Quantum Quenching, Annealing and Computation

314 Pages·2010·8.43 MB·English

by  R.D. Somma

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A.K. Chandra
A. Das
B.K. Chakrabarti (Eds.)
Quantum Quenching,
Annealing and Computation
ABC
AnjanKumarChandra ArnabDas
SahaInstituteofNuclear AbdusSalamInternational
Physics CentreforTheoreticalPhysics
CentreforApplied (ICTP)
Mathematics& StradaCostiera,11
ComputationalScience 34014Trieste
1/AFBidhannagar Italy
Kolkata-700064 [email protected]
India
[email protected]
BikasK.Chakrabarti
SahaInstituteofNuclear
Physics
CentreforApplied
Mathematics&
ComputationalScience
1/AFBidhannagar
Kolkata-700064
India
[email protected]
Chandra A.K., Das A., Chakrabarti B.K. (Eds.): Quantum Quenching, Annealing and
Computation,Lect.NotesPhys.802(Springer,BerlinHeidelberg2010),
DOI10.1007/978-3-642-11470-0
LectureNotesinPhysicsISSN 0075-8450 e-ISSN 1616-6361
ISBN 978-3-642-11469-4 e-ISBN 978-3-642-11470-0
DOI10.1007/978-3-642-11470-0
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Preface
The process of realizing the ground state of some typical (frustrated) quantum
many-bodysystems,startingfromthe‘disordered’orexcitedstates,canbeformally
mappedtothesearchofsolutionsforcomputationallyhardproblems.Thedynamics
throughthecriticalpoint,inbetween,arethereforeextremelycrucial.Inthecontext
of such computational optimization problems, the dynamics (of rapid quenching
orslowannealing),whiletuningtheappropriatefieldsorfluctuations,inparticular
while crossing the quantum critical point, are extremely intriguing and are being
investigated these days intensively. Several successful methods and tricks are now
wellestablished.
This volume gives a collection of introductory reviews on such developments
written by well-known experts. It concentrates on quantum phase transitions and
their dynamics as the transition or critical points are crossed. Both the quenching
andannealingdynamicsareextensivelycovered.
Wehopethesetimelyreviewswillinspiretheyoungresearcherstojoinandcon-
tribute to this fast-growing, intellectually challenging, as well as technologically
demanding field. We are extremely thankful to the contributors for their intensive
work and pleasant cooperations. We are also very much indebted to Kausik Das
for his help in compiling this book. Finally, we express our gratitude to Johannes
Zittartz,SeriesEditor,LNP,andChristianCaronofphysicseditorialdepartmentof
Springerfortheirencouragementandsupport.
TriesteandKolkata,India AnjanKumarChandra
ArnabDas
BikasK.Chakrabarti
v

Contents
1 Quantum Approach to Classical Thermodynamics
andOptimization .............................................. 1
R.D.SommaandG.Ortiz
1.1 Introduction .............................................. 1
1.2 QuantumAnnealingStrategiesforOptimization................ 4
1.2.1 Classical-to-QuantumMapping ...................... 4
1.2.2 AdiabaticStatePreparation:RatesofConvergence ...... 7
1.2.3 QuantumMonteCarloMethodsforQuantumAnnealing . 10
1.2.4 QuantumQuadraticSpeedup ........................ 11
1.2.5 Efficient Quantum Gibbs State Preparation
ofOne-DimensionalSystems........................ 15
1.3 Quantum Algorithms for Numerical Integration
withNo ImportanceSampling .............................. 16
1.4 Conclusions .............................................. 19
References ..................................................... 19
2 Non-equilibriumDynamicsofQuantumSystems:OrderParameter
Evolution,DefectGeneration,andQubitTransfer ................. 21
S.Mondal,D.Sen,andK.Sengupta
2.1 Introduction .............................................. 21
2.2 QuenchDynamics ......................................... 23
2.2.1 UltracoldAtomsinanOpticalLattice ................. 23
2.2.2 InfiniteRangeIsingModelinaTransverseField ........ 29
2.3 Non-adiabaticDynamics.................................... 36
2.3.1 QuenchingAcrossaCriticalSurface .................. 39
2.3.2 Non-linearQuenchingAcrossaCriticalPoint .......... 43
2.3.3 ExperimentalRealizations........................... 49
2.4 QuantumCommunication................................... 51
2.5 Discussion ............................................... 53
References ..................................................... 55
vii
viii Contents
3 DefectProductionDuetoQuenchingThroughaMulticriticalPoint
andAlongaGaplessLine ....................................... 57
U.Divakaran,V.Mukherjee,A.Dutta,andD.Sen
3.1 Introduction .............................................. 57
3.2 ASpinModel:TransverseandAnisotropicQuenching .......... 59
3.3 QuenchingThroughaMulticriticalPoint ...................... 66
3.4 QuenchingAlongaGaplessLine ............................ 68
3.5 Conclusions .............................................. 72
References ..................................................... 72
4 AdiabaticPerturbationTheory:FromLandau–ZenerProblem
toQuenchingThroughaQuantumCriticalPoint .................. 75
C.DeGrandiandA.Polkovnikov
4.1 Introduction .............................................. 75
4.2 AdiabaticPerturbationTheory............................... 77
4.2.1 ApplicationtotheLandau–ZenerProblem ............. 80
4.3 Adiabatic Dynamics in Gapless Systems with
QuasiparticleExcitations ................................... 83
4.4 AdiabaticDynamicsNearaQuantumCriticalPoint............. 92
4.4.1 ScalingAnalysis................................... 93
4.4.2 Examples......................................... 95
4.5 SuddenQuenchesNearQuantumCriticalPoints................103
4.6 EffectoftheQuasiparticleStatisticsintheFiniteTemperature
Quenches ................................................ 106
4.7 Conclusions ..............................................109
References .....................................................113
5 QuenchDynamicsofQuantumandClassicalIsingChains:From
theViewpointoftheKibble–ZurekMechanism ...................115
S.Suzuki
5.1 Introduction ..............................................115
5.2 Kibble–ZurekMechanism ..................................118
5.3 QuenchinthePureIsingChain ..............................120
5.3.1 ClassicalQuench ..................................121
5.3.2 QuantumQuench ..................................125
5.4 QuenchintheRandomIsingChain...........................133
5.4.1 ClassicalQuench ..................................134
5.4.2 QuantumQuench ..................................138
5.5 Conclusion ...............................................140
5.5.1 SummaryoftheResults.............................140
5.5.2 DiscussionandFutureProblems......................141
References .....................................................142
Contents ix
6 QuantumPhaseTransitionintheSpinBosonModel ...............145
S.Florens,D.Venturelli,andR.Narayanan
6.1 Introduction ..............................................145
6.2 TheSpinBosonModel .....................................148
6.2.1 TheModel........................................148
6.2.2 ThePhasesoftheSBM .............................149
6.3 TheSBMUsingtheMajoranaRepresentation..................151
6.4 PerturbativeRenormalizationGroupintheLocalizedRegime.....152
6.5 AnalyzingtheRGFlow ....................................154
6.5.1 TheRGEquationsfortheOhmicCase(s=1)..........154
6.5.2 TheRGEquationsfortheSuper-ohmicCase(s>1) ....154
6.5.3 TheRGEquationsfortheSub-ohmicCase(0<s<1) ..155
6.6 MappingtoaLong-RangedIsingModel ......................156
6.7 ASpecialIdentity .........................................159
6.8 Shiba’sRelationfortheSub-ohmicSpinBosonModel ..........160
6.9 OnthePossibleBreakdownofQuantumtoClassicalMapping ...161
References .....................................................162
7 InfluenceofLocalMomentFluctuationsontheMottTransition .....163
C.Janani,S.Florens,T.Gupta,andR.Narayanan
7.1 Introduction ..............................................163
7.2 TheHeisenbergModel .....................................164
7.2.1 Single-SiteRepresentationoftheHeisenbergModel.....164
7.2.2 TheAuxiliaryFermionRepresentation ................167
7.3 TheHubbard–HeisenbergModel.............................168
7.4 NumericalResults .........................................170
7.4.1 NumericalResultsforJ =0.........................171
7.4.2 NumericalResultsforNon-zeroJ ....................171
7.5 Conclusions,FutureOutlook,andOpenProblems ..............175
References .....................................................175
8 Signatures of Quantum Phase Transitions via Quantum
InformationTheoreticMeasures ................................177
I.BoseandA.Tribedi
8.1 Introduction ..............................................177
8.2 EntanglementandFidelityMeasuresProbingQPTs .............179
8.3 QPTsinModelSpinSystems................................183
8.4 SignaturesofQPTsinaSpinLadder .........................190
8.5 ConcludingRemarks.......................................197
References .....................................................199
9 HowEntangledIsaMany-ElectronState? ........................201
V.Subrahmanyam
9.1 Introduction ..............................................201