Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information

358 Pages·2015·10.752 MB·English

by  Amit Dutta, Gabriel Aeppli, Bikas K. Chakrabarti, Uma Divakaran, Thomas F. Rosenbaum, Diptiman Sen

Quantum Phase Transitions
in Transverse Field Spin Models
FromStatisticalPhysicstoQuantumInformation
AmitDutta
GabrielAeppli
BikasK.Chakrabarti
UmaDivakaran
ThomasF.Rosenbaum
DiptimanSen
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(cid:13)c AmitDutta,GabrielAeppli,BikasK.Chakrabarti,UmaDivakaran,
ThomasF.RosenbaumandDiptimanSen2015
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Dutta,Amit,1968-
Quantumphasetransitionsintransversefieldspinmodels:From
StatisticalPhysicstoQuantumInformation/AmitDutta[and5others].
pagescm
Includesbibliographicalreferencesandindex.
Summary:”Discussesthefundamentalconnectionsbetweenthephysicsof
quantumphasetransitionsandthetechnologicalpromiseofquantum
information,non-equilibriumquantumdynamicsandadiabaticquantum
computations”–Providedbypublisher.
ISBN978-1-107-06879-7(hardback)
1.Phasetransformations(Statisticalphysics)2.Phaseruleandequilibrium.I.Title.
QC175.16.P5D882015
530.4’74–dc23
2014027793
ISBN978-1-107-06879-7Hardback
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Toourparents
DollDuttaandSukomalDutta
DorotheeAeppliandAlfredAeppli
PratimaChakrabartiandBimalK.Chakrabarti
SavithryE.S.andA.P.Divakaran
HannaL.RosenbaumandMartinM.Rosenbaum
GeetiSenandAmiyaKumarSen

Contents
ListofFiguresandTables ix
Preface xv
Acknowledgements xvii
I AnIntroductiontoQuantumPhaseTransitions,
InformationandDynamics 1
1. QuantumPhaseTransitions 3
1.1 AimandScopeofthisBook 3
1.2 ABriefIntroductiontoClassicalPhaseTransitions: NotionofUniversality 12
1.3 QuantumPhaseTransitions 14
1.4 TransverseIsingandXYModels 17
1.5 Quantum-ClassicalCorrespondenceandScaling 22
1.6 QuantumRotorModels 28
1.7 JosephsonJunctionArrays 29
1.8 MatrixProductStates 30
1.9 ChapterSummary 31
2. InformationTheoreticMeasuresClosetoaQuantumCriticalPoint 32
2.1 EntanglementEntropy 32
2.2 Concurrence 34
2.3 QuantumFidelity 36
2.4 ChapterSummary 42
3. Non-EquilibriumDynamicsacrossQuantumCriticalPoints 44
3.1 DefectGeneration: Kibble–Zurek(KZ)Scaling 45
3.2 AdiabaticPerturbationTheory: SlowandSuddenQuenches 49
3.3 Thermalization 53
3.4 ChapterSummary 54
II TransverseFieldModels: Statics 57
4. TransverseIsingModelsinHigherDimensions 59
4.1 MeanFieldTheories 59
vi Contents
(A) EquivalentSingleSpinModel 59
(B) InfiniteRangeInteractions 60
(C) LargeSpinLimits: TransverseXYSpinChain 62
4.2 ChapterSummary 64
5. TransverseFieldModelsinOneDimension 65
5.1 ExactSolutioninOneDimension: Jordan−WignerTransformation 65
5.2 ConnectiontoConformalFieldTheory 68
5.3 QuantumSpinChainsCoupledtoaBath 72
5.4 ChapterSummary 74
6. QuantumPhaseTransitionsinRelatedModels 75
6.1 SomeExactlySolvableModelsRelatedtoTransverseIsingand
XYModels 75
6.2 TheDickeModel 78
6.3 TopologicalQuantumPhaseTransitions 80
6.4 ExactSolutionoftheKitaevModel 80
(A) One-DimensionalModel 81
(B) Two-DimensionalKitaevModel 84
6.5 One-Dimensional p-WaveSuperconductingChain: MajoranaFermions 89
6.6 QuantumPhaseTransitioninDiracHamiltonians: Grapheneand
TopologicalInsulators 93
6.7 ChapterSummary 98
7. RoleofQuenchedDisorder 100
7.1 AModifiedHarrisCriterion 100
7.2 QuantumIsingSpinGlass(QISG) 101
7.3 GriffithsSingularitiesandActivatedDynamics 104
7.4 AGeneralizedRandomTransverseFieldIsingSpinChain 107
7.5 HigherDimensionalRealizationofInfiniteRandomnessFixedPoint
(IRFP) 108
7.6 QuantumIsingModelinaRandomLongitudinalField 110
7.7 ChapterSummary 111
8. RelatedModelswithFrustration 112
8.1 QuantumANNNIModel 112
8.2 QuantumLifshitzPoint 119
8.3 ModelswithLong-RangeAntiferromagneticInteractions 120
8.4 ChapterSummary 121
9. QuantumInformationTheoreticMeasures: TransverseFieldand
RelatedModels 122
Contents vii
9.1 Concurrence 122
9.2 EntanglementEntropy 124
9.3 QuantumDiscord 127
9.4 QuantumFidelity 129
9.5 ScalingoftheGeometricPhaseClosetoaQCP 133
9.6 QuantumCriticalEnvironment: DecoherenceandLoschmidtEcho 135
9.7 RoleofMarginalityintheScalingofFidelity: 2DDiracPoint 139
9.8 FidelitySusceptibilityforaModelwithνd >2 144
9.9 ChapterSummary 147
III TransverseFieldandRelatedModels: Non-Equilibrium
Dynamics 149
10. Non-EquilibriumDynamicsAcrossQuantumCriticalPoints:
SlowQuenching 151
10.1 LinearandNon-LinearSlowEvolutionthroughCriticalPoints 151
10.2 QuenchingthroughMulticriticalPoints 156
10.3 GeneralizedQuenchingSchemes 158
10.4 QuenchingthroughGaplessPhases 161
10.5 QuenchingthroughanAnisotropicQuantumCriticalPoint 164
10.6 GeneralizedKibble–ZurekScalingRelation 164
10.7 QuenchingofaDisorderedChain 166
10.8 QuenchingwithCouplingtoaBath 167
10.9 KZMechanismandAdiabaticEvolution 168
10.10 KZMechanisminSpace 170
10.11 ChapterSummary 171
11. FurtherStudiesonNon-EquilibriumDynamics 174
11.1 SuddenQuenches: DefectandHeatDensity 174
11.2 SuddenQuenches: SemiclassicalTheory 176
11.3 LocalQuench 180
11.4 DynamicsFollowingaRapidQuenchingthroughaQCP 182
11.5 EffectofTopologyonQuantumQuenching 186
11.6 StudiesofTomonaga–LuttingerLiquids 191
11.7 ChapterSummary 192
12. QuenchingandQuantumInformation 194
12.1 SlowQuenches 194
12.2 SuddenQuenches 200
12.3 DynamicsofDecoherence 202
viii Contents
12.4 ChapterSummary 205
13. SomeRecentDevelopmentsinInformationandDynamics 207
13.1 TransitionlessDynamics 207
13.2 Quenches,LoschmidtEchoandNonanalyticities 210
13.3 PeriodicDrivingandFloquetTheory: TransverseIsingModel 213
13.4 DynamicalGenerationofEdgeStates 216
13.5 DynamicsofaChainofHardCoreBosons: DynamicalLocalization 220
13.6 ChapterSummary 225
IV ExperimentalRealizationsofTransverseFieldModels 229
14. ExperimentalRealizationsofTransverseFieldIsingSystems 231
14.1 SingletGroundStateMagnets 231
14.2 Order/DisorderTransitionsinHydrogen-BondedandOther
Ferro/AntiferroelectricSystems 233
14.3 Low-DimensionalMagneticRealizationsoftheTransverse
FieldIsingModel 235
14.4 IsingDoubletsinExternalMagneticFieldsinThree-Dimensions 237
14.5 DisorderandtheTransverseFieldIsingModel 239
14.6 ChapterSummary 244
V QuantumAnnealingandAdiabaticQuantumComputations
UsingTransverseFieldModels 245
15. AdiabaticQuantumComputationsandTransverseFieldModels 247
15.1 TheoreticalStudies 249
15.2 ExperimentalStudies 254
15.3 AQuantumComputerintheMaking? 261
15.4 ChapterSummary 263
16. ConcludingComments 265
AppendixA DerivationofaMatrixProductHamiltonian 267
AppendixB FromJordan−WignertoBosonization: Tomonaga−LuttingerLiquids 269
AppendixC CalculationoftheEntanglementEntropyforaTransverseIsingChain 278
AppendixD TheLoschmidtEchofortheOne-DimensionalXYModel 281
AppendixE Landau−ZenerTunneling: CalculationofNon-AdiabaticTransition
Probability 283
AppendixF ANoteontheTheoreticalStudiesofHysteresis 292
Bibliography 295
Index 329
ColorPlates 331