliza- tion group (TERG) [2]. Then, we use TERG to setup a variational optimization of the ground state of the O(2) quantum rotor model in a square lattice. I. INTRODUCTION mean-field theory cannot be applied in systems with long- range entanglement, such as phases found beyond Lan- Tensor-renormalization group (TRG) is a real- dau's symmetry breaking paradigm [2]. In some cases, space renormalization technique used to study any 2- \Tensor Product States"(TPS) are possible trial ground dimensional classical lattice systems with local interac- state wave functions. To define a TPS, consider a system tions [1]. It stems from the idea that any such system with quantum numbers mi 2 1; 2;:::;M residing over a can be rewritten as a tensor-network model. To describe square lattice. A TPS of this square lattice is a tensor-network model, we assign a p-rank tensor T ij::: X to every site of a lattice, where p- is the number of bonds m1 m2 jΨ(fmig)i = TijkmTemfg ::: (2) in each site. Each index fi; j; : : : g is D dimensional, i;j::: and can be seen as residing on an adjacent bond of the site (Fig. 1). For a classical lattice model Hamiltonian mi where Tijkm are complex numbers, fmig are the physical −βH = −βH(i; j; : : : ) with local interactions, we can M-dimensional indexes, and fi; j; : : : g are the indexes find a tensor T such that with bond dimensionality D.
    [Show full text]
  • Zero Temperature Phase Transitions in Quantum
    Zero temperature phase transitions in quantum Heisenberg ferromagnets Subir Sachdev and T. Senthil Department of Physics, P.O.Box 208120, Yale University, New Haven, CT 06520-8120 (February 5, 1996) Abstract The purpose of this work is to understand the zero temperature phases, and the phase transitions, of Heisenberg spin systems which can have an extensive, spontaneous magnetic moment; this entails a study of quantum transitions with an order parameter which is also a non-abelian conserved charge. To this end, we introduce and study a new class of lattice models of quantum rotors. We compute their mean-field phase diagrams, and present continuum, quantum field-theoretic descriptions of their low energy properties in different regimes. We argue that, in spatial dimension d = 1, the phase transitions in itinerant Fermi systems are in the same universality class as the corresponding transitions in certain rotor models. We discuss implications of our results for itinerant fermions systems in higher d, and for other physical systems. arXiv:cond-mat/9602028 6 Feb 96 Typeset using REVTEX 1 I. INTRODUCTION Despite the great deal of attention lavished recently on magnetic quantum critical phe- nomena, relatively little work has been done on systems in which one of the phases has an extensive, spatially averaged, magnetic moment. In fact, the simple Stoner mean-field theory [1] of the zero temperature transition from an unpolarized Fermi liquid to a ferro- magnetic phase is an example of such a study, and is probably also the earliest theory of a quantum phase transition in any system. What makes such phases, and the transitions between them, interesting is that the order parameter is also a conserved charge; in sys- tems with a Heisenberg O(3) symmetry this is expected to lead to strong constraints on the critical field theories [2].
    [Show full text]
  • Quantum Phase Transitions in Disordered Superconductors and Detection of Modulated Superfluidity in Imbalanced Fermi Gases
    Quantum phase transitions in disordered superconductors and detection of modulated superfluidity in imbalanced Fermi gases DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Mason Swanson, B.S. Graduate Program in Physics The Ohio State University 2014 Dissertation Committee: Professor Nandini Trivedi, Advisor Professor Eric Braaten Professor Thomas Lemberger Professor Mohit Randeria c Copyright by Mason Swanson 2014 Abstract Ultracold atomic gas experiments have emerged as a new testing ground for finding elusive, exotic states of matter. One such state that has eluded detection is the Larkin-Ovchinnikov (LO) phase predicted to exist in a system with unequal populations of up and down fermions. This phase is characterized by periodic domain walls across which the order parameter changes sign and the excess polarization is localized Despite fifty years of theoretical and experimental work, there has so far been no unambiguous observation of an LO phase. In this thesis, we propose an experiment in which two fermion clouds, prepared with unequal population imbalances, are allowed to expand and interfere. We show that a pattern of staggered fringes in the interference is unequivocal evidence of LO physics. Finally, we study the superconductor-insulator quantum phase transition. Both super- conductivity and localization stand on the shoulders of giants { the BCS theory of super- conductivity and the Anderson theory of localization. Yet, when their combined effects are considered, both paradigms break down, even for s-wave superconductors. In this work, we calculate the dynamical quantities that help guide present and future experiments.
    [Show full text]
  • Quantum Phase Transitions in Disordered Boson Systems
    University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations Dissertations and Theses October 2018 QUANTUM PHASE TRANSITIONS IN DISORDERED BOSON SYSTEMS Zhiyuan Yao University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/dissertations_2 Part of the Atomic, Molecular and Optical Physics Commons, Condensed Matter Physics Commons, and the Quantum Physics Commons Recommended Citation Yao, Zhiyuan, "QUANTUM PHASE TRANSITIONS IN DISORDERED BOSON SYSTEMS" (2018). Doctoral Dissertations. 1396. https://doi.org/10.7275/12500150 https://scholarworks.umass.edu/dissertations_2/1396 This Open Access Dissertation is brought to you for free and open access by the Dissertations and Theses at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. QUANTUM PHASE TRANSITIONS IN DISORDERED BOSON SYSTEMS A Dissertation Presented by ZHIYUAN YAO Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY September 2018 Department of Physics c Copyright by Zhiyuan Yao 2018 All Rights Reserved QUANTUM PHASE TRANSITIONS IN DISORDERED BOSON SYSTEMS A Dissertation Presented by ZHIYUAN YAO Approved as to style and content by: Nikolay V. Prokof'ev, Chair Boris V. Svistunov, Member Tigran Sedrakyan, Member Murugappan Muthukumar, Member Narayanan Menon, Department Chair Department of Physics ACKNOWLEDGMENTS First and foremost, my deepest appreciation goes to my advisors, Nikolay Prokof'ev and Boris Svistunov. Whenever I went to their office for help, they would immediately stop their work and patiently answer my questions.
    [Show full text]
  • The Quantum Hall Effect: Novel Excitations and Broken Symmetries
    COURSE 2 THE QUANTUM HALL EFFECT: NOVEL EXCITATIONS ANO BROKEN SYMMETRIES S.M. GIRVIN Indiana University, Department of Physics, Bloomington, IN 47405, U.S.A. Contents 1 The quantum Hali effect 55 1.1 Introduction ..... 55 1.2 Why 2D is important . 57 1.3 Constructing the 2DEG 57 1.4 Why is disorder and-localization important? . 58 1.5 Classical dynamics ........... 61 1.6 Semi-classical approximation . 64 1. 7 Quantum Dynamics in Strong B Fields . 65 1.8 IQHE edge states . 72 1.9 Semiclassical percolation picture 76 1.10 Fractional QHE . 80 1.11 The v = 1 many-body state 85 1.12 Neutra! collective excitations 94 1.13 Charged excitations ... 104 1.14 FQHE edge states .... 113 1.15 Quantum hall ferromagnets 116 1.16 Coulomb exchange .. 118 1.17 Spin wave excitations 119 1.18. Effective action .... 124 1.19 Topologica! excitations . 129 1.20 Skyrmion dynamics . 141 1.21 Skyrme lattices . 147 1.22 Double-layer quantum hall ferromagnets 152 1.23 Pseudospin analogy . 154 1.24 Experimental background . 156 1.25 Interlayer phase coherence . 160 1.26 Interlayer tunneling and tilted field effects 162 Appendix 165 A Lowest Landau level projection 165 B Berry's phase and adiabatic transport 168 THE QUANTUM HALL EFFECT: NOVEL EXCITATIONS ANO BROKEN SYMMETRIES S.M. Girvin 1 The quantum Hali effect 1.1 lntroduction The Quantum Hall Effect (QHE) is one of the most remarkable condensed­ matter phenomena discovered in the second half of the 20th century. It rivals superconductivity in its fundamental significance as a manifestation of quantum mechanics on macroscopic scales.
    [Show full text]
  • Monte Carlo Study of Lattice Compact Quantum Electrodynamics with Fermionic Matter: the Parent State of Quantum Phases
    PHYSICAL REVIEW X 9, 021022 (2019) Monte Carlo Study of Lattice Compact Quantum Electrodynamics with Fermionic Matter: The Parent State of Quantum Phases – † ‡ Xiao Yan Xu,1,* Yang Qi,2 4, Long Zhang,5 Fakher F. Assaad,6 Cenke Xu,7 and Zi Yang Meng8,9,10,11, 1Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China 2Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China 3State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China 4Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China 5Kavli Institute for Theoretical Sciences and CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China 6Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074, Würzburg, Germany 7Department of Physics, University of California, Santa Barbara, California 93106, USA 8Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 9Department of Physics, The University of Hong Kong, China 10CAS Center of Excellence in Topological Quantum Computation and School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China 11Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China (Received 1 August 2018; revised manuscript received 19 March 2019; published 2 May 2019) The interplay between lattice gauge theories and fermionic matter accounts for fundamental physical phenomena ranging from the deconfinement of quarks in particle physics to quantum spin liquid with fractionalized anyons and emergent gauge structures in condensed matter physics. However, except for certain limits (for instance, a large number of flavors of matter fields), analytical methods can provide few concrete results.
    [Show full text]
  • Quantum Phases and Phase Transitions in Disordered Low
    Quantum Phases and Phase Transitions In Disordered Low-Dimensional Systems: Thin Film Superconductors, Bilayer Two-dimensional Electron Systems, And One-dimensional Optical Lattices Thesis by Yue Zou In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2011 (Submitted 2010) ii ⃝c 2011 Yue Zou All Rights Reserved iii In memory of my grandfather iv Acknowledgments I would like to express my deep gratitude to my advisor Gil Refael. I always consider myself extremely lucky to have him as my advisor, who is more than an advisor but also my role model and personal good friend. I want to thank him for his amazing insights in physics, his encouragement and inspiration, and his patient guidance and support. I would also like to thank many professors I have interactions with: Jim Eisenstein, Ady Stern, and Jongsoo Yoon, for all the exciting and fruitful collab- orations; Nai-Chang Yeh and Alexei Kitaev, for many insightful conversations and generous help I received from them; Olexei Motrunich and Matthew Fisher, for many eye-opening discussions and comments. I enjoyed very much working closely with Israel Klich and Ryan Barnett on Chapter 2 and 5 of this thesis during their stay at Caltech. I would also like to thank Waheb Bishara, for being my academic big brother and personal good friend. I also had the privilege to discuss and interact with many great scientists here at Caltech: Jason Alicea, Doron Bergman, Gregory Fiete, Karol Gregor, Oleg Kogan, Tami Pereg-Barnea, Heywood Tam, Jing Xia, and Ke Xu, among others.
    [Show full text]
  • Duality in Low Dimensional Quantum Field Theories
    Chapter 13 DUALITY IN LOW DIMENSIONAL QUANTUM FIELD THEORIES Matthew P.A. Fisher Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106–4030 [email protected] Abstract In some strongly correlated electronic materials Landau’s quasiparticle concept appears to break down, suggesting the possibility of new quantum ground states which support particle-like excitations carrying fractional quantum numbers. Theoretical descriptions of such exotic ground states can be greatly aided by the use of duality transforma- tions which exchange the electronic operators for new quantum fields. This chapter gives a brief and self-contained introduction to duality transformations in the simplest possible context - lattice quantum field theories in one and two spatial dimensions with a global Ising or XY symmetry. The duality transformations are expressed as exact oper- ator change of variables performed on simple lattice Hamiltonians. A Hamiltonian version of the Z2 gauge theory approach to electron frac- tionalization is also reviewed. Several experimental systems of current interest for which the ideas of duality might be beneficial are briefly discussed. Keywords: Duality, quantum Ising model, quantum XY model, rotors, Z2 gauge theory, fractionalization 1. Introduction At the heart of quantum mechanics is the wave-particle dualism. Quantum particles such as electrons when detected “are” particles, but exhibit many wavelike characteristics such as diffraction and interfer- ence. In condensed matter physics one is often interested in the collective behavior of 1023 electrons, which must be treated quantum mechanically 419 D. Baeriswyl and L. Degiorgi (eds.), Strong Interactions in Low Dimensions, 419–438. © 2004 by Kluwer Academic Publishers, Printed in the Netherlands.
    [Show full text]