UnderstandingQuantum PhaseTransitions K110133_FM.indd 1 9/13/10 1:28:15 PM 2011 by Taylor and Francis Group, LLCSeries in Condensed Matter PhysicsSeries Editor:D R VijDepartment of Physics, Kurukshetra University, IndiaOther titles in the series include:Magnetic Anisotropies in Nanostructured MatterPeter WeinbergerAperiodic Structures in Condensed Matter: Fundamentals and Applications Enrique Maci BarberThermodynamics of the Glassy State Luca Leuzzi, Theo M NieuwenhuizenOne- and Two-Dimensional Fluids: Properties of Smectic, Lamellar and Columnar Liquid Crystals A Jkli, A SaupeTheory of Superconductivity: From Weak to Strong Coupling A S AlexandrovThe Magnetocaloric Effect and Its Applications A M Tishin, Y I SpichkinField Theories in Condensed Matter Physics Sumathi RaoNonlinear Dynamics and Chaos in Semiconductors K AokiPermanent Magnetism R Skomski, J M D CoeyModern Magnetooptics and Magnetooptical Materials A K Zvezdin, V A KotovSeries in Condensed Matter PhysicsLincoln D. CarrA TAYLOR & FRANCI S BOOKCRC Press is an imprint of theTaylor & Francis Group, an informa businessBoca Raton London New YorkUnderstandingQuantum PhaseTransitionsK110133_FM.indd 2 9/13/10 1:28:15 PM 2011 by Taylor and Francis Group, LLCSeries in Condensed Matter PhysicsSeries Editor:D R VijDepartment of Physics, Kurukshetra University, IndiaOther titles in the series include:Magnetic Anisotropies in Nanostructured MatterPeter WeinbergerAperiodic Structures in Condensed Matter: Fundamentals and Applications Enrique Maci BarberThermodynamics of the Glassy State Luca Leuzzi, Theo M NieuwenhuizenOne- and Two-Dimensional Fluids: Properties of Smectic, Lamellar and Columnar Liquid Crystals A Jkli, A SaupeTheory of Superconductivity: From Weak to Strong Coupling A S AlexandrovThe Magnetocaloric Effect and Its Applications A M Tishin, Y I SpichkinField Theories in Condensed Matter Physics Sumathi RaoNonlinear Dynamics and Chaos in Semiconductors K AokiPermanent Magnetism R Skomski, J M D CoeyModern Magnetooptics and Magnetooptical Materials A K Zvezdin, V A KotovSeries in Condensed Matter PhysicsLincoln D. CarrA TAYLOR & FRANCI S BOOKCRC Press is an imprint of theTaylor & Francis Group, an informa businessBoca Raton London New YorkUnderstandingQuantum PhaseTransitionsK110133_FM.indd 3 9/13/10 1:28:15 PM 2011 by Taylor and Francis Group, LLCCRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742 2011 by Taylor and Francis Group, LLCCRC Press is an imprint of Taylor & Francis Group, an Informa businessNo claim to original U.S. Government worksPrinted in the United States of America on acid-free paper10 9 8 7 6 5 4 3 2 1International Standard Book Number: 978-1-4398-0251-9 (Hardback)This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, includ-ing photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.Library of Congress CataloginginPublication DataUnderstanding quantum phase transitions / [edited by] Lincoln Carr.p. cm. -- (Condensed matter physics)Summary: Exploring a steadily growing field, this book focuses on quantum phase transitions (QPT), frontier area of research. It takes a look back as well as a look forward to the future and the many open problems that remain. The book covers new concepts and directions in QPT and specific models and systems closely tied to particular experimental realization or theoretical methods. Although mainly theoretical, the book includes experimental chapters that make the discussion of QPTs meaningful. The book also presents recent advances in the numerical methods used to study QPTs-- Provided by publisher.Includes bibliographical references and index.ISBN 978-1-4398-0251-9 (hardback)1. Phase transformations (Statistical physics) 2. Transport theory. 3. Quantum statistics. I. Carr, Lincoln. II. Title. III. Series.QC175.16.P5U53 2010530.474--dc22 2010034921Visit the Taylor & Francis Web site athttp://www.taylorandfrancis.comand the CRC Press Web site athttp://www.crcpress.com K110133_FM.indd 4 9/13/10 1:28:15 PM 2011 by Taylor and Francis Group, LLCDedicationTo Badia, Samuel, and HalimFor their patience and loveAnd to the three magical childrenWho appeared in my life as I completed this bookAhmed, Oumaima, and Yassmina 2011 by Taylor and Francis Group, LLCContributorsSami AmashaStanford University, U.S.A.George G. BatrouniUniversite de Nice - SophiaAntipolis, FranceImmanuel BlochLudwig-Maximilians-Universit at,GermanyMark A. CaprioUniversity of Notre Dame, U.S.A.Lincoln D. CarrColorado School of Mines, U.S.A.Claudio CastelnovoOxford University, U.K.Sudip ChakravartyUniversity of California Los Angeles,U.S.A.Ignacio CiracMax-Planck-Institut f urQuantenoptik, GermanyJ.C. DavisCornell University, U.S.A.Brookhaven National Laboratory,U.S.A.University of St. Andrews, ScotlandPhilipp GegenwartUniversity of G ottingen, GermanyThierry GiamarchiUniversity of Geneva, SwitzerlandDavid Goldhaber-GordonStanford University, U.S.A.Andrew D. GreentreeUniversity of Melbourne, AustraliaVladimir GritsevUniversity of Fribourg, SwitzerlandSean HartnollHarvard University, U.S.A.Tetsuo HatsudaUniversity of Tokyo, JapanLloyd C. L. HollenbergUniversity of Melbourne, AustraliaFrancesco IachelloYale University, U.S.A.Tetsuaki ItouKyoto University, JapanRina KanamotoOchanomizu University, JapanReizo KatoRIKEN, JapanYuki KawaguchiUniversity of Tokyo, Japanvii 2011 by Taylor and Francis Group, LLCviiiEun-Ah KimCornell University, U.S.A.Sergey KravchenkoNortheastern University, U.S.A.Michael J. LawlerThe State University of New York atBinghamton, U.S.A.Cornell University, U.S.A.Karyn Le HurYale University, U.S.A.Kenji MaedaThe University of Tokyo, JapanAndrew J. MillisColumbia University, U.S.A.Valentin MurgMax-Planck-Institut f urQuantenoptik, GermanyYuval OregWeizmann Institute of Science, IsraelGerardo OrtizIndiana University, U.S.A.Masaki OshikawaUniversity of Tokyo, JapanAnatoli PolkovnikovBoston University, U.S.A.Nikolay Prokof evUniversity of Massachusetts,Amherst, U.S.A.Ileana G. RauStanford University, U.S.A.Subir SachdevHarvard University, U.S.A.Richard T. ScalettarUniversity of California, Davis,U.S.A.Ulrich SchollwockUniversity of Munich, GermanyAlexander ShashkinInstitute of Solid State Physics,RussiaQimiao SiRice University, U.S.A.Frank SteglichMax Planck Institute for ChemicalPhysics of Solids, GermanyBoris SvistunovUniversity of Massachusetts,Amherst, U.S.A.Simon TrebstUniversity of California, SantaBarbara, U.S.A.Matthias TroyerETH Zurich, SwitzerlandMasahito UedaUniversity of Tokyo, JapanFrank VerstraeteUniversit at Wien, AustriaGuifre VidalThe University of Queensland,AustraliaPhilipp WernerETH Zurich, Switzerland 2011 by Taylor and Francis Group, LLCEditorLincoln D. Carr is a the-oretical physicist who worksprimarily in quantum many-body theory, articial mate-rials, and nonlinear dynam-ics. He obtained his B.A.in physics at the Univer-sity of California, Berkeleyin 1994. He attended theUniversity of Washington inSeattle from 1996 to 2001,where he received both hisM.S. and Ph.D. in physics.He was a Distinguished In-ternational Fellow of the Na-tional Science Foundation from 2001-2004 at the Ecole normale superieure inParis and a professional research associate at JILA in Boulder, Colorado from2003-2005. He joined the faculty in the physics department at the ColoradoSchool of Mines in 2005, where he is presently an associate professor. He isan Associate of the National Institute of Standards and Technology and hasbeen a visiting researcher at the Max Planck Institute for the Physics of Com-plex Systems in Dresden, Germany, the Kavli Institute of Theoretical Physicsin Santa Barbara, California, the Institute Henri Poincare at the UniversitePierre et Marie Curie in Paris, and the Kirchho Institute for Physics at theUniversity of Heidelberg.ix 2011 by Taylor and Francis Group, LLCPrefacePhase transitions occur in all elds of the physical sciences and are crucialin engineering as well; abrupt changes from one state of matter to anotherare apparent everywhere we look, from the freezing of rivers to the steamrising up from the tea kettle. But why should it be only temperature andpressure that drive such abrupt transitions? In fact, quantum uctuationscan replace thermal uctuations, a phase transition can occur even at zerotemperature, and the concept of a phase transition turns out to be a lot moregeneral than it is made out to be in elementary thermodynamics. Over thelast twenty or so years the eld of quantum phase transitions (QPTs) hasseen steady growth. This book focuses especially on the latter half of thisdevelopment. There are now so many experimental examples of QPTs thatwe hardly have space to include them all in a single volume. New numericalmethods have opened up quantum many-body problems thought impossibleto solve or understand. We can treat open and closed systems; we begin tounderstand the role of entanglement; we nd or predict QPTs in naturallyoccurring systems ranging from chunks of matter to neutron stars, as well asengineered ones like quantum dots.There are now almost ve thousand papers devoted to QPTs. This bookgives us a chance to pause and look back as well as to look forward to thefuture and the many open problems that remain. QPTs are a frontier area ofresearch in many-body quantum mechanics, particularly in condensed matterphysics. While we emphasize condensed matter, we include an explicit sectionat the end on QPTs across physics, and connections to other elds appearthroughout the text. The book is divided into ve parts, each containing fromfour to seven chapters.Part I is intended to be somewhat more accessible to advanced gradu-ate students and researchers entering the eld. Thus it includes four morepedagogical, slightly longer chapters, covering new concepts and directions inQPTs: nite temperature and transport, dissipation, dynamics, and topolog-ical phases. Each of these chapters leads the reader from simpler ideas andconcepts to the latest advances in these areas. The last two chapters of Part Icover entanglement, an important new tool for analysis of quantum many-body systems: rst from a quantum-information-theoretic perspective, thenfrom a geometrical picture tied to physical observables.Part II delves into specic models and systems, in seven chapters. Theseare more closely tied to particular experimental realizations or theoreticalmethods. The topics include topological order, the Kondo lattice, ultracoldxi 2011 by Taylor and Francis Group, LLCxiiquantum gases, dissipation and cavity quantum electrodynamics (QED), spinsystems and group theory, Hubbard models, and metastability and nite-sizeeects.Part III covers experiments, in six chapters. Although the book is mainlytheoretical, the experimental chapters are key to making our whole discussionof QPTs meaningful; there are many observations now supporting the theo-ries laid out in these pages. We present a selection covering a range of suchexperiments, including quantum dots, 2D electron systems, high-Tc materials,molecular systems, heavy fermions, and ultracold quantum gases in opticallattices.Part IV presents recent advances in the key numerical methods used tostudy QPTS, in ve chapters. These include the worm algorithm for quan-tum Monte Carlo, cluster Monte Carlo for dissipative QPTs, time-dependentdensity matrix renormalization group methods, new ideas in matrix productstate methods, and dynamical mean eld theory.Finally, Part V presents a selection of QPTs in elds besides condensedmatter physics, in four chapters. These include neutron stars and the quark-gluon plasma, cavity QED, nuclei, and a new mapping, now used by manystring theorists, from classical gravitational theories (anti-de Sitter space) toconformal quantum eld theories.You can read this book by skipping around from topic to topic; that is howI edited it. However, in retrospect, I strongly recommend spending some timein Part I before delving into whichever topics catch your interest in the restof the book. I also recommend reading thoroughly one or two experimentalchapters early on in your perusing of this text, as it puts the rest in perspective.This book tells its own story, and besides a few words of thanks, I wontdelay you further with my remarks.First and foremost, I thank the authors, who wrote amazing chapters fromwhich I learned a tremendous amount. It is their writing that made the twoyears of eort I spent taking this book from conception to completion worthevery last minute. The layout of the book and topic choices, although ulti-mately my own choice and my own responsibility, received useful input frommany of the authors, for which I am also thankful.I am grateful to the Aspen Center for Physics, which hosted a number ofauthors of this book, including myself, while we wrote our respective chapters.I am grateful to the Kirchho Institute for Physics and the Graduate Schoolfor Fundamental Physics at the University of Heidelberg, for hosting me duringan important initial phase of the book.I thank my post-doc and graduate students who oered a student per-spective on these chapters, ensuring the text would be useful for physicists atlevels ranging from graduate student to emeritus professor: Dr. Miguel-AngelGarca-March, Laith Haddad, Dr. David Larue, Scott Strong, and MichaelWall. I thank Jim McNeil and Chip Durfee for their perspectives on nuclearphysics and quantum optics, respectively, which they brought to bear in sup-plemental reviews for Part V, and Jim Bernard and David Wood for their 2011 by Taylor and Francis Group, LLCxiiioverall comments as well. I thank John Navas and Sarah Morris from Tay-lor & Francis, for doing a spectacular job in bringing the book to a nishedproduct.My wife and children were very, very patient with me throughout theprocess. I thank them for their love and support.Last but not least, I am grateful to Je and Jean at Higher Grounds Cafe,where I did a good part of the detailed work on this book.This work was supported by the National Science Foundation under GrantPHY-0547845 as part of the NSF CAREER program. 2011 by Taylor and Francis Group, LLCContentsI New Directions and New Concepts in QuantumPhase Transitions 11 Finite Temperature Dissipation and Transport Near Quan-tum Critical Points 3Subir Sachdev1.1 Model Systems and Their Critical Theories . . . . . . . . . . 41.1.1 Coupled Dimer Antiferromagnets . . . . . . . . . . . . 41.1.2 Deconned Criticality . . . . . . . . . . . . . . . . . . 61.1.3 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.4 Spin Density Waves . . . . . . . . . . . . . . . . . . . 91.2 Finite Temperature Crossovers . . . . . . . . . . . . . . . . . 111.3 Quantum Critical Transport . . . . . . . . . . . . . . . . . . 151.4 Exact Results for Quantum Critical Transport . . . . . . . . 171.5 Hydrodynamic Theory . . . . . . . . . . . . . . . . . . . . . 211.5.1 Relativistic Magnetohydrodynamics . . . . . . . . . . 211.5.2 Dyonic Black Hole . . . . . . . . . . . . . . . . . . . . 231.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 231.6 The Cuprate Superconductors . . . . . . . . . . . . . . . . . 25Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Dissipation, Quantum Phase Transitions, and Measurement 31Sudip Chakravarty2.1 Multiplicity of Dynamical Scales and Entropy . . . . . . . . 322.2 Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3 Quantum Phase Transitions . . . . . . . . . . . . . . . . . . 362.3.1 Innite Number of Degrees of Freedom . . . . . . . . . 362.3.2 Broken Symmetry . . . . . . . . . . . . . . . . . . . . 382.3.2.1 Unitary Inequivalence . . . . . . . . . . . . . 382.4 Measurement Theory . . . . . . . . . . . . . . . . . . . . . . 392.4.1 Coleman-Hepp Model . . . . . . . . . . . . . . . . . . 392.4.2 Tunneling Versus Coherence . . . . . . . . . . . . . . . 412.4.3 Quantum-to-Classical Transition . . . . . . . . . . . . 422.5 Von Neumann Entropy . . . . . . . . . . . . . . . . . . . . . 432.5.1 A Warmup Exercise: Damped Harmonic Oscillator . . 442.5.2 Double Well Coupled to a Dissipative Heat Bath . . . 452.5.3 Disordered Systems . . . . . . . . . . . . . . . . . . . 46xv 2011 by Taylor and Francis Group, LLCxvi2.5.3.1 Anderson Localization . . . . . . . . . . . . . 472.5.3.2 Integer Quantum Hall Plateau Transitions . 482.5.3.3 Innite Randomness Fixed Point . . . . . . . 492.6 Disorder and First Order Quantum Phase Transitions . . . . 512.7 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 Universal Dynamics Near Quantum Critical Points 59Anatoli Polkovnikov and Vladimir Gritsev3.1 Brief Review of the Scaling Theory for Second Order PhaseTransitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2 Scaling Analysis for Dynamics near Quantum Critical Points 653.3 Adiabatic Perturbation Theory . . . . . . . . . . . . . . . . . 733.3.1 Sketch of the Derivation . . . . . . . . . . . . . . . . . 733.3.2 Applications to Dynamics near Critical Points . . . . 763.3.3 Quenches at Finite Temperatures, and the Role ofQuasi-particle Statistics . . . . . . . . . . . . . . . . . 793.4 Going Beyond Condensed Matter . . . . . . . . . . . . . . . 813.4.1 Adiabaticity in Cosmology . . . . . . . . . . . . . . . 813.4.2 Time Evolution in a Singular Space-Time . . . . . . . 843.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . 86Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884 Fractionalization and Topological Order 91Masaki Oshikawa4.1 Quantum Phases and Orders . . . . . . . . . . . . . . . . . . 914.2 Conventional Quantum Phase Transitions: Transverse IsingModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.3 Haldane-Gap Phase and Topological Order . . . . . . . . . . 934.3.1 Quantum Antiferromagnets . . . . . . . . . . . . . . . 934.3.2 Quantum Antiferromagnetic Chains and the ValenceBonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.3 AKLT State and the Haldane Gap . . . . . . . . . . . 964.3.4 Haldane Phase and Topological Order . . . . . . . . . 984.3.5 Edge States . . . . . . . . . . . . . . . . . . . . . . . . 994.4 RVB Quantum Spin Liquid and Topological Order . . . . . . 1004.4.1 Introduction to RVB States . . . . . . . . . . . . . . . 1004.4.2 Quantum Dimer Model . . . . . . . . . . . . . . . . . 1014.4.3 Commensurability and Spin Liquids . . . . . . . . . . 1024.4.4 Topological Degeneracy of the RVB Spin Liquid . . . 1034.4.5 Fractionalization in the RVB Spin Liquid . . . . . . . 1054.5 Fractionalization and Topological Order . . . . . . . . . . . . 1064.5.1 What is Topological Order? . . . . . . . . . . . . . . . 1064.5.2 Fractionalization: General Denition . . . . . . . . . . 1064.5.3 Fractionalization Implies Topological Degeneracy . . . 108 2011 by Taylor and Francis Group, LLCxvii4.5.4 Implications . . . . . . . . . . . . . . . . . . . . . . . . 1104.6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135 Entanglement Renormalization: An Introduction 115Guifre Vidal5.1 Coarse Graining and Ground State Entanglement . . . . . . 1165.1.1 A Real-Space Coarse-Graining Transformation . . . . 1175.1.2 Ground State Entanglement . . . . . . . . . . . . . . . 1195.1.3 Accumulation of Short-Distance Degrees of Freedom . 1215.2 Entanglement Renormalization . . . . . . . . . . . . . . . . . 1225.2.1 Disentanglers . . . . . . . . . . . . . . . . . . . . . . . 1235.2.2 Ascending and Descending Superoperators . . . . . . . 1235.2.3 Multi-scale Entanglement Renormalization Ansatz . . 1255.3 The Renormalization Group Picture . . . . . . . . . . . . . . 1275.3.1 A Real-Space Renormalization-Group Map . . . . . . 1275.3.2 Properties of the Renormalization-Group Map . . . . 1285.3.3 Fixed Points of Entanglement Renormalization . . . . 1295.4 Quantum Criticality . . . . . . . . . . . . . . . . . . . . . . . 1305.4.1 Scaling Operators and Critical Exponents . . . . . . . 1305.4.2 Correlators and the Operator Product Expansion . . . 1325.4.3 Surface Critical Phenomena . . . . . . . . . . . . . . . 1335.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . 135Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376 The Geometry of Quantum Phase Transitions 139Gerardo Ortiz6.1 Entanglement and Quantum Phase Transitions . . . . . . . . 1416.1.1 Entanglement 101 . . . . . . . . . . . . . . . . . . . . 1416.1.2 Generalized Entanglement . . . . . . . . . . . . . . . . 1426.1.3 Quantifying Entanglement: Purity . . . . . . . . . . . 1436.1.3.1 A Simple Example . . . . . . . . . . . . . . . 1446.1.4 Statics of Quantum Phase Transitions . . . . . . . . . 1456.1.5 Dynamics of Quantum Phase Transitions . . . . . . . 1486.2 Topological Quantum Numbers and Quantum Phase Transi-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506.2.1 Geometric Phases and Response Functions . . . . . . 1516.2.2 The Geometry of Response Functions . . . . . . . . . 1546.2.3 The Geometry of Quantum Information . . . . . . . . 1576.2.4 Phase Diagrams and Topological Quantum Numbers . 1586.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606.4 Appendix: Generalized Coherent States . . . . . . . . . . . . 162Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 2011 by Taylor and Francis Group, LLCxviiiII Progress in Model Hamiltonians and in SpecicSystems 1677 Topological Order and Quantum Criticality 169Claudio Castelnovo, Simon Trebst, and Matthias Troyer7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1697.1.1 The Toric Code . . . . . . . . . . . . . . . . . . . . . . 1707.2 Quantum Phase Transitions . . . . . . . . . . . . . . . . . . 1737.2.1 Lorentz-Invariant Transitions . . . . . . . . . . . . . . 1757.2.1.1 Other Hamiltonian Deformations . . . . . . . 1787.2.2 Conformal Quantum Critical Points . . . . . . . . . . 1787.2.2.1 Microscopic Model for Wavefunction Deforma-tion . . . . . . . . . . . . . . . . . . . . . . . 1797.2.2.2 Dimensionality Reduction and the 2D IsingModel . . . . . . . . . . . . . . . . . . . . . . 1807.2.2.3 Topological Entropy . . . . . . . . . . . . . . 1817.2.2.4 Topological Entropy along the WavefunctionDeformation . . . . . . . . . . . . . . . . . . 1837.3 Thermal Transitions . . . . . . . . . . . . . . . . . . . . . . . 1847.3.1 Non-local Order Parameters at Finite Temperature . . 1857.3.2 Topological Entropy at Finite Temperature . . . . . . 1867.3.3 Fragile vs. Robust Behavior: A Matter of(De)connement . . . . . . . . . . . . . . . . . . . . . 1877.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1918 Quantum Criticality and the Kondo Lattice 193Qimiao Si8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1948.1.1 Quantum Criticality: Competing Interactions in Many-Body Systems . . . . . . . . . . . . . . . . . . . . . . . 1948.1.2 Heavy Fermion Metals . . . . . . . . . . . . . . . . . . 1968.1.3 Quantum Critical Point in Antiferromagnetic HeavyFermions . . . . . . . . . . . . . . . . . . . . . . . . . 1988.2 Heavy Fermi Liquid of Kondo Lattices . . . . . . . . . . . . 1998.2.1 Single-Impurity Kondo Model . . . . . . . . . . . . . . 1998.2.2 Kondo Lattice and Heavy Fermi Liquid . . . . . . . . 2008.3 Quantum Criticality in the Kondo Lattice . . . . . . . . . . 2038.3.1 General Considerations . . . . . . . . . . . . . . . . . 2038.3.2 Microscopic Approach Based on the Extended Dynam-ical Mean-Field Theory . . . . . . . . . . . . . . . . . 2048.3.3 Spin-Density-Wave Quantum Critical Point . . . . . . 2058.3.4 Local Quantum Critical Point . . . . . . . . . . . . . . 2068.4 Antiferromagnetism and Fermi Surfaces in Kondo Lattices . 2078.5 Towards a Global Phase Diagram . . . . . . . . . . . . . . . 208 2011 by Taylor and Francis Group, LLCxix8.5.1 How to Melt a Kondo-Destroyed Antiferromagnet . . 2088.5.2 Global Phase Diagram . . . . . . . . . . . . . . . . . . 2098.6 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 2108.6.1 Quantum Criticality . . . . . . . . . . . . . . . . . . . 2108.6.2 Global Phase Diagram . . . . . . . . . . . . . . . . . . 2118.7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . 2128.7.1 Kondo Lattice . . . . . . . . . . . . . . . . . . . . . . 2128.7.2 Quantum Criticality . . . . . . . . . . . . . . . . . . . 2128.7.3 Global Phase Diagram . . . . . . . . . . . . . . . . . . 2138.7.4 Superconductivity . . . . . . . . . . . . . . . . . . . . 213Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2139 Quantum Phase Transitions in Spin-Boson Systems: Dissipa-tion and Light Phenomena 217Karyn Le Hur9.1 Dissipative Transitions for the Two-State System . . . . . . 2179.1.1 Ohmic Case . . . . . . . . . . . . . . . . . . . . . . . . 2189.1.2 Exact Results . . . . . . . . . . . . . . . . . . . . . . . 2199.1.3 Spin Dynamics and Entanglement . . . . . . . . . . . 2219.1.4 Sub-ohmic Case . . . . . . . . . . . . . . . . . . . . . 2239.1.5 Realizations . . . . . . . . . . . . . . . . . . . . . . . . 2249.2 Dissipative Spin Array . . . . . . . . . . . . . . . . . . . . . 2259.2.1 Boson-Mediated Magnetic Interaction . . . . . . . . . 2259.2.2 Solvable Dissipative Model . . . . . . . . . . . . . . . 2269.2.3 Dissipative 4Theory . . . . . . . . . . . . . . . . . . 2279.2.4 Critical Exponents . . . . . . . . . . . . . . . . . . . . 2279.2.5 Realizations . . . . . . . . . . . . . . . . . . . . . . . . 2289.3 One-Mode Superradiance Model . . . . . . . . . . . . . . . . 2299.3.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 2299.3.2 Normal Phase . . . . . . . . . . . . . . . . . . . . . . . 2309.3.3 Superradiant Phase . . . . . . . . . . . . . . . . . . . 2309.3.4 Second-Order Quantum Phase Transition . . . . . . . 2319.3.5 Realizations . . . . . . . . . . . . . . . . . . . . . . . . 2329.4 Jaynes-Cummings Lattice . . . . . . . . . . . . . . . . . . . . 2329.4.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 2339.4.2 Mott Insulator-Superuid Transition . . . . . . . . . . 2339.4.3 Spin-1/2 Mapping for the Polaritons . . . . . . . . . . 2359.4.4 Field Theory Approach to the Transition . . . . . . . 2359.4.5 Realizations . . . . . . . . . . . . . . . . . . . . . . . . 2369.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 2011 by Taylor and Francis Group, LLCxx10 Topological Excitations in Superuids with Internal Degreesof Freedom 241Yuki Kawaguchi and Masahito Ueda10.1 Quantum Phases and Symmetries . . . . . . . . . . . . . . . 24210.1.1 Group-Theoretic Characterization of the Order Param-eter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24210.1.2 Symmetries and Order Parameters of Spinor BECs . . 24410.1.2.1 Spin-1 . . . . . . . . . . . . . . . . . . . . . . 24410.1.2.2 Spin-2 . . . . . . . . . . . . . . . . . . . . . . 24510.1.3 Order-Parameter Manifold . . . . . . . . . . . . . . . . 24610.2 Homotopy Classication of Defects . . . . . . . . . . . . . . . 24710.3 Topological Excitations . . . . . . . . . . . . . . . . . . . . . 25010.3.1 Line Defects . . . . . . . . . . . . . . . . . . . . . . . . 25110.3.1.1 Nonquantized Circulation . . . . . . . . . . . 25110.3.1.2 Fractional Vortices . . . . . . . . . . . . . . . 25310.3.2 Point Defects . . . . . . . . . . . . . . . . . . . . . . . 25410.3.2.1 t Hooft-Polyakov Monopole (Hedgehog) . . 25410.3.2.2 Dirac Monopole . . . . . . . . . . . . . . . . 25410.3.3 Particle-like Solitons . . . . . . . . . . . . . . . . . . . 25510.4 Special Topics . . . . . . . . . . . . . . . . . . . . . . . . . . 25710.4.1 The Kibble-Zurek Mechanism . . . . . . . . . . . . . . 25710.4.2 Knot Soliton . . . . . . . . . . . . . . . . . . . . . . . 25810.5 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . 261Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26311 Quantum Monte Carlo Studies of the Attractive HubbardHamiltonian 265Richard T. Scalettar and George G. Batrouni11.1 Quantum Monte Carlo Methods . . . . . . . . . . . . . . . . 26711.2 Pseudogap Phenomena . . . . . . . . . . . . . . . . . . . . . 26911.2.1 Chemical Potential and Magnetic Susceptibility . . . . 26911.2.2 Scaling of NMR Relaxation Rate . . . . . . . . . . . . 27111.3 The Eect of Disorder . . . . . . . . . . . . . . . . . . . . . . 27211.3.1 Real Space Pair Correlation Function . . . . . . . . . 27311.3.2 Superuid Stiness . . . . . . . . . . . . . . . . . . . . 27511.3.3 Density of States . . . . . . . . . . . . . . . . . . . . . 27611.4 Imbalanced Populations . . . . . . . . . . . . . . . . . . . . . 27811.4.1 FFLO Pairing in 1D . . . . . . . . . . . . . . . . . . . 28011.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28412 Quantum Phase Transitions in Quasi-One-Dimensional Sys-tems 289Thierry Giamarchi12.1 Spins: From Luttinger Liquids to Bose-Einstein Condensates 290 2011 by Taylor and Francis Group, LLCxxi12.1.1 Coupled Spin-1/2 Chains . . . . . . . . . . . . . . . . 29112.1.2 Dimer or Ladder Coupling . . . . . . . . . . . . . . . . 29212.2 Bosons: From Mott Insulators to Superuids . . . . . . . . . 29712.2.1 Coupled Superuid: Dimensional Crossover . . . . . . 29812.2.2 Coupled Mott Chains: Deconnement Transition . . . 29912.3 Fermions: Dimensional Crossover and Deconnement . . . . 30012.3.1 Dimensional Crossover . . . . . . . . . . . . . . . . . . 30212.3.2 Deconnement Transition . . . . . . . . . . . . . . . . 30412.4 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . 306Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30713 Metastable Quantum Phase Transitions in a One-DimensionalBose Gas 311Lincoln D. Carr, Rina Kanamoto, and Masahito Ueda13.1 Fundamental Considerations . . . . . . . . . . . . . . . . . . 31413.2 Topological Winding and Unwinding: Mean-Field Theory . . 31713.3 Finding the Critical Boundary: Bogoliubov Analysis . . . . . 31913.4 Weakly-Interacting Many-Body Theory: Exact Diagonalization 32213.5 Strongly-Interacting Many-Body Theory: Tonks-GirardeauLimit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32713.6 Bridging All Regimes: Finite-Size Bethe Ansatz . . . . . . . 33013.7 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . 335Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336III Experimental Realizations of Quantum Phasesand Quantum Phase Transitions 33914 Quantum Phase Transitions in Quantum Dots 341Ileana G. Rau, Sami Amasha, Yuval Oreg, and David Goldhaber-Gordon14.1 The Kondo Eect and Quantum Dots: Theory . . . . . . . . 34414.1.1 Brief History of the Kondo Eect . . . . . . . . . . . . 34414.1.2 Theory of Conductance through Quantum Dots . . . . 34614.1.3 Examples of Conductance Scaling Curves . . . . . . . 34714.1.3.1 G(V, T) in the Two-Channel Kondo Case . . 34814.1.3.2 G(V, T) in the Single-Channel Kondo Case . 34814.2 Kondo and Quantum Dots: Experiments . . . . . . . . . . . 34914.2.1 The Two-Channel Kondo Eect in a Double QuantumDot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34914.2.2 The Two-Channel Kondo Eect in Other Quantum DotGeometries . . . . . . . . . . . . . . . . . . . . . . . . 35314.2.3 The Two-Channel Kondo Eect in Graphene Sheets . 35414.2.4 The Two-Impurity Kondo Eect in a Double QuantumDot Geometry . . . . . . . . . . . . . . . . . . . . . . 35514.2.5 The Two-Impurity Kondo Eect in a Quantum Dot atthe Singlet-triplet Transition . . . . . . . . . . . . . . 356 2011 by Taylor and Francis Group, LLCxxii14.3 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . 35814.3.1 Inuence of Channel Asymmetry and Magnetic Field onthe Two-Channel Kondo Eect . . . . . . . . . . . . . 35914.3.2 Multiple Sites . . . . . . . . . . . . . . . . . . . . . . . 36014.3.3 Dierent Types of Reservoirs . . . . . . . . . . . . . . 36114.3.3.1 Superconducting Leads and Graphene at theDirac Point . . . . . . . . . . . . . . . . . . . 36114.3.3.2 The Bose-Fermi Kondo Model in QuantumDots . . . . . . . . . . . . . . . . . . . . . . . 362Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36315 Quantum Phase Transitions in Two-Dimensional ElectronSystems 369Alexander Shashkin and Sergey Kravchenko15.1 Strongly and Weakly Interacting 2D Electron Systems . . . . 36915.2 Proof of the Existence of Extended States in the Landau Levels 37115.3 Metal-Insulator Transitions in Perpendicular Magnetic Fields 37315.3.1 Floating-Up of Extended States . . . . . . . . . . . . . 37315.3.2 Similarity of the Insulating Phase and Quantum HallPhases . . . . . . . . . . . . . . . . . . . . . . . . . . . 37615.3.3 Scaling and Thermal Broadening . . . . . . . . . . . . 37915.4 Zero-Field Metal-Insulator Transition . . . . . . . . . . . . . 38115.5 Possible Ferromagnetic Transition . . . . . . . . . . . . . . . 38415.6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38716 Local Observables for Quantum Phase Transitions inStrongly Correlated Systems 393Eun-Ah Kim, Michael J. Lawler, and J.C. Davis16.1 Why Use Local Probes? . . . . . . . . . . . . . . . . . . . . . 39416.1.1 Nanoscale Heterogeneity . . . . . . . . . . . . . . . . . 39416.1.2 Quenched Impurity as a Tool . . . . . . . . . . . . . . 39516.1.3 Interplay between Inhomogeneity and Dynamics . . . 39516.1.4 Guidance for Suitable Microscopic Models . . . . . . . 39616.2 What are the Challenges? . . . . . . . . . . . . . . . . . . . . 39616.3 Searching for Quantum Phase Transitions Using STM . . . . 39716.3.1 STM Hints towards Quantum Phase Transitions . . . 39816.3.2 Theory of the Nodal Nematic Quantum Critical Pointin Homogeneous d-wave Superconductors . . . . . . . 40216.4 Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . 409Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41417 Molecular Quasi-Triangular Lattice Antiferromagnets 419Reizo Kato and Tetsuaki Itou17.1 Anion Radical Salts of Pd(dmit)2 . . . . . . . . . . . . . . . 420 2011 by Taylor and Francis Group, LLCxxiii17.2 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . 42017.3 Electronic Structure: Molecule, Dimer, and Crystal . . . . . 42217.4 Long-Range Antiferromagnetic Order vs. Frustration . . . . 42417.5 Quantum Spin-Liquid State in the EtMe3Sb Salt . . . . . . . 42517.6 Other Ground States: Charge Order and Valence Bond Solid 43017.6.1 Charge Order Transition in the Et2Me2Sb Salt . . . . 43017.6.2 Valence-Bond Solid State in the EtMe3P Salt . . . . . 43217.6.3 Intra- and Inter-Dimer Valence Bond Formations . . . 43317.7 Pressure-Induced Mott Transition . . . . . . . . . . . . . . . 43317.7.1 Pressure-Induced Metallic State in the Solid-CrossingColumn System . . . . . . . . . . . . . . . . . . . . . . 43417.7.2 Phase Diagram for the EtMe3P Salt: Superconductivityand Valence-Bond Solid . . . . . . . . . . . . . . . . . 43417.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44018 Probing Quantum Criticality and its Relationship with Su-perconductivity in Heavy Fermions 445Philipp Gegenwart and Frank Steglich18.1 Heavy Fermions . . . . . . . . . . . . . . . . . . . . . . . . . 44518.2 Heavy Fermi Liquids and Antiferromagnets . . . . . . . . . . 44718.3 Heavy-Fermion Superconductors . . . . . . . . . . . . . . . . 44718.4 Spin-Density-Wave-Type Quantum Criticality . . . . . . . . 45118.5 Quantum Criticality Beyond the Conventional Scenario . . . 45318.6 Interplay between Quantum Criticality and Unconventional Su-perconductivity . . . . . . . . . . . . . . . . . . . . . . . . . 45718.7 Conclusions and Open Questions . . . . . . . . . . . . . . . . 459Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46219 Strong Correlation Eects with Ultracold Bosonic Atoms inOptical Lattices 469Immanuel Bloch19.1 Optical Lattices . . . . . . . . . . . . . . . . . . . . . . . . . 46919.1.1 Optical Potentials . . . . . . . . . . . . . . . . . . . . 46919.1.2 Optical Lattices . . . . . . . . . . . . . . . . . . . . . 47119.1.2.1 Band Structure . . . . . . . . . . . . . . . . . 47319.1.3 Time-of-Flight Imaging and Adiabatic Mapping . . . . 47519.1.3.1 Sudden Release . . . . . . . . . . . . . . . . 47519.1.3.2 Adiabatic Mapping . . . . . . . . . . . . . . 47619.2 Many-Body Eects in Optical Lattices . . . . . . . . . . . . 47719.2.1 Bose-Hubbard Model . . . . . . . . . . . . . . . . . . . 47819.2.2 Superuid-Mott-Insulator Transition . . . . . . . . . . 47919.2.2.1 Superuid Phase . . . . . . . . . . . . . . . . 47919.2.2.2 Mott-Insulating Phase . . . . . . . . . . . . . 48019.2.2.3 Phase Diagram . . . . . . . . . . . . . . . . . 481 2011 by Taylor and Francis Group, LLCxxiv19.2.2.4 In-Trap Density Distribution . . . . . . . . . 48319.2.2.5 Phase Coherence Across the SF-MI Transition 48419.2.2.6 Excitation Spectrum . . . . . . . . . . . . . . 48719.2.2.7 Number Statistics . . . . . . . . . . . . . . . 48719.2.2.8 Dynamics near Quantum Phase Transitions . 48819.2.2.9 Bose-Hubbard Model with Finite Current . . 49019.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493IV Numerical Solution Methods for Quantum PhaseTransitions 49720 Worm Algorithm for Problems of Quantum and ClassicalStatistics 499Nikolay Prokof ev and Boris Svistunov20.1 Path-Integrals in Discrete and Continuous Space . . . . . . . 49920.2 Loop Representations for Classical High-Temperature Expan-sions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50220.3 Worm Algorithm: The Concept and Realizations . . . . . . . 50320.3.1 Discrete Conguration Space: Classical High-Tem-perature Expansions . . . . . . . . . . . . . . . . . . . 50420.3.2 Continuous Time: Quantum Lattice Systems . . . . . 50520.3.3 Bosons in Continuous Space . . . . . . . . . . . . . . . 50820.3.4 Momentum Conservation in Feynman Diagrams . . . 50920.4 Illustrative Applications . . . . . . . . . . . . . . . . . . . . . 51020.4.1 Optical-Lattice Bosonic Systems . . . . . . . . . . . . 51020.4.2 Supersolidity of Helium-4 . . . . . . . . . . . . . . . . 51220.4.3 The Problem of Deconned Criticality and the Flow-gram Method . . . . . . . . . . . . . . . . . . . . . . . 51620.5 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . 520Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52121 Cluster Monte Carlo Algorithms for Dissipative QuantumPhase Transitions 523Philipp Werner and Matthias Troyer21.1 Dissipative Quantum Models . . . . . . . . . . . . . . . . . . 52321.1.1 The Caldeira-Leggett Model . . . . . . . . . . . . . . . 52321.1.2 Dissipative Quantum Spin Chains . . . . . . . . . . . 52521.1.3 Resistively Shunted Josephson Junction . . . . . . . . 52521.1.4 Single Electron Box . . . . . . . . . . . . . . . . . . . 52721.2 Importance Sampling and the Metropolis Algorithm . . . . . 52821.3 Cluster Algorithms for Classical Spins . . . . . . . . . . . . . 53021.3.1 The Swendsen-Wang and Wol Cluster Algorithms . . 53021.3.2 Ecient Treatment of Long-Range Interactions . . . . 53221.4 Cluster Algorithm for Resistively Shunted Josephson Junctions 534 2011 by Taylor and Francis Group, LLCxxv21.4.1 Local Updates in Fourier Space . . . . . . . . . . . . . 53521.4.2 Cluster Updates . . . . . . . . . . . . . . . . . . . . . 53521.5 Winding Number Sampling . . . . . . . . . . . . . . . . . . . 53821.5.1 Path-Integral Monte Carlo . . . . . . . . . . . . . . . . 53921.5.2 Transition Matrix Monte Carlo . . . . . . . . . . . . . 53921.6 Applications and Open Questions . . . . . . . . . . . . . . . 54221.6.1 Single Spins Coupled to a Dissipative Bath . . . . . . 54221.6.2 Dissipative Spin Chains . . . . . . . . . . . . . . . . . 54221.6.3 The Single Electron Box . . . . . . . . . . . . . . . . . 54321.6.4 Resistively Shunted Josephson Junctions . . . . . . . . 543Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54422 Current Trends in Density Matrix Renormalization GroupMethods 547Ulrich Schollwock22.1 The Density Matrix Renormalization Group . . . . . . . . . 54722.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 54722.1.2 Innite-System and Finite-System Algorithms . . . . . 54922.2 DMRG and Entanglement . . . . . . . . . . . . . . . . . . . 55222.3 Density Matrix Renormalization Group and Matrix ProductStates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55322.3.1 Matrix Product States . . . . . . . . . . . . . . . . . . 55322.3.2 Density Matrix Renormalization in Matrix ProductState Language . . . . . . . . . . . . . . . . . . . . . . 55522.3.3 Matrix Product Operators . . . . . . . . . . . . . . . . 55522.4 Time-Dependent Simulation: Extending the Range . . . . . . 55822.4.1 Basic Algorithms . . . . . . . . . . . . . . . . . . . . . 55822.4.1.1 Time Evolution at Finite Temperatures . . . 55822.4.2 Linear Prediction and Spectral Functions . . . . . . . 55922.5 Density Matrix and Numerical Renormalization Groups . . . 56222.5.1 Wilsons Numerical Renormalization Group and MatrixProduct States . . . . . . . . . . . . . . . . . . . . . . 56222.5.2 Going Beyond the Numerical Renormalization Group 564Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56623 Simulations Based on Matrix Product States and ProjectedEntangled Pair States 571Valentin Murg, Ignacio Cirac, and Frank Verstraete23.1 Time Evolution using Matrix Product States . . . . . . . . . 57223.1.1 Variational Formulation of Time Evolution with MPS 57223.1.2 Time-Evolving Block-Decimation . . . . . . . . . . . . 57523.1.3 Finding Ground States by Imaginary-Time Evolution 57623.1.4 Innite Spin Chains . . . . . . . . . . . . . . . . . . . 57623.2 PEPS and Ground States of 2D Quantum Spin Systems . . . 57823.2.1 Construction and Calculus of PEPS . . . . . . . . . . 579 2011 by Taylor and Francis Group, LLCxxvi23.2.2 Calculus of PEPS . . . . . . . . . . . . . . . . . . . . 58123.2.3 Variational Method with PEPS . . . . . . . . . . . . . 58223.2.4 Time Evolution with PEPS . . . . . . . . . . . . . . . 58423.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 58723.2.6 PEPS and Fermions . . . . . . . . . . . . . . . . . . . 59123.2.7 PEPS on Innite Lattices . . . . . . . . . . . . . . . . 59323.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59524 Continuous-Time Monte Carlo Methods for Quantum Impu-rity Problems and Dynamical Mean Field Calculations 597Philipp Werner and Andrew J. Millis24.1 Quantum Impurity Models . . . . . . . . . . . . . . . . . . . 59724.2 Dynamical Mean Field Theory . . . . . . . . . . . . . . . . . 59924.3 Continuous-Time Impurity Solvers . . . . . . . . . . . . . . . 60024.3.1 General Recipe for Diagrammatic Quantum MonteCarlo . . . . . . . . . . . . . . . . . . . . . . . . . . . 60124.3.2 Weak-Coupling Approach . . . . . . . . . . . . . . . . 60224.3.2.1 Monte Carlo Congurations . . . . . . . . . 60224.3.2.2 Sampling Procedure and Detailed Balance . 60324.3.2.3 Determinant Ratios and Fast Matrix Updates 60424.3.2.4 Measurement of the Greens Function . . . . 60524.3.2.5 Expansion Order and Role of the Parameter K 60524.3.3 Strong-Coupling Approach: Expansion in the Impurity-Bath Hybridization . . . . . . . . . . . . . . . . . . . 60624.3.3.1 Monte Carlo Congurations . . . . . . . . . 60624.3.3.2 Sampling Procedure and Detailed Balance . 60924.3.3.3 Measurement of the Greens Function . . . . 60924.3.3.4 Generalization: Matrix Formalism . . . . . . 61024.3.4 Comparison Between the Two Approaches . . . . . . . 61124.4 Application: Phase Transitions in Multi-Orbital Systems withRotationally Invariant Interactions . . . . . . . . . . . . . . . 61224.4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 61324.4.2 Metal-Insulator Phase Diagram of the Three-OrbitalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 61324.4.3 Spin-Freezing Transition in the Paramagnetic MetallicState . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61424.4.4 Crystal Field Splittings and Orbital Selective MottTransitions . . . . . . . . . . . . . . . . . . . . . . . . 61624.4.5 High-Spin to Low-Spin Transition in a Two-OrbitalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 61724.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619V Quantum Phase Transitions Across Physics 621 2011 by Taylor and Francis Group, LLCxxvii25 Quantum Phase Transitions in Dense QCD 623Tetsuo Hatsuda and Kenji Maeda25.1 Introduction to QCD . . . . . . . . . . . . . . . . . . . . . . 62325.1.1 Symmetries in QCD . . . . . . . . . . . . . . . . . . . 62525.1.2 Dynamical Breaking of Chiral Symmetry . . . . . . . 62725.2 QCD Matter at High Temperature . . . . . . . . . . . . . . 62725.3 QCD Matter at High Baryon Density . . . . . . . . . . . . . 62925.3.1 Neutron-Star Matter and Hyperonic Matter . . . . . . 63025.3.2 Quark Matter . . . . . . . . . . . . . . . . . . . . . . . 63125.4 Superuidity in Neutron-Star Matter . . . . . . . . . . . . . 63225.5 Color Superconductivity in Quark Matter . . . . . . . . . . . 63325.5.1 The Gap Equation . . . . . . . . . . . . . . . . . . . . 63325.5.2 Tightly Bound Cooper Pairs . . . . . . . . . . . . . . 63425.6 QCD Phase Structure . . . . . . . . . . . . . . . . . . . . . . 63525.6.1 Ginzburg-Landau Potential for Hot/Dense QCD . . . 63725.6.2 Possible Phase Structure for Realistic Quark Masses . 63925.7 Simulating Dense QCD with Ultracold Atoms . . . . . . . . 64025.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64426 Quantum Phase Transitions in Coupled Atom-Cavity Sys-tems 647Andrew D. Greentree and Lloyd C. L. Hollenberg26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 64826.2 Photon-Photon Interactions in a Single Cavity . . . . . . . . 64926.2.1 Jaynes-Cummings Model . . . . . . . . . . . . . . . . 65026.2.2 The Giant Kerr Nonlinearity in Four-State Systems . 65326.2.3 Many-Atom Schemes . . . . . . . . . . . . . . . . . . . 65626.2.4 Other Atomic Schemes . . . . . . . . . . . . . . . . . . 65626.3 The Jaynes-Cummings-Hubbard Model . . . . . . . . . . . . 65726.3.1 The Bose-Hubbard Model . . . . . . . . . . . . . . . . 65726.3.2 Mean-Field Analysis of the JCH Model . . . . . . . . 65826.4 Few-Cavity Systems . . . . . . . . . . . . . . . . . . . . . . . 66226.5 Potential Physical Implementations . . . . . . . . . . . . . . 66526.5.1 Rubidium Microtrap Arrays . . . . . . . . . . . . . . . 66526.5.2 Diamond Photonic Crystal Structures . . . . . . . . . 66626.5.3 Superconducting Stripline Cavities: Circuit QED . . . 66726.6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66927 Quantum Phase Transitions in Nuclei 673Francesco Iachello and Mark A. Caprio27.1 QPTs and Excited-State QPTs in s-b Boson Models . . . . . 67427.1.1 Algebraic Structure of s-b Boson Models . . . . . . . . 67527.1.2 Geometric Structure of s-b Boson Models . . . . . . . 676 2011 by Taylor and Francis Group, LLCxxviii27.1.3 Phase Diagram and Phase Structure of s-b Boson Mod-els . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67827.2 s-b Models with Pairing Interaction . . . . . . . . . . . . . . 67927.3 Two-Level Bosonic and Fermionic Systems with Pairing Inter-actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68427.4 s-b Bosonic Systems with Generic Interactions: The Interacting-Boson Model of Nuclei . . . . . . . . . . . . . . . . . . . . . 68727.4.1 Algebraic Structure . . . . . . . . . . . . . . . . . . . 68727.4.2 Phase Structure and Phase Diagram . . . . . . . . . . 68727.4.3 Experimental Evidence . . . . . . . . . . . . . . . . . 69127.5 Two-Fluid Bosonic Systems . . . . . . . . . . . . . . . . . . . 69327.6 Bosonic Systems with Fermionic Impurities . . . . . . . . . . 69527.6.1 The Interacting Boson-Fermion Model . . . . . . . . . 69627.7 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . 697Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69828 Quantum Critical Dynamics from Black Holes 701Sean Hartnoll28.1 The Holographic Correspondence as a Tool . . . . . . . . . . 70228.1.1 The Basic Dictionary . . . . . . . . . . . . . . . . . . 70628.1.2 Finite Temperature . . . . . . . . . . . . . . . . . . . . 70928.1.3 Spectral Functions and Quasi-normal Modes . . . . . 71128.2 Finite Chemical Potential . . . . . . . . . . . . . . . . . . . . 71428.2.1 Bosonic Response and Superconductivity . . . . . . . 71628.2.2 Fermionic Response and Non-Fermi Liquids . . . . . . 71828.3 Current and Future Directions . . . . . . . . . . . . . . . . . 719Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721Index 725 2011 by Taylor and Francis Group, LLCPart INew Directions and NewConcepts in QuantumPhase Transitions 2011 by Taylor and Francis Group, LLC1Finite Temperature Dissipation andTransport Near Quantum Critical PointsSubir SachdevDepartment of Physics, Harvard University, Cambridge, MA 02138, U.S.A.The authors book [1] on quantum phase transitions has an extensive dis-cussion on the dynamic and transport properties of a variety of systems atnon-zero temperatures above a zero temperature quantum critical point. Thepurpose of this chapter is to briey review some basic material, and to thenupdate the earlier discussion with a focus on experimental and theoreticaldevelopments in the decade since the book was written. We note other recentreviews [2] from which portions of this chapter have been adapted.We will begin in Section 1.1 by introducing a variety of model systems andtheir quantum critical points; these are motivated by recent experimental andtheoretical developments. We will use these systems to introduce basic ideason nite temperature crossovers near quantum critical points in Section 1.2.In Section 1.3, we will focus on the important nite temperature quantumcritical region and present a general discussion of its transport properties. Animportant recent development has been the complete exact solution of thedynamic and transport properties in the quantum critical region of a varietyof (supersymmetric) model systems in two and higher dimensions: this willbe described in Section 1.4. The exact solutions are found to agree with theearlier general ideas discussed in Section 1.3. Quite remarkably, the exactsolution proceeds via a mapping to the theory of black holes in one higherspatial dimension: we will only briey mention this mapping here, and referthe reader to Chap. 28 for more information. As has often been the case in thehistory of physics, the existence of a new class of solvable models leads to newand general insights which apply to a much wider class of systems, almost allof which are not exactly solvable. This has also been the case here, as we willreview in Section 1.5: a hydrodynamic theory of the low frequency transportproperties has been developed, and has led to new relations between a varietyof thermo-electric transport co-ecients. Finally, in Section 1.6 we will turn tothe cuprate high temperature superconductors, and present recent proposalson how ideas from the theory of quantum phase transition may help unravelthe very complex phase diagram of these important materials.3 2011 by Taylor and Francis Group, LLC4 Understanding Quantum Phase TransitionscFIGURE 1.1The coupled dimer antiferromagnet. The full lines represent an exchange in-teraction J, while the dashed lines have exchange J/. The ellipses representa singlet valence bond of spins ([ ) [ ))/2. The two sides of the guresketch the Neel and dimerized quantum phases, respectively.1.1 Model Systems and Their Critical Theories1.1.1 Coupled Dimer AntiferromagnetsSome of the best studied examples of quantum phase transitions arise in in-sulators with unpaired S = 1/2 electronic spins residing on the sites, i, of aregular lattice. Using Sai (a = x, y, z) to represent the spin S = 1/2 opera-tor on site i, the low energy spin excitations are described by the Heisenbergexchange HamiltonianHJ =

i 0 is the antiferromagnetic exchange interaction. We will beginwith a simple realization of this model as illustrated in Fig. 1.1. The S = 1/2spins reside on the sites of a square lattice, and have nearest neighbor exchangeequal to either J or J/. Here 1 is a tuning parameter which induces aquantum phase transition in the ground state of this model.At = 1, the model has full square lattice symmetry, and this case isknown to have a Neel ground state which breaks spin rotation symmetry.This state has a checkerboard polarization of the spins, just as found in theclassical ground state, and as illustrated on the left side of Fig. 1.1. It can becharacterized by a vector order parameter awhich measures the staggeredspin polarizationa= iSai (1.2) 2011 by Taylor and Francis Group, LLCFinite Temperature Dissipation and Transport Near QCPs 5where i = 1 on the two sublattices of the square lattice. In the Neel statewe have a) ,= 0, and we expect that the low energy excitations can bedescribed by long wavelength uctuations of a eld a(x, ) over space, x,and imaginary time, .On the other hand, for 1 it is evident from Fig. 1.1 that the groundstate preserves all symmetries of the Hamiltonian: it has total spin S = 0 andcan be considered to be a product of nearest neighbor singlet valence bondson the J links. It is clear that this state cannot be smoothly connected to theNeel state, and so there must be at least one quantum phase transition as afunction of .Extensive quantum Monte Carlo simulations [35] on this model haveshown there is a direct phase transition between these states at a critical c,as in Fig. 1.1. The value of c is known accurately, as are the critical expo-nents characterizing a second-order quantum phase transition. These criticalexponents are in excellent agreement with the simplest proposal for the criti-cal eld theory [5], which can be obtained via conventional Landau-Ginzburgarguments. Given the vector order parameter a, we write down the action ind spatial and one time dimension,oLG =_ ddr d_12_(a)2+v2(a)2+s(a)2+ u4_(a)22_, (1.3)as the simplest action expanded in gradients and powers of awhich is con-sistent with all the symmetries of the lattice antiferromagnet. The transitionis now tuned by varying s ( c). Notice that this model is identicalto the Landau-Ginzburg theory for the thermal phase transition in a d + 1dimensional ferromagnet, because time appears as just another dimension. Asan example of this agreement, the critical exponent of the correlation length,, has the same value, = 0.711 . . ., to three signicant digits in a quantumMonte Carlo study of the coupled dimer antiferromagnet [5] and in a 5-loopanalysis [6] of the renormalization group xed point of oLG in d = 2. Similarexcellent agreement is obtained for the double-layer antiferromagnet [7, 8] andthe coupled-plaquette antiferromagnet [9].In experiments, the best studied realization of the coupled-dimer antiferro-magnet is TlCuCl3. In this crystal, the dimers are coupled in all three spatialdimensions, and the transition from the dimerized state to the Neel statecan be induced by application of pressure. Neutron scattering experiments byRuegg and collaborators [10] have clearly observed the transformation in theexcitation spectrum across the transition, as is described by a simple uctu-ations analysis about the mean eld saddle point of oLG. In the dimerizedphase (s > 0), a triplet of gapped excitations is observed, corresponding tothe three normal modes of aoscillating about a= 0; as expected, thistriplet gap vanishes upon approaching the quantum critical point. In a meaneld analysis, valid for d 3, the eld theory in Eq. (1.3) has a triplet gap ofs. In the Neel phase, the neutron scattering detects two gapless spin wavesand one gapped longitudinal mode [11]. This is described by oLG for s < 0, 2011 by Taylor and Francis Group, LLC6 Understanding Quantum Phase Transitions0 0.5 1 1.5 200.20.40.60.811.21.4L (p < pc)L (p > pc)Q=(0 4 0)L,T1 (p < pc)L (p > pc)Q=(0 0 1)E(p < pc)unscaledEnergy 2*E(p < pc), E(p > pc) [meV]Pressure |(p pc)| [kbar]TlCuCl3pc = 1.07 kbarT = 1.85 KFIGURE 1.2Energies of the gapped collective modes across the pressure (p) tuned quan-tum phase transition in TlCuCl3 observed by Ruegg et al. [10]. We test thedescription by the action oLG in Eq. (1.3) with s (pc p) by comparing2 times the energy gap for p < pc with the energy of the longitudinal modefor p > pc. The lines are the ts to a _[p pc[ dependence, testing the 1/2exponent.where aexperiences an inverted Mexican hat potential with a minimum at[a[ = _[s[/v. Expanding about this minimum we nd that in addition tothe gapless spin waves, there is a mode involving amplitude uctuations of[a[ which has an energy gap of _2[s[. These mean eld predictions for theenergy of the gapped modes on the two sides of the transition are tested inFig. 1.2: the observations are in good agreement with the 1/2 exponent andthe predicted 2 ratio [12], providing a non-trivial experimental test of theoLG eld theory.1.1.2 Deconned CriticalityWe now consider an analog of the transition discussed in Section 1.1.1, but fora Hamiltonian H = H0 + H1 which has full square lattice symmetry at all. For H0, we choose a form of HJ such that Jij = J for all nearest neighborlinks. Thus at = 0 the ground state has Neel order, as in the left panel ofFig. 1.1. We now want to choose H1 so that increasing leads to a spin singletstate with spin rotation symmetry restored. A large number of choices have 2011 by Taylor and Francis Group, LLCFinite Temperature Dissipation and Transport Near QCPs 7been made in the literature, and the resulting ground state invariably [13]has valence bond solid (VBS) order. The VBS state is supercially similarto the dimer singlet state in the right panel of Fig. 1.1: the spins primarilyform valence bonds with near-neighbor sites. However, because of the squarelattice symmetry of the Hamiltonian, a columnar arrangement of the valencebonds as in Fig. 1.1 breaks the square lattice rotation symmetry; there are4 equivalent columnar states, with the valence bond columns running alongdierent directions. More generally, a VBS state is a spin singlet state, with anon-zero degeneracy due to a spontaneously broken lattice symmetry. Thus adirect transition between the Neel and VBS states involves two distinct brokensymmetries: spin rotation symmetry, which is broken only in the Neel state,and a lattice rotation symmetry, which is broken only in the VBS state. Therules of Landau-Ginzburg-Wilson theory imply that there can be no genericsecond-order transition between such states.It has been argued that a second-order Neel-VBS transition can indeedoccur [14], but the critical theory is not expressed directly in terms of eitherorder parameter. It involves a fractionalized bosonic spinor z ( =, ), andan emergent gauge eld A. The key step is to express the vector eld ainterms of z bya= zaz (1.4)where aare the 2 2 Pauli matrices. Note that this mapping from ato zis redundant. We can make a spacetime-dependent change in the phase of thez by the eld (x, )z eiz (1.5)and leave aunchanged. All physical properties must therefore also be invari-ant under Eq. (1.5), and so the quantum eld theory for z has a U(1) gaugeinvariance, much like that found in quantum electrodynamics. The eectiveaction for the z therefore requires introduction of an emergent U(1) gaugeeld A (where = x, is a three-component spacetime index). The eldA is unrelated to the electromagnetic eld, but is an internal eld whichconveniently describes the couplings between the spin excitations of the anti-ferromagnet. As we did for oLG, we can write down the quantum eld theoryfor z and A by the constraints of symmetry and gauge invariance, whichnow yieldsoz =_ d2r d_[(iA)z[2+s[z[2+u([z[2)2+ 12g2(A)2_. (1.6)For brevity, we have now used a relativistically invariant notation, andscaled away the spin-wave velocity v; the values of the couplings s, u aredierent from, but related to, those in oLG. The Maxwell action for A isgenerated from short distance z uctuations, and it makes A a dynamicaleld; its coupling g is unrelated to the electron charge. The action oz is avalid description of the Neel state for s < 0 (the critical upper value of s willhave uctuation corrections away from 0), where the gauge theory enters a 2011 by Taylor and Francis Group, LLC8 Understanding Quantum Phase TransitionsHiggs phase with z) ,= 0. This description of the Neel state as a Higgs phasehas an analogy with the Weinberg-Salam theory of the weak interactions. Inthe latter case it is hypothesized that the condensation of a Higgs boson givesa mass to the W and Z gauge bosons, whereas here the condensation of zquenches the A gauge boson. As written, the s > 0 phase of oz is a spinliquid state with an S = 0 collective gapless excitation associated with theA photon. Non-perturbative eects [13] associated with the monopoles in A(not discussed here) show that this spin liquid is ultimately unstable to theappearance of VBS order.Numerical studies of the Neel-VBS transition have focused on a speciclattice antiferromagnet proposed by Sandvik [1517]. There is strong evidencefor VBS order proximate to the Neel state, along with persuasive evidence ofa second-order transition. However, some studies [18, 19] support a very weakrst order transition.1.1.3 GrapheneThe last few years have seen an explosion in experimental and theoreticalstudies [20] of graphene: a single hexagonal layer of carbon atoms. At thecurrently observed temperatures, there is no evident broken symmetry in theelectronic excitations, and so it is not conventional to think of graphene asbeing in the vicinity of a quantum critical point. However, graphene doesindeed undergo a bona de quantum phase transition, but one without anyorder parameters or broken symmetry. This transition may be viewed as beingtopological in character, and is associated with a change in the nature of theFermi surface as a function of carrier density.Pure, undoped graphene has a conical electronic dispersion spectrum attwo points in the Brillouin zone, with the Fermi energy at the particle-holesymmetric point at the apex of the cone. So there is no Fermi surface, justa Fermi point, where the electronic energy vanishes, and pure graphene is asemi-metal. By applying a bias voltage, the Fermi energy can move away fromthis symmetric point, and a circular Fermi surface develops, as illustrated inFig. 1.3. The Fermi surface is electron-like for one sign of the bias, and hole-likefor the other sign. This change from electron to hole character as a functionof bias voltage constitutes the quantum phase transition in graphene. As wewill see below, with regard to its dynamic properties near zero bias, graphenebehaves in almost all respects like a canonical quantum critical system.The eld theory for graphene involves fermionic degrees of freedom. Rep-resenting the electronic orbitals near one of the Dirac points by the two-component fermionic spinor a, where a is a sublattice index (we suppress 2011 by Taylor and Francis Group, LLCFinite Temperature Dissipation and Transport Near QCPs 9FIGURE 1.3Dirac dispersion spectrum for graphene showing a topological quantum phasetransition from a hole Fermi surface for < 0 to an electron Fermi surface for > 0.spin and valley indices), we have the eective electronic actiono =_ d2r_ d a [( +iA )ab +ivFxabx +ivFyaby] b+ 12g2_ d2q42_ d q2 [A(q, )[2, (1.7)where iab are Pauli matrices in the sublattice space, is the chemical potential,vF is the Fermi velocity, and A is the scalar potential mediating the Coulombinteraction with coupling g2= e2/ ( is a dielectric constant). This theoryundergoes a quantum phase transition as a function of , at = 0, similarin many ways to that of oLG as a function of s. The interaction between thefermionic excitations here has coupling g2, which is the analog of the non-linearity u in oLG. However, while u is scaled to a non-zero xed point valueunder the renormalization group ow, g ows logarithmically slowly to zero.For many purposes, it is safe to ignore this ow, and to set g equal to a xedvalue; this value is characterized by the dimensionless ne structure constant = g2/(vF) which is of order unity in graphene.1.1.4 Spin Density WavesFinally, we consider the onset of Neel order, as in Section 1.1.1, but in a metalrather than an insulator. It is conventional to refer to such metallic Neel statesas having spin density wave (SDW) order. Our discussion here is motivatedby application to the cuprate superconductors: there is good evidence [21, 22] 2011 by Taylor and Francis Group, LLC10 Understanding Quantum Phase Transitionsthat the transition we describe below is present in the electron-doped cuprates,and proposals of its application to the hole-doped cuprates will be discussedin Section 1.6.We begin with the band structure describing the cuprates in the over-doped region, well away from the Mott insulator. Here the electrons ci aredescribed by the HamiltonianHc =

i c (or s > 0 in oLG),the excitations consist of a triplet of S = 1 particles, or triplons, which can beunderstood perturbatively in the large expansion as an excited S = 1 stateon a dimer, hopping between dimers (see Fig. 1.5). The mean eld theory tellsus that the excitation energy of this dimer vanishes as s upon approachingthe quantum critical point. Fluctuations beyond mean eld, described by oLG,show that the exponent is modied to sz, where z = 1 is the dynamic criticalexponent, and is the correlation length exponent. Now imagine turning ona non-zero temperature. As long as T is smaller than the triplon gap, i.e.,T < sz, we expect a description in terms of a dilute gas of thermally excitedtriplon particles. This leads to the behavior shown on the right-hand side ofFig. 1.5, delimited by the crossover indicted by the dashed line. Note that thecrossover line approaches T = 0 only at the quantum critical point.Now let us look at the complementary behavior at T > 0 on the Neel-ordered side of the transition, with s < 0. In two spatial dimensions, thermaluctuations prohibit the breaking of a non-Abelian symmetry at all T > 0, 2011 by Taylor and Francis Group, LLC12 Understanding Quantum Phase TransitionsClassicalspinwavesDilutetriplongasQuantumcriticalFIGURE 1.5Finite temperature crossovers of the coupled dimer antiferromagnet in Fig. 1.1.and so spin rotation symmetry is immediately restored. Nevertheless, thereis an exponentially large spin correlation length, , and at distances shorterthan we can use the ordered ground state to understand the nature of theexcitations. Along with the spin waves, we also found the longitudinal Higgsmode with energy 2s in mean eld theory. Thus, just as was this case fors > 0, we expect this spin-wave+Higgs picture to apply at all temperatureslower than the natural energy scale, i.e., for T < (s)z. This leads to thecrossover boundary shown on the left-hand side of Fig. 1.5.Having delineated the physics on the two sides of the transition, we areleft with the crucial quantum critical region in the center of Fig. 1.5. Thisis present for T > [s[z, i.e., at higher temperatures in the vicinity of thequantum critical point. To the left of the quantum critical region, we have adescription of the dynamics and transport in terms of an eectively classicalmodel of spin waves: this is the renormalized classical regime of Ref. [28]. Tothe right of the quantum critical region, we again have a regime of classicaldynamics, but now in terms of a Boltzmann equation for the triplon particles.A key property of the quantum critical region is that there is no description interms of either classical particles or classical waves at times on the order of thetypical relaxation time r of thermal excitations [29]. Instead, quantum andthermal eects are equally important, and involve the non-trivial dynamics ofthe xed-point theory describing the quantum critical point. Note that whilethe xed-point theory applies only at a single point ( = c) at T = 0, itsinuence broadens into the quantum critical region at T > 0. Because there 2011 by Taylor and Francis Group, LLCFinite Temperature Dissipation and Transport Near QCPs 13-1 -0.5 0 0.5 1100200300400500600-1 -0.5 0 0.5 1100200300400500600Quantum criticalDirac liquidElectronFermi liquidHoleFermi liquidFIGURE 1.6Finite temperature crossovers of graphene as a function of electron density n(tuned) by in Eq. (1.7) and temperature T. Adapted from Ref. [30].is no characteristic energy scale associated with the xed-point theory, kBTis the only energy scale available to determine r at non-zero temperatures.Thus, in the quantum critical region [29]r = ( kBT , (1.9)where ( is a universal constant dependent only upon the universality class ofthe xed point theory, i.e., it is a universal number just like the critical ex-ponents. This value of r determines the friction coecients associated withthe dissipative relaxation of spin uctuations in the quantum critical region.It is also important for the transport co-ecients associated with conservedquantities, and this will be discussed in Section 1.3.Let us now consider the similar T > 0 crossovers for the other models ofSection 1.1.The Neel-VBS transition of Section 1.1.2 has crossovers very similar tothose in Fig. 1.5, with one important dierence. The VBS state breaks adiscrete lattice symmetry, and this symmetry remains broken for a nite rangeof non-zero temperatures. Thus, within the right-hand triplon gas regime ofFig. 1.5, there is a phase transition line at a critical temperature TVBS. Thevalue of TVBS vanishes very rapidly as s 0+, and is controlled by the non-perturbative monopole eects which were briey noted in Section 1.1.2.For graphene, the discussion above applied to Fig. 1.3 leads to the crossoverdiag