Sachdev, Quantum Phase Transitions, 2ed, Springer, 2011[1].pdf

1、This page intentionally left blankQuantumPhaseTransitionsSecondEditionThis is the first book to describe the physical properties of quantum materials near criticalpoints with long-range many-body quantum entanglement.Readers are introduced to thebasic theory of quantum phases,their phase transitions

2、,and their observable properties.This second edition begins with nine chapters,six of them new,suitable for an introduc-tory course on quantum phase transitions,assuming no prior knowledge of quantum fieldtheory.There are several new chapters covering important recent advances,such as theFermi gas n

3、ear unitarity,Dirac fermions,Fermi liquids and their phase transitions,quan-tum magnetism,and solvable models obtained from string theory.After introducing thebasic theory,it moves on to a detailed description of the canonical quantum-critical phasediagram at nonzero temperatures.Finally,a variety o

4、f more complex models is explored.This book is ideal for graduate students and researchers in condensed matter physics andparticle and string theory.SubirSachdev is Professor of Physics at Harvard University and holds a DistinguishedResearch Chair at the Perimeter Institute for Theoretical Physics.H

5、is research has focusedon a variety of quantum materials,and especially on their quantum phase transitions.QuantumPhaseTransitionsSecondEditiontSUBIRSACHDEVHarvardUniversityCAMBRIDGE UNIVERSITY PRESSCambridge,New York,Melbourne,Madrid,Cape Town,Singapore,So Paulo,Delhi,TokyoCambridge University Pres

6、sThe Edinburgh Building,Cambridge CB2 8RU,UKPublished in the United States of America by Cambridge University Press,New Yorkwww.cambridge.orgInformation on this title:www.cambridge.org/9780521514682c?S.Sachdev 2011This publication is in copyright.Subject to statutory exceptionand to the provisions o

7、f relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.First published 2011Printed in the United Kingdom at the University Press,CambridgeA catalog record for this publication is available from the British Lib

8、raryLibrary of Congress Cataloging-in-Publication DataSachdev,Subir,1961-Quantum phase transitions/Subir Sachdev.Second edition.p.cmIncludes bibliographical references and index.ISBN 978-0-521-51468-2(Hardback)1.Phase transformations(Statistical physics)2.Quantum theory.I.Title.QC175.16.P5S23 201153

9、0.4?74dc222010050328ISBN 978-0-521-51468-2 HardbackCambridge University Press has no responsibility for the persistence oraccuracy of URLs for external or third-party internet websites referred toin this publication,and does not guarantee that any content on suchwebsites is,or will remain,accurate o

10、r appropriate.TomyparentsandMenaka,Monisha,andUshaContentsFrom the Preface to the first editionpage xiiiPreface to the second editionxviiPartI Introduction11 Basicconcepts31.1 What is a quantum phase transition?31.2 Nonzero temperature transitions and crossovers51.3 Experimental examples81.4 Theoret

11、ical models91.4.1 Quantum Ising model101.4.2 Quantum rotor model121.4.3 Physical realizations of quantum rotors142 Overview182.1 Quantum field theories212.2 Whats different about quantum transitions?24PartII Afirstcourse273 Classicalphasetransitions293.1 Mean-field theory303.2 Landau theory333.3 Flu

12、ctuations and perturbation theory343.3.1 Gaussian integrals363.3.2 Expansion for susceptibility39Exercises424 Therenormalizationgroup454.1 Gaussian theory464.2 Momentum shell RG484.3 Field renormalization534.4 Correlation functions54Exercises56viiviiiContents5 ThequantumIsingmodel585.1 Effective Ham

13、iltonian method585.2 Large-g expansion595.2.1 One-particle states605.2.2 Two-particle states615.3 Small-g expansion645.3.1 d=2645.3.2 d=1665.4 Review675.5 The classical Ising chain675.5.1 The scaling limit705.5.2 Universality715.5.3 Mapping to a quantum model:Ising spin ina transverse field725.6 Map

14、ping of the quantum Ising chain to a classical Ising model74Exercises776 Thequantumrotormodel796.1 Large-g expansion796.2 Small-g expansion806.3 The classical XY chain and an O(2)quantum rotor826.4 The classical Heisenberg chain and an O(3)quantum rotor886.5 Mapping to classical field theories896.6

15、Spectrum of quantum field theory906.6.1 Paramagnet916.6.2 Quantum critical point926.6.3 Magnetic order92Exercises957 Correlations,susceptibilities,andthequantumcriticalpoint967.1 Spectral representation977.1.1 Structure factor987.1.2 Linear response997.2 Correlations across the quantum critical poin

16、t1017.2.1 Paramagnet1017.2.2 Quantum critical point1037.2.3 Magnetic order104Exercises1078 Brokensymmetries1088.1 Discrete symmetry and surface tension1088.2 Continuous symmetry and the helicity modulus1108.2.1 Order parameter correlations112ixContents8.3 The London equation and the superfluid densi

17、ty1128.3.1 The rotor model115Exercises1159 BosonHubbardmodel1179.1 Mean-field theory1199.2 Coherent state path integral1239.2.1 Boson coherent states1259.3 Continuum quantum field theories126Exercises130PartIII Nonzerotemperatures13310 TheIsingchaininatransversefield13510.1 Exact spectrum13710.2 Con

18、tinuum theory and scaling transformations14010.3 Equal-time correlations of the order parameter14610.4 Finite temperature crossovers14910.4.1 Low T on the magnetically ordered side,?0,T?15110.4.2 Low T on the quantum paramagnetic side,?gc17511.2.2 Critical point,g=gc17711.2.3 Magnetically ordered gr

19、ound state,g gc,T?+18611.3.2 High T,T?+,?18611.3.3 Low T on the magnetically ordered side,g gc,T?18711.4 Numerical studies18812 Thed=1,O(N3)rotormodels19012.1 Scaling analysis at zero temperature19212.2 Low-temperature limit of the continuum theory,T?+19312.3 High-temperature limit of the continuum

20、theory,?+?T?J199xContents12.3.1 Field-theoretic renormalization group20112.3.2 Computation of u20512.3.3 Dynamics20612.4 Summary21113 Thed=2,O(N3)rotormodels21313.1 Low T on the magnetically ordered side,T?s21513.1.1 Computation of c21613.1.2 Computation of 22013.1.3 Structure of correlations22213.2

21、 Dynamics of the quantum paramagnetic and high-T regions22513.2.1 Zero temperature22713.2.2 Nonzero temperatures23113.3 Summary23414 Physicsclosetoandabovetheupper-criticaldimension23714.1 Zero temperature23914.1.1 Tricritical crossovers23914.1.2 Field-theoretic renormalization group24014.2 Statics

22、at nonzero temperatures24214.2.1 d 324814.3 Order parameter dynamics in d=225014.4 Applications and extensions25715 Transportind=226015.1 Perturbation theory26415.1.1 I26815.1.2 I I26915.2 Collisionless transport equations26915.3 Collision-dominated transport27315.3.1?expansion27315.3.2 Large-N limi

23、t27915.4 Physical interpretation28115.5 The AdS/CFT correspondence28315.5.1 Exact results for quantum critical transport28515.5.2 Implications28815.6 Applications and extensions289PartIV Othermodels29116 DiluteFermiandBosegases29316.1 The quantum XX model296xiContents16.2 The dilute spinless Fermi g

24、as29816.2.1 Dilute classical gas,kBT?|,030116.2.3 High-T limit,kBT?|30416.3 The dilute Bose gas30516.3.1 d gcand for g gc(g 0.It turns out that working outward from the quantum critical point at g=gcand T=0 is apowerful way of understanding and describing the thermodynamic and dynamic propertiesof n

25、umerous systems over a broad range of values of|g gc|and T.Indeed,it is not evennecessary that the system of interest ever have its microscopic couplings reach a valuesuch that g=gc:it can still be very useful to argue that there is a quantum critical pointat a physically inaccessible coupling g=gca

26、nd to develop a description in the deviation|ggc|.It is one of the purposes of this book to describe the physical perspective that suchan approach offers,and to contrast it with more conventional expansions about very weak(say g 0)or very strong couplings(say g ).1.2 Nonzerotemperaturetransitionsand

27、crossoversLet us now discuss some basic aspects of the T 0 phase diagram.First,let us ask onlyabout the presence of phase transitions at nonzero T.With this limited criterion,there aretwo important possibilities for the T 0 phase diagram of a system near a quantum criticalpoint.These are shown in Fi

28、g.1.2,and we will meet examples of both kinds in this book.In the first,shown in Fig.1.2a,the thermodynamic singularity is present only at T=0,andall T 0 properties are analytic as a function of g near g=gc.In the second,shown ingT0T0gcgcg(a)(b)tFig.1.2Twopossiblephasediagramsofasystemnearaquantumph

29、asetransition.Inbothcasesthereisaquantumcriticalpointatg=gcandT=0.In(b),thereisalineofT 0second-orderphasetransitionsterminatingatthequantumcriticalpoint.Thetheoryofphasetransitionsinclassicalsystemsdrivenbythermalfluctuationscanbeappliedwithintheshadedregionof(b).6BasicconceptsFig.1.2b,there is a l

30、ine of T 0 second-order phase transitions(this is a line at which thethermodynamic free energy is not analytic)that terminates at the T=0 quantum criticalpoint at g=gc.Moving beyond phase transitions,let us ask some basic questions about the dynamicsof the system.A very general way to characterize t

31、he dynamics at T 0 is in terms of thethermal equilibration time eq.This is the characteristic time in which local thermal equi-librium is established after imposition of a weak external perturbation(say,a heat pulse).Here we are excluding equilibration with respect to globally conserved quantities(s

32、uch asenergy or charge)which will take a long time to equilibrate,dependent upon the lengthscale of the perturbation:hence the emphasis on local equilibration.Global equilibrationis described by the equations of hydrodynamics,and we expect such equations to applyin all cases at times much larger tha

33、n eq.We focus here on the value of eqas a functionof g gcand T.From the energy scales discussed in Section 1.1,we can immediatelydraw an important distinction between two regimes of the phase diagram.We character-ized the ground state by the energy?in(1.1).At nonzero temperature,we have a secondenergy scale,kBT.Comparing the values of?and kBT,we are immediately led to theimportant phase diagram in