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 StateLuca 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 ApplicationsA M Tishin, Y I SpichkinField Theories in Condensed Matter PhysicsSumathi RaoNonlinear Dynamics and Chaos in SemiconductorsK AokiPermanent MagnetismR Skomski,J M D CoeyModern Magnetooptics and Magnetooptical Materials A K Zvezdin, V A KotovSeries in Condensed Matter PhysicsLincoln D. CarrATAYLOR & 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 StateLuca 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 ApplicationsA M Tishin, Y I SpichkinField Theories in Condensed Matter PhysicsSumathi RaoNonlinear Dynamics and Chaos in SemiconductorsK AokiPermanent MagnetismR Skomski,J M D CoeyModern Magnetooptics and Magnetooptical Materials A K Zvezdin, V A KotovSeries in Condensed Matter PhysicsLincoln D. CarrATAYLOR & 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.ExceptaspermittedunderU.S.CopyrightLaw,nopartofthisbookmaybereprinted,reproduced,transmitted, orutilizedinanyformbyanyelectronic,mechanical,orothermeans,nowknownorhereafterinvented,includ-ing photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.Forpermissiontophotocopyorusematerialelectronicallyfromthiswork,pleaseaccesswww.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--dc222010034921Visit 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, LLCContributorsSamiAmashaStanford University, U.S.A.GeorgeG.BatrouniUniversite de Nice - SophiaAntipolis, FranceImmanuelBlochLudwig-Maximilians-Universit at,GermanyMarkA.CaprioUniversity of Notre Dame, U.S.A.LincolnD.CarrColorado School of Mines, U.S.A.ClaudioCastelnovoOxford University, U.K.Sudip ChakravartyUniversity of California Los Angeles,U.S.A.IgnacioCiracMax-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, GermanyThierryGiamarchiUniversity of Geneva, SwitzerlandDavidGoldhaber-GordonStanford University, U.S.A.AndrewD.GreentreeUniversity of Melbourne, AustraliaVladimir GritsevUniversity of Fribourg, SwitzerlandSeanHartnollHarvard University, U.S.A.TetsuoHatsudaUniversity of Tokyo, JapanLloydC.L.HollenbergUniversity of Melbourne, AustraliaFrancescoIachelloYale University, U.S.A.TetsuakiItouKyoto University, JapanRinaKanamotoOchanomizu University, JapanReizoKatoRIKEN, JapanYukiKawaguchiUniversity of Tokyo, Japanvii 2011 by Taylor and Francis Group, LLCviiiEun-AhKimCornell University, U.S.A.SergeyKravchenkoNortheastern University, U.S.A.MichaelJ.LawlerThe State University of New York atBinghamton, U.S.A.Cornell University, U.S.A.KarynLeHurYale University, U.S.A.KenjiMaedaThe University of Tokyo, JapanAndrewJ.MillisColumbia University, U.S.A.ValentinMurgMax-Planck-Institut f urQuantenoptik, GermanyYuvalOregWeizmann Institute of Science, IsraelGerardoOrtizIndiana University, U.S.A.MasakiOshikawaUniversity of Tokyo, JapanAnatoliPolkovnikovBoston University, U.S.A.NikolayProkof evUniversity of Massachusetts,Amherst, U.S.A.IleanaG.RauStanford University, U.S.A.SubirSachdevHarvard University, U.S.A.RichardT.ScalettarUniversity of California, Davis,U.S.A.UlrichSchollwockUniversity of Munich, GermanyAlexanderShashkinInstitute of Solid State Physics,RussiaQimiaoSiRice University, U.S.A.FrankSteglichMax Planck Institute for ChemicalPhysics of Solids, GermanyBorisSvistunovUniversity of Massachusetts,Amherst, U.S.A.SimonTrebstUniversity of California, SantaBarbara, U.S.A.MatthiasTroyerETH Zurich, SwitzerlandMasahitoUedaUniversity of Tokyo, JapanFrankVerstraeteUniversit at Wien, AustriaGuifreVidalThe University of Queensland,AustraliaPhilippWernerETH Zurich, Switzerland 2011 by Taylor and Francis Group, LLCEditorLincolnD. Carrisathe-oretical physicist whoworksprimarily in quantum many-bodytheory, articial mate-rials, andnonlineardynam-ics. He obtained his B.A.in physics at the Univer-sity of California, Berkeleyin 1994. He attended theUniversity ofWashingtoninSeattle from1996 to 2001,where he receivedbothhisM.S. andPh.D. inphysics.Hewas aDistinguishedIn-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. Hejoinedthefacultyinthephysics departmentattheColoradoSchoolofMinesin2005,whereheispresentlyanassociateprofessor. Heisan 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 PhysicsinSanta Barbara, California, theInstituteHenriPoincare attheUniversitePierre et Marie Curie in Paris, and the Kirchho Institute for Physics at theUniversity of Heidelberg.ix 2011 by Taylor and Francis Group, LLCPrefacePhasetransitionsoccurinall eldsofthephysical sciencesandarecrucialinengineeringaswell; abruptchangesfromonestateofmattertoanotherareapparenteverywherewelook, fromthefreezingof riverstothesteamrisingupfromtheteakettle. Butwhyshoulditbeonlytemperatureandpressurethat drivesuchabrupttransitions?Infact, quantumuctuationscanreplacethermal uctuations, aphasetransitioncanoccurevenatzerotemperature, and the concept of a phase transition turns out to be a lot moregeneralthanitismadeouttobeinelementarythermodynamics.Overthelasttwentyorsoyearstheeldof quantumphasetransitions (QPTs)hasseensteadygrowth. Thisbookfocusesespeciallyonthelatterhalf of thisdevelopment. TherearenowsomanyexperimentalexamplesofQPTsthatwe hardly have space to include them all in a single volume. New numericalmethodshave opened upquantum many-body problems thoughtimpossibletosolveorunderstand.Wecantreatopenandclosedsystems;webegintounderstandtheroleofentanglement; wendorpredictQPTsinnaturallyoccurring 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 bookgivesusachancetopauseandlookbackaswell astolookforwardtothefuture 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 sectionattheendonQPTsacrossphysics, andconnectionstoothereldsappearthroughout the text. The book is divided into ve parts, each containing fromfour to seven chapters.PartI isintendedtobesomewhatmoreaccessibletoadvancedgradu-atestudentsandresearchersenteringtheeld. Thusitincludesfourmorepedagogical, slightly longer chapters, covering new concepts and directions inQPTs: nite temperature and transport, dissipation, dynamics, and topolog-ical phases.Eachofthesechaptersleadsthereaderfromsimplerideasandconcepts to the latest advances in these areas. The last two chapters of Part Icoverentanglement, animportantnewtool foranalysisof quantummany-bodysystems: rstfromaquantum-information-theoreticperspective, thenfrom a geometrical picture tied to physical observables.Part II delves into specic models and systems, in seven chapters. Thesearemorecloselytiedtoparticular experimental realizationsor theoreticalmethods. Thetopicsincludetopological order,theKondolattice, 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 thetheo-ries laidoutinthesepages. Wepresent a selectioncovering arange ofsuchexperiments, including quantum dots, 2D electron systems, high-Tc materials,molecularsystems, heavyfermions, andultracoldquantumgasesinopticallattices.PartIVpresentsrecentadvancesinthekeynumerical methodsusedtostudyQPTS,invechapters. Theseincludethewormalgorithmforquan-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, PartVpresentsaselectionofQPTsineldsbesidescondensedmatter physics, in four chapters. These include neutron stars and the quark-gluonplasma,cavityQED, nuclei, andanewmapping, nowusedbymanystring 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 restofthebook.Ialsorecommendreadingthoroughlyoneortwoexperimentalchapters 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 wortheverylastminute. Thelayoutofthebookandtopicchoices,althoughulti-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.Ithankmypost-docandgraduatestudentswhooeredastudentper-spective on these chapters, ensuring the text would be useful for physicists atlevels ranging from graduate student to emeritus professor: Dr. Miguel-AngelGarca-March, LaithHaddad, Dr. DavidLarue, ScottStrong, andMichaelWall. I thank Jim McNeil and Chip Durfee for their perspectives on nuclearphysics and quantum optics, respectively, which they brought to bear in sup-plemental reviewsforPartV, andJimBernardandDavidWoodfortheir 2011 by Taylor and Francis Group, LLCxiiioverallcommentsaswell. IthankJohnNavasandSarahMorrisfromTay-lor & Francis, for doing a spectacular job inbringing the book to a nishedproduct.Mywifeandchildrenwerevery, verypatient withmethroughouttheprocess. 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 NewDirections and NewConcepts in QuantumPhaseTransitions 11 FiniteTemperatureDissipationandTransportNearQuan-tumCriticalPoints 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, QuantumPhaseTransitions,andMeasurement 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 UniversalDynamicsNearQuantumCriticalPoints 59Anatoli Polkovnikovand VladimirGritsev3.1 Brief Reviewof theScalingTheoryforSecondOrderPhaseTransitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 FractionalizationandTopological Order 91Masaki Oshikawa4.1 Quantum Phases and Orders . . . . . . . . . . . . . . . . . . 914.2 Conventional QuantumPhase Transitions: Transverse IsingModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.3 Haldane-Gap Phase and Topological Order . . . . . . . . . . 934.3.1 Quantum Antiferromagnets . . . . . . . . . . . . . . . 934.3.2 QuantumAntiferromagnetic Chains andthe 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: AnIntroduction 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 TheGeometry ofQuantumPhaseTransitions 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 QuantumNumbers andQuantumPhaseTransi-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 inModel Hamiltonians andinSpecicSystems 1677 Topological OrderandQuantumCriticality 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 DimensionalityReductionandthe 2DIsingModel . . . . . . . . . . . . . . . . . . . . . . 1807.2.2.3 Topological Entropy. . . . . . . . . . . . . . 1817.2.2.4 Topological EntropyalongtheWavefunctionDeformation . . . . . . . . . . . . . . . . . . 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 QuantumCriticalityandtheKondoLattice 193Qimiao Si8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1948.1.1 Quantum Criticality: Competing Interactions in Many-Body Systems . . . . . . . . . . . . . . . . . . . . . . . 1948.1.2 Heavy Fermion Metals . . . . . . . . . . . . . . . . . . 1968.1.3 QuantumCritical 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-tionandLightPhenomena 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 Dissipative4Theory . . . . . . . . . . . . . . . . . . 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 InternalDegreesof Freedom 241Yuki Kawaguchi and Masahito Ueda10.1Quantum 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 . . . . . . . . . . . . . . . . 24610.2Homotopy Classication of Defects . . . . . . . . . . . . . . . 24710.3Topological 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 . . . . . . . . . . . . . . . . . . . 25510.4Special Topics . . . . . . . . . . . . . . . . . . . . . . . . . . 25710.4.1 The Kibble-Zurek Mechanism. . . . . . . . . . . . . . 25710.4.2 Knot Soliton . . . . . . . . . . . . . . . . . . . . . . . 25810.5Conclusion and Discussion . . . . . . . . . . . . . . . . . . . 261Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26311 QuantumMonteCarloStudies of theAttractive HubbardHamiltonian 265Richard T. Scalettar and George G. Batrouni11.1Quantum Monte Carlo Methods . . . . . . . . . . . . . . . . 26711.2Pseudogap Phenomena . . . . . . . . . . . . . . . . . . . . . 26911.2.1 Chemical Potential and Magnetic Susceptibility . . . . 26911.2.2 Scaling of NMR Relaxation Rate . . . . . . . . . . . . 27111.3The Eect of Disorder . . . . . . . . . . . . . . . . . . . . . . 27211.3.1 Real Space Pair Correlation Function . . . . . . . . . 27311.3.2 Superuid Stiness. . . . . . . . . . . . . . . . . . . . 27511.3.3 Density of States . . . . . . . . . . . . . . . . . . . . . 27611.4Imbalanced Populations . . . . . . . . . . . . . . . . . . . . . 27811.4.1 FFLO Pairing in 1D. . . . . . . . . . . . . . . . . . . 28011.5Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28412 QuantumPhaseTransitionsinQuasi-One-DimensionalSys-tems 289Thierry Giamarchi12.1Spins: 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 . . . . . . . . . . . . . . . . 29212.2Bosons: From Mott Insulators to Superuids . . . . . . . . . 29712.2.1 Coupled Superuid: Dimensional Crossover . . . . . . 29812.2.2 Coupled Mott Chains: Deconnement Transition . . . 29912.3Fermions: Dimensional Crossover and Deconnement . . . . 30012.3.1 Dimensional Crossover. . . . . . . . . . . . . . . . . . 30212.3.2 Deconnement Transition . . . . . . . . . . . . . . . . 30412.4Conclusions and Perspectives . . . . . . . . . . . . . . . . . . 306Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30713 Metastable Quantum Phase Transitions in a One-DimensionalBose Gas 311Lincoln D. Carr, Rina Kanamoto, and Masahito Ueda13.1Fundamental Considerations . . . . . . . . . . . . . . . . . . 31413.2Topological Winding and Unwinding: Mean-Field Theory . . 31713.3Finding the Critical Boundary: Bogoliubov Analysis . . . . . 31913.4Weakly-Interacting Many-Body Theory: Exact Diagonalization 32213.5Strongly-Interacting Many-Body Theory: Tonks-GirardeauLimit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32713.6Bridging All Regimes: Finite-Size Bethe Ansatz . . . . . . . 33013.7Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . 335Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336III Experimental Realizations of QuantumPhasesandQuantumPhaseTransitions 33914 QuantumPhaseTransitionsin QuantumDots 341Ileana G. Rau, Sami Amasha, Yuval Oreg, and David Goldhaber-Gordon14.1The 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 . 34814.2Kondo 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, LLCxxii14.3Looking 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 SuperconductingLeadsandGrapheneattheDirac Point. . . . . . . . . . . . . . . . . . . 36114.3.3.2 The Bose-Fermi Kondo Model inQuantumDots . . . . . . . . . . . . . . . . . . . . . . . 362Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36315 QuantumPhase Transitions inTwo-Dimensional ElectronSystems 369Alexander Shashkin and Sergey Kravchenko15.1Strongly and Weakly Interacting 2D Electron Systems . . . . 36915.2Proof of the Existence of Extended States in the Landau Levels 37115.3Metal-Insulator Transitions in Perpendicular Magnetic Fields 37315.3.1 Floating-Up of Extended States . . . . . . . . . . . . . 37315.3.2 Similarityof theInsulatingPhaseandQuantumHallPhases . . . . . . . . . . . . . . . . . . . . . . . . . . . 37615.3.3 Scaling and Thermal Broadening . . . . . . . . . . . . 37915.4Zero-Field Metal-Insulator Transition . . . . . . . . . . . . . 38115.5Possible Ferromagnetic Transition . . . . . . . . . . . . . . . 38415.6Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38716 Local Observables for Quantum Phase Transitions inStrongly CorrelatedSystems 393Eun-Ah Kim, MichaelJ. Lawler, and J.C. Davis16.1Why 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. . . . . . . 39616.2What are the Challenges? . . . . . . . . . . . . . . . . . . . . 39616.3Searching for Quantum Phase Transitions Using STM . . . . 39716.3.1 STM Hints towards Quantum Phase Transitions . . . 39816.3.2 TheoryoftheNodalNematicQuantumCriticalPointin Homogeneousd-wave Superconductors . . . . . . . 40216.4Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . 409Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41417 Molecular Quasi-Triangular LatticeAntiferromagnets 419Reizo Kato and Tetsuaki Itou17.1Anion Radical Salts of Pd(dmit)2. . . . . . . . . . . . . . . 420 2011 by Taylor and Francis Group, LLCxxiii17.2Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . 42017.3Electronic Structure: Molecule, Dimer, and Crystal . . . . . 42217.4Long-Range Antiferromagnetic Order vs. Frustration . . . . 42417.5Quantum Spin-Liquid State in the EtMe3Sb Salt . . . . . . . 42517.6Other 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 . . . 43317.7Pressure-Induced Mott Transition . . . . . . . . . . . . . . . 43317.7.1 Pressure-InducedMetallicStateintheSolid-CrossingColumn System . . . . . . . . . . . . . . . . . . . . . . 43417.7.2 Phase Diagram for the EtMe3P Salt: Superconductivityand Valence-Bond Solid . . . . . . . . . . . . . . . . . 43417.8Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44018 ProbingQuantumCriticalityanditsRelationshipwithSu-perconductivityinHeavyFermions 445Philipp Gegenwart and Frank Steglich18.1Heavy Fermions . . . . . . . . . . . . . . . . . . . . . . . . . 44518.2Heavy Fermi Liquids and Antiferromagnets . . . . . . . . . . 44718.3Heavy-Fermion Superconductors . . . . . . . . . . . . . . . . 44718.4Spin-Density-Wave-Type Quantum Criticality . . . . . . . . 45118.5Quantum Criticality Beyond the Conventional Scenario . . . 45318.6Interplay between Quantum Criticality and Unconventional Su-perconductivity . . . . . . . . . . . . . . . . . . . . . . . . . 45718.7Conclusions and Open Questions . . . . . . . . . . . . . . . . 459Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46219 StrongCorrelationEectswithUltracoldBosonicAtomsinOpticalLattices 469Immanuel Bloch19.1Optical 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 . . . . . . . . . . . . . . 47619.2Many-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. . 49019.3Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493IV NumericalSolutionMethodsforQuantumPhaseTransitions 49720 WormAlgorithmfor Problems of QuantumandClassicalStatistics 499Nikolay Prokof ev and Boris Svistunov20.1Path-Integrals in Discrete and Continuous Space . . . . . . . 49920.2LoopRepresentations forClassical High-TemperatureExpan-sions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50220.3Worm 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 . . . 50920.4Illustrative Applications . . . . . . . . . . . . . . . . . . . . . 51020.4.1 Optical-Lattice Bosonic Systems . . . . . . . . . . . . 51020.4.2 Supersolidity of Helium-4 . . . . . . . . . . . . . . . . 51220.4.3 TheProblemof DeconnedCriticalityandtheFlow-gram Method. . . . . . . . . . . . . . . . . . . . . . . 51620.5Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . 520Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52121 Cluster Monte CarloAlgorithms for Dissipative QuantumPhaseTransitions 523Philipp Werner and Matthias Troyer21.1Dissipative 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 . . . . . . . . . . . . . . . . . . . 52721.2Importance Sampling and the Metropolis Algorithm . . . . . 52821.3Cluster Algorithms for Classical Spins . . . . . . . . . . . . . 53021.3.1 The Swendsen-Wang and Wol Cluster Algorithms. . 53021.3.2 Ecient Treatment of Long-Range Interactions . . . . 53221.4Cluster 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 . . . . . . . . . . . . . . . . . . . . . 53521.5Winding Number Sampling . . . . . . . . . . . . . . . . . . . 53821.5.1 Path-Integral Monte Carlo . . . . . . . . . . . . . . . . 53921.5.2 Transition Matrix Monte Carlo . . . . . . . . . . . . . 53921.6Applications 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 inDensityMatrixRenormalizationGroupMethods 547Ulrich Schollwock22.1The Density Matrix Renormalization Group . . . . . . . . . 54722.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 54722.1.2 Innite-System and Finite-System Algorithms . . . . . 54922.2DMRG and Entanglement . . . . . . . . . . . . . . . . . . . 55222.3DensityMatrixRenormalizationGroupandMatrixProductStates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55322.3.1 Matrix Product States . . . . . . . . . . . . . . . . . . 55322.3.2 Density Matrix Renormalization in Matrix ProductState Language. . . . . . . . . . . . . . . . . . . . . . 55522.3.3 Matrix Product Operators . . . . . . . . . . . . . . . . 55522.4Time-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 . . . . . . . 55922.5Density 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 SimulationsBasedonMatrixProductStatesandProjectedEntangledPairStates 571Valentin Murg, Ignacio Cirac, and Frank Verstraete23.1Time 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 . . . . . . . . . . . . . . . . . . . 57623.2PEPS 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 . . . . . . . . . . . . . . . . 59323.3Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59524 Continuous-Time Monte Carlo Methods for Quantum Impu-rityProblemsandDynamicalMeanField Calculations 597Philipp Werner and Andrew J. Millis24.1Quantum Impurity Models . . . . . . . . . . . . . . . . . . . 59724.2Dynamical Mean Field Theory . . . . . . . . . . . . . . . . . 59924.3Continuous-Time Impurity Solvers . . . . . . . . . . . . . . . 60024.3.1 General Recipe for Diagrammatic QuantumMonteCarlo . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 K60524.3.3 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 . . . . . . . 61124.4Application:Phase Transitions in Multi-Orbital Systems withRotationally Invariant Interactions . . . . . . . . . . . . . . . 61224.4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 61324.4.2 Metal-Insulator Phase Diagramof the Three-OrbitalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 61324.4.3 Spin-FreezingTransitionintheParamagnetic MetallicState . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61424.4.4 Crystal Field Splittings and Orbital Selective MottTransitions . . . . . . . . . . . . . . . . . . . . . . . . 61624.4.5 High-Spin to Low-Spin Transition in a Two-OrbitalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 61724.5Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619V QuantumPhaseTransitionsAcrossPhysics 621 2011 by Taylor and Francis Group, LLCxxvii25 QuantumPhaseTransitionsinDenseQCD 623Tetsuo Hatsuda and Kenji Maeda25.1Introduction to QCD . . . . . . . . . . . . . . . . . . . . . . 62325.1.1 Symmetries in QCD . . . . . . . . . . . . . . . . . . . 62525.1.2 Dynamical Breaking of Chiral Symmetry . . . . . . . 62725.2QCD Matter at High Temperature . . . . . . . . . . . . . . 62725.3QCD Matter at High Baryon Density . . . . . . . . . . . . . 62925.3.1 Neutron-Star Matter and Hyperonic Matter. . . . . . 63025.3.2 Quark Matter . . . . . . . . . . . . . . . . . . . . . . . 63125.4Superuidity in Neutron-Star Matter . . . . . . . . . . . . . 63225.5Color Superconductivity in Quark Matter . . . . . . . . . . . 63325.5.1 The Gap Equation . . . . . . . . . . . . . . . . . . . . 63325.5.2 Tightly Bound Cooper Pairs . . . . . . . . . . . . . . 63425.6QCD Phase Structure . . . . . . . . . . . . . . . . . . . . . . 63525.6.1 Ginzburg-Landau Potential for Hot/Dense QCD . . . 63725.6.2 Possible Phase Structure for Realistic Quark Masses . 63925.7Simulating Dense QCD with Ultracold Atoms . . . . . . . . 64025.8Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64426 QuantumPhase Transitions inCoupledAtom-CavitySys-tems 647Andrew D. Greentree and Lloyd C. L. Hollenberg26.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 64826.2Photon-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 . . . . . . . . . . . . . . . . . . 65626.3The Jaynes-Cummings-Hubbard Model . . . . . . . . . . . . 65726.3.1 The Bose-Hubbard Model . . . . . . . . . . . . . . . . 65726.3.2 Mean-Field Analysis of the JCH Model . . . . . . . . 65826.4Few-Cavity Systems . . . . . . . . . . . . . . . . . . . . . . . 66226.5Potential Physical Implementations . . . . . . . . . . . . . . 66526.5.1 Rubidium Microtrap Arrays . . . . . . . . . . . . . . . 66526.5.2 Diamond Photonic Crystal Structures . . . . . . . . . 66626.5.3 Superconducting Stripline Cavities: Circuit QED. . . 66726.6Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66927 QuantumPhaseTransitionsinNuclei 673Francesco Iachello and Mark A. Caprio27.1QPTs and Excited-State QPTs ins-b Boson Models . . . . . 67427.1.1 Algebraic Structure ofs-b Boson Models. . . . . . . . 67527.1.2 Geometric Structure ofs-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 . . . . . . . . . . . . . . 67927.3Two-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 . . . . . . . . . . . . . . . . . 69127.5Two-Fluid Bosonic Systems . . . . . . . . . . . . . . . . . . . 69327.6Bosonic Systems with Fermionic Impurities . . . . . . . . . . 69527.6.1 The Interacting Boson-Fermion Model . . . . . . . . . 69627.7Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . 697Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69828 QuantumCriticalDynamicsfromBlackHoles 701Sean Hartnoll28.1The 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 . . . . . 71128.2Finite Chemical Potential . . . . . . . . . . . . . . . . . . . . 71428.2.1 Bosonic Response and Superconductivity . . . . . . . 71628.2.2 Fermionic Response and Non-Fermi Liquids . . . . . . 71828.3Current and Future Directions . . . . . . . . . . . . . . . . . 719Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721Index 725 2011 by Taylor and Francis Group, LLCPartINewDirectionsandNewConceptsinQuantumPhaseTransitions 2011 by Taylor and Francis Group, LLC1FiniteTemperatureDissipationandTransportNearQuantumCriticalPointsSubirSachdevDepartment of Physics, Harvard University, Cambridge, MA 02138, U.S.A.Theauthorsbook[1] onquantumphasetransitionshasanextensivedis-cussiononthedynamicandtransportpropertiesofavarietyofsystemsatnon-zero temperatures above a zero temperature quantum critical point. Thepurpose of this chapter is to briey review some basic material, and to thenupdatetheearlierdiscussionwithafocusonexperimental andtheoreticaldevelopments 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.InSection1.3, wewill focusontheimportantnitetemperaturequantumcritical regionand present a general discussion of its transport properties. Animportantrecentdevelopmenthasbeenthecompleteexactsolutionof thedynamic and transport properties in the quantum critical region of a varietyof(supersymmetric)model systemsintwoandhigherdimensions: thiswillbe described in Section 1.4. The exact solutions are found to agree with theearlier general ideas discussedinSection1.3. Quiteremarkably, theexactsolutionproceedsviaamappingtothetheoryofblackholesinonehigherspatial dimension: we will only briey mention thismapping 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 TransitionscFIGURE1.1The coupled dimer antiferromagnet. The full lines represent an exchange in-teractionJ, 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 SystemsandTheirCriticalTheories1.1.1 CoupledDimerAntiferromagnetsSome of the best studied examples of quantum phase transitions arise in in-sulators with unpairedS= 1/2 electronic spins residing on thesites, i, of aregularlattice.UsingSai(a=x, y, z)torepresentthespinS=1/2opera-tor on sitei, the low energy spin excitations are described by the Heisenbergexchange HamiltonianHJ=i0istheantiferromagneticexchangeinteraction. Wewill beginwith a simple realization of this model as illustrated in Fig. 1.1. TheS = 1/2spins reside on the sites of a square lattice, and have nearest neighbor exchangeequal to eitherJorJ/.Here 1 is a tuning parameter which induces aquantum phase transition in the ground state of this model.At =1, themodel hasfull squarelatticesymmetry, andthiscaseisknowntohaveaNeel groundstatewhichbreaksspinrotationsymmetry.This state has a checkerboard polarization of the spins, justas found in theclassical ground state, and as illustrated on the left side of Fig. 1.1. It can becharacterized bya vector order parameterawhichmeasures thestaggeredspin polarizationa= iSai(1.2) 2011 by Taylor and Francis Group, LLCFinite Temperature Dissipation and Transport Near QCPs 5wherei = 1 on the two sublattices of the square lattice. In the Neel statewehave 'a` =0, andweexpectthat thelowenergyexcitationscanbedescribedbylongwavelengthuctuationsof aelda(x, )overspace, x,and imaginary time,.On the other hand, for1 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 theJlinks. 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 quantumMonte Carlosimulations [35] onthis model haveshown there is a direct phase transition between these states at a criticalc,as inFig. 1.1. The valueofcis 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 d12

(a)2+v2(a)2+s(a)2

+u4

(a)2

2

, (1.3)as the simplest action expanded in gradients and powers ofawhich is con-sistent with all the symmetries of the lattice antiferromagnet. The transitionisnowtunedbyvaryings ( c). Noticethatthismodel isidenticaltotheLandau-Ginzburgtheoryforthethermal phasetransitioninad + 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 oLGind = 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, andthetransitionfromthedimerizedstatetotheNeel 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-ationsanalysisaboutthemeaneldsaddlepointof oLG. Inthedimerizedphase(s>0),atripletofgappedexcitationsisobserved, correspondingtothethreenormal modesof aoscillatingabout a=0; asexpected, 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]. Thisis described by oLGfors< 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 KFIGURE1.2Energies of the gapped collective modes across the pressure (p) tuned quan-tumphase transition inTlCuCl3observed byRuegg etal.[10]. Wetest thedescriptionbytheaction oLGinEq.(1.3)withs (pc p)bycomparing2 times the energy gap forp < pc with the energy of the longitudinal modeforp>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. Expandingaboutthisminimumwendthatinadditiontothegaplessspinwaves,thereisamodeinvolvingamplitudeuctuationsof[a[ which has an energy gap of 2[s[. These mean eld predictions for theenergy ofthegappedmodesonthetwo sidesofthetransition are testedinFig. 1.2: the observations are in good agreement with the 1/2 exponent andthepredicted 2ratio [12],providing anon-trivial experimentaltestoftheoLGeld theory.1.1.2 DeconnedCriticalityWe now consider an analog of the transition discussed in Section 1.1.1, but fora HamiltonianH=H0 + H1which has full square lattice symmetry at all. ForH0, we choose a form ofHJsuch thatJij = Jfor 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 withspinrotation symmetryrestored. Alarge numberof choices have 2011 by Taylor and Francis Group, LLCFinite Temperature Dissipation and Transport Near QCPs 7beenmadeintheliterature, andtheresultinggroundstateinvariably[13]hasvalencebondsolid(VBS)order. TheVBSstateissuperciallysimilartothedimersingletstateintherightpanelofFig.1.1:thespinsprimarilyform valence bonds with near-neighbor sites. However, because of the squarelattice symmetry of the Hamiltonian, a columnar arrangement of the valencebondsasinFig.1.1breaksthesquarelatticerotationsymmetry; thereare4equivalentcolumnarstates,withthevalencebondcolumnsrunningalongdierent 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.Ithasbeenarguedthatasecond-orderNeel-VBStransitioncanindeedoccur [14], but the critical theory is not expressed directly in terms of eitherorder parameter. It involves a fractionalized bosonic spinorz( =, ), andan emergent gauge eldA. The key step is to express the vector eldainterms ofz bya= zaz(1.4)whereaare the 2 2 Pauli matrices. Note that this mapping fromatozis redundant. We can make a spacetime-dependent change in the phase of thezby 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 forz has a U(1) gaugeinvariance,muchlikethatfoundinquantumelectrodynamics.Theeectiveaction for theztherefore requires introduction of an emergent U(1) gaugeeldA(where=x, isathree-componentspacetimeindex). TheeldAisunrelatedtotheelectromagneticeld, butisaninternal eldwhichconveniently describes the couplings between the spin excitations of the anti-ferromagnet. As we did for oLG, we can write down the quantum eld theoryforzandAbytheconstraintsofsymmetryandgaugeinvariance,whichnow yieldsoz =

d2r d[(iA)z[2+s[z[2+u([z[2)2+12g2(A)2

. (1.6)For brevity, wehavenowusedarelativisticallyinvariant notation, andscaledawaythe spin-wavevelocityv; the values of thecouplings s, uaredierentfrom, butrelatedto, thosein oLG. TheMaxwell actionfor Aisgenerated from short distancezuctuations, and it makesAa dynamicaleld; itscouplinggisunrelatedtotheelectroncharge. Theaction ozisavalid description of the Neel state fors < 0 (the critical upper value ofs willhaveuctuationcorrectionsawayfrom0), wherethegaugetheoryentersa 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 givesamass totheWandZgauge bosons, whereas here thecondensation of zquenchestheAgaugeboson.Aswritten, thes>0phaseof ozisaspinliquidstatewithanS= 0collective gaplessexcitationassociated withtheA photon. Non-perturbative eects [13] associated with the monopoles in A(notdiscussedhere) showthatthisspinliquidisultimatelyunstableto theappearance of VBS order.Numerical studiesoftheNeel-VBStransitionhavefocusedonaspeciclattice 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 GrapheneThelastfewyearshaveseenanexplosioninexperimental andtheoreticalstudies[20] of graphene: asinglehexagonal layerof carbonatoms. Atthecurrently observed temperatures, there is no evident broken symmetry in theelectronicexcitations, andsoitisnotconventionaltothinkofgrapheneasbeinginthevicinityof aquantumcritical point. However, graphenedoesindeedundergo abonadequantumphasetransition,butonewithoutanyorder 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, undopedgraphenehasaconicalelectronicdispersionspectrumattwopointsintheBrillouinzone,withtheFermienergyattheparticle-holesymmetricpointattheapexofthecone.SothereisnoFermisurface,justa Fermi point,where theelectronic energy vanishes, andpure 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-resentingtheelectronicorbitals near oneof theDiracpoints bythetwo-componentfermionicspinora, whereaisasublatticeindex(wesuppress 2011 by Taylor and Francis Group, LLCFinite Temperature Dissipation and Transport Near QCPs 9FIGURE1.3Dirac 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

dq2 [A(q, )[2, (1.7)where iab are Pauli matrices in the sublattice space, is the chemical potential,vF is the Fermi velocity, and Ais the scalar potential mediating the Coulombinteractionwithcouplingg2=e2/(isadielectricconstant).Thistheoryundergoesaquantumphasetransitionasafunctionof ,at=0,similarin many ways to that of oLGas a function ofs. The interaction between thefermionicexcitationsherehascouplingg2, whichistheanalogofthenon-linearityu in oLG. However, whileu is scaled to a non-zero xed point valueunder the renormalization group ow,gows logarithmically slowly to zero.For many purposes, it is safe to ignore this ow, and to setg 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 SpinDensityWavesFinally, 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 spindensitywave (SDW)order. Ourdiscussion hereismotivatedby 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.Webeginwiththebandstructuredescribingthecupratesintheover-dopedregion, wellaway fromtheMott insulator. Heretheelectronsciaredescribed by the HamiltonianHc = i c (ors > 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 excitedS = 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, andis the correlation length exponent. Now imagine turning onanon-zerotemperature. AslongasTissmallerthanthetriplongap, 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.NowletuslookatthecomplementarybehavioratT >0ontheNeel-ordered side of the transition, withs < 0. In two spatial dimensions, thermaluctuationsprohibitthebreaking ofanon-Abeliansymmetryatall T> 0, 2011 by Taylor and Francis Group, LLC12 Understanding Quantum Phase TransitionsClassicalspinwavesDilutetriplongasQuantumcriticalFIGURE1.5Finite temperature crossovers of the coupled dimer antiferromagnet in Fig. 1.1.andsospinrotationsymmetryisimmediatelyrestored.Nevertheless, thereisan exponentiallylarge spincorrelation length, ,andatdistances shorterthanwe 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, weexpectthisspin-wave+Higgs pictureto apply atall temperatureslowerthanthenaturalenergyscale,i.e., forT [s[z, i.e., athigher temperaturesinthevicinityof 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]. Totheright ofthequantumcritical region, we again have a regime ofclassicaldynamics, 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 timerof thermal excitations [29]. Instead, quantumandthermal eects are equally important, and involve the non-trivial dynamics ofthe xed-point theory describing the quantum critical point. Note that whilethexed-pointtheoryappliesonlyatasinglepoint(=c)atT=0,itsinuence broadens into the quantum critical region atT> 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 liquidFIGURE1.6Finite temperature crossovers of graphene as a function of electron densityn(tuned) by in Eq. (1.7) and temperatureT. Adapted from Ref. [30].is no characteristic energy scale associated with thexed-point theory, kBTistheonlyenergy scaleavailable todetermineratnon-zero temperatures.Thus, in the quantum critical region [29]r = (

kBT, (1.9)where (is a universal constant dependent only upon the universality class ofthexedpointtheory,i.e.,itisauniversal numberjustlikethecritical ex-ponents. This value of rdetermines the friction coecients associated withthe dissipative relaxation of spin uctuations in the quantum critical region.Itisalso important forthetransport co-ecients associated withconservedquantities, and this will be discussed in Section 1.3.Let us now consider the similarT> 0 crossovers for the other models ofSection 1.1.TheNeel-VBStransitionof Section1.1.2hascrossoversverysimilartothoseinFig. 1.5, withoneimportant dierence. TheVBSstatebreaksadiscrete 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 temperatureTVBS. Thevalue ofTVBSvanishes very rapidly ass 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 crossoverdiagramshown in Fig. 1.6, as noted by Sheehy and Schmalian [30]. We have theFermi liquid regimes of the electron- and hole-like Fermi surfaces on either side 2011 by Taylor and Francis Group, LLC14 Understanding Quantum Phase TransitionsFluctuatingFermipocketsLargeFermisurfaceStrangeMetalSpin density wave (SDW)FIGURE1.7Finitetemperaturecrossovers neartheSDWordering transition ofFig.1.4.Here, anticipating application to the cuprates to be discussed in Section 1.6,we have assumed that the carrier density, x, tunes the system across the SDWtransition.of the critical point, along with an intermediate quantum critical Dirac liquid.Anewfeaturehereisrelatedtothelogarithmicowof thedimensionlessnestructureconstant controllingtheCoulombinteractions, whichwasnotedinSection1.1.3. Inthequantumcritical region, this constanttakesthetypical value 1/ ln(1/T). Consequently, fortherelaxationtimeinEq. (1.9) we have ( ln2(1/T). This time determines both the width of theelectron spectral functionsand also thetransport coecients, as we will seein Section 1.3.Finally, we turn to the SDW transition of Section 1.1.4. From the evolutionin Fig. 1.4 and the discussion above, we have the crossover phase diagram [31]in Fig. 1.7. As in Fig. 1.5, there is no transition at non-zero temperatures, andnow the crossover is between the topologically distinct Fermi surface cong-urations of Fig. 1.4. The Hertz action for the SDW uctuations in Eq. (1.8)predicts [26,31] logarithmic corrections to the leading scaling behavior, similarto those for graphene. The interplay of such SDW uctuations with the topo-logical change in the Fermi surface conguration is not fully understood [32],and is labeled as the strange metal regime in Fig. 1.7. 2011 by Taylor and Francis Group, LLCFinite Temperature Dissipation and Transport Near QCPs 151.3 QuantumCriticalTransportWe now turn to the transport properties in the quantum critical region: weconsider the response functions associated with any globally conserved quan-tity. For the antiferromagnetic systems in Sections 1.1.1 and 1.1.2, this requiresconsideration of the transport of total spin, and the associated spin conductiv-itiesand diusivities.Forgraphene, we can considercharge and momentumtransport. Ourdiscussionbelowwill alsoapplytothesuperuid-insulatortransition: for bosons in a periodic potential, this transition is described [33]by a eld theory closely related to that in Eq. (1.3). However, we will primar-ily use a language appropriate to charge transport in graphene below. We willdescribethepropertiesofagenericstrongly-coupledquantumcriticalpointandmention,where appropriate, thechanges dueto thelogarithmic owofthe coupling in graphene.Intraditional condensedmatterphysics, transportisdescribedbyiden-tifyingthelow-lyingexcitationsof thequantumgroundstate, andwritingdown transport equations for the conserved charges carried by them. Often,theseexcitationshaveaparticle-likenature,suchasthetriplonparticlesofFig. 1.5 or the electron or hole quasiparticles of the Fermi liquids in Fig. 1.6.Inothercases, thelow-lyingexcitationsarewaves, suchasthespin-wavesinFig. 1.5, andtheirtransportisdescribedbyanon-linearwaveequation(such as the Gross-Pitaevski equation). However, as we have discussed in Sec-tion 1.2 neither description is possible in the quantum critical region, becausethe excitations do not have a particle-like or wave-like character.Despite the absence of an intuitive description of the quantum critical dy-namics, weexpectthatthetransportpropertieshaveauniversal characterdetermined by the quantum eld theory of the quantum critical point. In ad-ditiontodescribingsingleexcitations, thiseldtheoryalsodeterminestheS-matrix of these excitations by the renormalization group xed-point valueof the couplings, and these should be sucient to determine transport proper-ties [34]. The transport coecients and the relaxation time to local equilibriumare not proportional to a mean free scattering time between the excitations,as isthecase intheBoltzmanntheory of quasiparticles. Such atimewouldtypically depend upon the interaction strength between the particles. Rather,thesystembehaveslikeaperfectuidinwhichtherelaxationtimeisasshort as possible, and is determined universally by the absolute temperature,asindicatedinEq.(1.9).Indeed,itwasconjecturedinRef.[1]thatthere-laxationtimeinEq.(1.9)isagenericlowerboundforinteractingquantumsystems. Thus the non-quantum-critical regimes of all the phase diagrams inSection 1.2 have relaxation times which are all longer than Eq. (1.9).The transport coecients of this quantum-critical perfect uid also do notdepend upon the interaction strength, and can be connected to the fundamen-talconstants ofnature.Inparticular, theelectrical conductivity, ,isgiven 2011 by Taylor and Francis Group, LLC16 Understanding Quantum Phase Transitionsby (in two spatial dimensions) [34]Q =e2h, (1.10)whereisauniversaldimensionlessconstantoforderunity, andwehaveadded the subscript Q to emphasize that this is the conductivity for the case ofgraphene with the Fermi level at the Dirac point (for the superuid-insulatortransition, this would correspond to bosons at integer lling) with no impurityscattering, and at zero magnetic eld. Here e is the charge of the carriers: fora superuid-insulatortransition ofCooperpairs, wehavee= 2e, while forgraphene we have e = e. The renormalization group ow of the ne structureconstant of graphene to zero at asymptotically low Tallows an exact com-putationinthiscase[35]: 0.05 ln2(1/T). Forthesuperuid-insulatortransition, isT-independent(thisisthegenericsituationwithnon-zeroxed point values of the interaction1) but it has only been computed [1, 34] toleading order in expansions in 1/Nand in 3d, where Nis the number of or-der parameter components and d is the spatial dimensionality. However, bothexpansions are neither straightforward nor rigorous, and require a physicallymotivatedresummationofthebareperturbativeexpansiontoall orders.Itwould therefore be valuable to have exact solutions of quantum critical trans-port where the above results can be tested, and we turn to such solutions inSec. 1.4.In addition to charge transport, we can also consider momentum transport.This was treated in the context of applications to the quark-gluon plasma [37];application of the analysis of Ref. [34] shows that the viscosity, , is given bys=

kB, (1.11)wheres is the entropy density, and again is a universal constant of orderunity. Thevalueof hasrecentlybeencomputed[40] forgraphene, andagainhasalogarithmicTdependencebecauseof themarginallyirrelevantinteraction: 0.008 ln2(1/T).We conclude this section by discussing some subtle aspects of the physicsbehind the seemingly simple result in Eq. (1.10). For simplicity, we will con-sider the case of a relativistically invariant quantum critical point in 2+1 di-mensions (such as the eld theories of Section 1.1.1 and 1.1.2), but marginallyviolated by graphene, a subtlety we ignore below). Consider the retarded corre-lation function of the charge density, (k, ), where k = [k[ is the wavevector,andisfrequency; thedynamicconductivity, (), isrelatedtobytheKubo formula,() =limk0ik2(k, ). (1.12)1For the case of neutral bosonsuperuids (but not chargedsystems like graphene),hydrodynamic long-time tails cause the constantsDand 2to acquire a weaklogarithmicdependenceon /kBTatsmallinasamplewithperfectmomentumconservation[36]. 2011 by Taylor and Francis Group, LLCFinite Temperature Dissipation and Transport Near QCPs 17It was argued in Ref. [34] that despite the absence of particle-like excitationsof thecritical groundstate, thecentral characteristicof thetransportisacrossover from collisionless to collision-dominated transport. At high frequen-ciesorlowtemperatures,thelimitingformforreducestothatatT= 0,which is completely determined by relativistic and scale invariance and currentconversion up to an overall constant(k, ) =e2hKk2

v2k2( +i)2, () =e2hK; kBT, (1.13)whereKisauniversal number[38]. However, phase-randomizingcollisionsare intrinsically present in any strongly interacting critical point (above onespatial dimension) and these lead to relaxation of perturbations to local equi-librium and the consequent emergence of hydrodynamic behavior. So at lowfrequencies,wehave insteadanEinsteinrelation whichdeterminesthecon-ductivity with(k, ) = e2cDk2Dk2i, () = e2cD =e2h12; 0ablackhole appears, resulting in an AdS-Schwarzschild spacetime, andTis also theHawking temperature of the black hole; the real time solutions also extend toT> 0.The results of a full computation [42] of the density correlation function,(k, ),are shown inFig.1.8 and 1.9. Themostimportant featureoftheseresults is that the expected limiting forms in the collisionless (Eq. (1.13)) and 2011 by Taylor and Francis Group, LLCFinite Temperature Dissipation and Transport Near QCPs 190.25 0.5 0.75 1 1.25 1.5 1.75 224681012q = 0.2, 0.5, 1.0FIGURE1.9As in Fig. 1.8, but for the collision-dominated regime.collision-dominated (Eq. (1.14)) are obeyed. Thus theresults do display thecollisionless tocollision-dominated crossover ata frequencyoforderkBT/,as was postulated in Section 1.3.An additional important feature of the solution is apparent upon describingthe full structure of both the density and current correlations. Using spacetimeindices (, =t, x, y) we can represent these as thetensor(k, ), wherethepreviouslyconsidered tt. At T >0, wedonotexpect toberelativistically covariant, and so can only constrain it by spatial isotropy anddensityconservation. Introducingaspacetimemomentump=(, k), andsetting the velocity v = 1, these two constraints lead to the most general form(k, ) =e2h

p2

PT KT(k, ) +PL KL(k, )

(1.16)wherep2= ppwith= diag(1, 1, 1), andPTandPLare orthogo-nal projectors dened byPT00 = PT0i = PTi0 = 0 , PTij= ijkikjk2, PL=

ppp2

PT, (1.17)with the indices i, j running over the 2 spatial components. The two functionsKT,L(k, )deneall thecorrelators ofthedensity and thecurrent, and theresults in Eqs. (1.14) and (1.13) are obtained by taking suitable limits of thesefunctions. We will also need below the general identityKT(0, ) = KL(0, ), (1.18) 2011 by Taylor and Francis Group, LLC20 Understanding Quantum Phase Transitionswhich follows from the analyticity of theT> 0 current correlations at k = 0.The relations of the previous paragraph are completely general and applytoanytheory.SpecializingtotheAdS-Schwarzschild solutionofSYM3,theresults were found to obey a simple and remarkable identity [42]:KL(k, )KT(k, ) = /2(1.19)where / is a known pure number, independent of andk. It was also shownthat such a relation applies to any theory which is equated to classical gravityon AdS4, and is a consequence of the electromagnetic self-duality of its four-dimensional Maxwell sector. The combination of Eqs. (1.18) and (1.19) fullydetermines thecorrelators at k = 0: we ndKL(0, ) =KT(0, ) = /,fromwhichitfollowsthatthek= 0conductivityisfrequencyindependentand that = 12 = K = /. These last features are believed to be specialtotheorieswhichareequivalenttoclassical gravity, andnottoholdmoregenerally.We can obtain further insight into the interpretation of Eq. (1.19) by con-sidering the eld theory of the superuid-insulator transition of lattice bosonsat integer lling. As wenotedearlier, this is givenbytheeldtheoryinEq. (1.3)withtheeldahaving2components. Itisknownthatthis2-component theory of relativistic bosons is equivalent to a dual relativistic the-ory, o, of vortices, under the well-known particle-vortex duality [43]. Ref. [42]considered theaction ofthisparticle-vortex dualityonthecorrelation func-tions in Eq. (1.16), and found the following interesting relations:KL(k, )KT(k, ) = 1 , KT(k, )KL(k, ) = 1 (1.20)where KL,Tdetermine the vortex current correlations in oas in Eq. (1.16).Unlike Eq. (1.19), Eq. (1.20) does not fully determine the correlation functionsat k = 0: it only serves to reduce the 4 unknown functionsKL,T, KL,Tto 2unknown functions. The key property here is that while the theories oLG andoare dual to each other, they are not equivalent, and the theory oLGis notself-dual.WenowseethatEq. (1.19)impliesthattheclassical gravitytheoryofSYM3isself-dual underananalogof particle-vortexduality[42]. Itisnotexpected that this self-duality will hold when quantum gravity corrections areincluded; equivalently, the SYM3 at niteNis expected to have a frequencydependence in its conductivity at k = 0. If we apply the AdS/CFT correspon-dence to the superuid-insulator transition, and approximate the latter theoryby classical gravity on AdS4, we immediately obtain the self-dual predictionfor the conductivity, = 1. This value is not far from that observed in nu-merous experiments, and we propose here that the AdS/CFT correspondenceoers a rationale for understanding such observations. 2011 by Taylor and Francis Group, LLCFinite Temperature Dissipation and Transport Near QCPs 211.5 HydrodynamicTheoryThe successful comparison between the general considerations of Section 1.3,andtheexactsolutionusingtheAdS/CFTcorrespondenceinSection1.4,emboldensustoseekamoregeneral theoryoflowfrequency( 1 of the nitetemperature incoherent tunneling rate between the wells seems to have some 2011 by Taylor and Francis Group, LLC54 Understanding Quantum Phase Transitionssupport from experiments [54]. This rate is1=e()22c()( + 1/2)kBTc

21. (2.44)Instead, the attention has shifted more to reducing dissipation as much as pos-sible in various Josephson devices to observe coherent oscillations of the doublewell [55, 56]. This is considered to be a miniature prototype for Schr odingerscat. Itcanbedebatedastotheextenttowhichthephasedierenceof aJosephson device or equivalently the ux variable in a superconducting inter-ference device can be considered a macroscopic variable, which appears tobeasemantic issue.These are indeedcollective degrees of freedom,buttheenergeticsaredeterminedbymicroscopicscales. Verylittleexperimentalattentionhasbeenpaidtoexploringthequantumcriticaldynamicsofhowoneofthebasicmodelsofquantummechanicsisinuencedbydissipation.As mentioned above, the inuence of dissipation in quantum tunneling froma metastable state does not count, because no quantum criticality is involved.Giventhat convincingexperimental demonstrations of quantumcriticalityaresofewandfarbetweenandgenerallysocomplex(duetocomplexma-terial issues), it seems hopeless to make further progress without a clear-cutstudyofthesimplestpossibleexample. Inanycase, asIhaveargued, thisis a well-studied example of a quantum-to-classical transition (FAPP) whosemathematics is rmly grounded at this time.Thesecond overarching themeofthischapterhasbeenhowdisorderin-uences QPTs. While much is understood for classical phase transitions, veryfewresultsare available fortransitions at zero temperature.Thisis again apity because there is considerable theoretical depth to this problem. In fact,practical applications aboundas well.Is itpossible thatmany experimentalsightingsof quantumcritical pointsareinrealityrstordertransitionsindisguise,roundedbydisorder?Ifso,howdoesitenrichourunderstanding?An interesting question is the uses of von Neumann entropy. There appear tobe some QPTs that exhibit no non-analyticity of the ground state energy andyettheirexistencecanbehardlydenied, astheysignifytransitionbetweentwo distinct states of matter. In this respect, it was quite remarkable that thevonNeumannentropyshouldexhibittherequisitenon-analyticity. Perhapsthe fundamental criterion should be replaced. So far we have only found suchexamples in disordered systems involving non-interacting electrons, Andersonlocalization in three dimensions, and plateau-to-plateau transition in integerquantumHallsystems.Aretheremore,especiallyinvolvinginteractingsys-tems? Is there a theorem? I conjecture that such results are only possible indisordered systems where the fundamental driving mechanism is uctuationsdriven by disorder, belonging to a dierent class from quantum uctuations,andtriggered byatuningcouplingconstant.Itwillbeinterestingtotacklethefractional quantumHall problemfromtheperspectiveoftheJaincon-struction [57], as this maps the problem to an essentially non-interacting one.Anotherprobleminvolvingquantumcriticalityindisorderedsystemsis 2011 by Taylor and Francis Group, LLCDissipation, Quantum Phase Transitions, and Measurement 55the innite randomness xed point. A number of important theoretical calcu-lations involving entanglement entropy have shown that the renormalizationgroup ow is to new xed points, dierent from the pure system xed points,but with universal amplitudes of logarithmic entanglement entropies. Do thesexedpointsreectthesamepropertiesofconformal invarianceofpuresys-tems?Or, arethemathematical underpinningsdierent?Aretherehigherdimensional problems that can be solved in a similar manner?Finally, a more pressing issue is the role of dissipative phase transitions ina number of important elds of current research, to name a few, phase slips innanowires [5864],c-axis conductivity [65], and local quantum criticality [66]in high temperature superconductors [67]. There is also much interest in low-temperature properties ofvery thinsuperconductingwires. Thekey processof interest is quantum phase slips, a virtual depletion of the superconductingdensitythatallowsthesystemtotunnel toadierentvalueof thesuper-current. Therateof thisprocessdependsnotonlyonthebarefugacityofthe phase slips, dened by the rate of an individual tunneling event, but alsoimportantly on the interaction between individual quantum phase slips. Thebackboneofthetheoreticalworkisthelogarithmicinteractionbetweenthequantum phase slips, whichserves almost as a paradigm to a whole class ofsimilarproblems[6870]. Interestingresultshavebeenobtained[69]:ithasbeenarguedthatforashortwirethereisnodistinctionbetweenasuper-conductor and an insulator. Even an insulator can support a weak Josephsoncurrent. Nonetheless, there is a range of parameters for which a short nanowirecan act as an insulator down to unobservably low temperatures.Acknowledgments I acknowledge a grant from the National Science Foun-dation (Grant No. DMR-0705092) and discussions with David Garcia-Aldea,Klaus Hepp, Joel Lebowitz, Chetan Nayak, Gil Refael, and David Schwab.Bibliography[1] A. J. Leggett, Journal of Physics-Condensed Matter 14, R415 (2002).[2] A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987).[3] S. Chakravarty, Phys. Rev. Lett. 49, 681 (1982).[4] K. Hepp, Helv. Phys. Acta 45, 237 (1972).[5] C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952).[6] A. Kopp, X. Jia, and S. Chakravarty, Ann. Phys. 322, 1466 (2007).[7] D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. 66, 261 (1994).[8] J. T. Edwards and D. J. Thouless, J. Phys. C 4, 453 (1971). 2011 by Taylor and Francis Group, LLC56 Understanding Quantum Phase Transitions[9] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrish-nan, Phys. Rev. Lett. 42, 673 (1979).[10] B. Huckestein, Rev. Mod. Phys. 67, 357 (1995).[11] P. Goswami, D. Schwab, andS. Chakravarty, Phys. Rev. Lett. 100,015703 (2008).[12] R. L. Greenblatt, M. Aizenman, andJ. L. Lebowitz, Phys. Rev. Lett.103, 197201 (2009).[13] M. Aizenman and J. Wehr, Comm. Math. Phys. 130, 489 (1990).[14] T. D. Lee and C. N. Yang, Phys. Rev. 87, 410 (1952).[15] K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986).[16] R. R. P. Singh and S. Chakravarty, Phys. Rev. Lett. 57, 245 (1986).[17] J. L. Lebowitz, arXiv:0709.0724 (2007).[18] S. Gold