Quantum simulations of complex many-body systems: from theory to algorithms

562 Pages·08.53 MB·English

by  Grotendorst, J. (Editor), Marx, D. (Editor), Muramatsu, A. (Editor)

Publication Series of the John von Neumann Institute for Computing (NIC)
NIC Series Volume 10

John von Neumann Institute for Computing (NIC)
Quantum Simulations of
Complex Many-Body Systems:
From Theory to Algorithms
edited by
Johannes Grotendorst
Dominik Marx
Alejandro Muramatsu
Winter School, 25 February - 1 March 2002
Rolduc Conference Centre, Kerkrade, The Netherlands
Lecture Notes
organized by
John von Neumann Institute for Computing
Ruhr-Universita¨t Bochum
Universita¨t Stuttgart
NIC Series Volume 10
ISBN3-00-009057-6
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Distributor: NIC-Secretariat
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c 2002 by John von Neumann Institute for Computing
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NIC Series Volume 10
ISBN 3-00-009057-6
Preface
This Winter School continues a series of schools and conferences in Computational
ScienceorganizedbytheJohnvonNeumannInstituteforComputing(NIC).Thetopicsof
theSchool,QuantumMonteCarloandQuantumMolecularDynamics,playanoutstanding
role in many NIC research projects which use the supercomputingfacilities providedby
theCentralInstituteforAppliedMathematics(ZAM)oftheResearchCentreJu¨lich. The
programmeoftheWinterSchoolcoversmodernquantumsimulationtechniquesandtheir
implementation on high-performance computers, in particular on parallel systems. The
focusclearly is on numericalmethodswhich are tailored to treat large quantumsystems
with many coupled degrees of freedom ranging from superfluid Helium to chemical
reactions.Amongothers,thefollowingtopicsaretreatedbytwenty-fivelectures:
DiffusionandGreen’sfunctionMonteCarlo
(cid:15)
PathintegralMonteCarloandMolecularDynamics
(cid:15)
Car-Parrinello/abinitioMolecularDynamics
(cid:15)
Real-timequantumdynamicsforlargesystems
(cid:15)
Latticeandcontinuumalgorithms
(cid:15)
Exchangestatisticsforbosonsandfermions/signproblem
(cid:15)
Parallelnumericaltechniquesandtools
(cid:15)
Numericalintegrationandrandomnumbers
(cid:15)
This strongly interdisciplinary School aims at bridging three “gaps” in the vast field of
large-scale quantum simulations. The first gap is between chemistry and physics, the
secondonebetweentypicalgraduatecoursesinthese fieldsandstate-of-the-artresearch,
and finally the one between the Monte Carlo and Molecular Dynamics communities.
The participants will benefit from this School by learning about recent methodological
advanceswithinandoutsidetheirfieldofspecialization. Inaddition,theygetinsightinto
recent software developments and implementation issues involved, in particular in the
contextofhigh-performancecomputing.
The lecturers of this Winter School come from chemistry, physics, mathematics and
computerscienceandthisistruefortheaudienceaswell. Participantsfromthirtymainly
European countries attend the NIC Winter School, and eighty contributions have been
submitted for the poster sessions. This overwhelming international resonance clearly
reflects the attractiveness of the programme and demonstrates the willingness of the
participantstoplayanactiveroleinthishigh-levelscientificSchool.
The scientific programme was worked out by Johannes Grotendorst (Research Centre
Ju¨lich),DominikMarx(Ruhr-Universita¨tBochum),andAlejandroMuramatsu(Universita¨t
Stuttgart). The programme structure consists of overview lectures on various important
fields, focus lectures on Quantum Monte Carlo and Quantum Molecular Dynamics
methods,andspeciallecturesonnumericalandcomputationaltechniques.
Many organizations and individuals have contributed significantly to the success of this
Winter School. Without the financial support of the European Commission within the
frameworkofthespecificresearchandtrainingprogramme“ImprovingHumanResearch
Potential” this one-week School on quantum simulation methods would not have been
possible. We are grateful for the generousfinancial supportby the Federal Ministry for
EducationandResearch(BMBF)andbytheResearchCentreJu¨lichaswellasforthehelp
providedbyitsConferenceServiceanditsCentralInstituteforAppliedMathematics.
WearegreatlyindebtedtothelocalorganizationcommitteeatForschungszentrumJu¨lich
who did the bigger part of the preparing work, namely Ru¨diger Esser (Finance), Bernd
Krahl-Urban (Accommodaton and Registration) and Monika Marx (Web Management,
Proceedings),andlastbutnotleasttheconferencesecretariesYasminAbdel-Fattah,Elke
Bielitza and Anke Reinartz. Special thanks go to Monika Marx for her tireless commit-
mentconcerningtheeditingandrealizationofthisbook. Furthermore,weappreciatethe
work of Stephan Bru¨ck who supported the difficult typesetting with great care. Finally,
wewouldliketothankboththeRuhr-Universita¨tBochumandtheUniversita¨tStuttgartfor
theirsupportofthisactivityintheareaofhigh-endscientificeducation.
ThenatureofaWinterSchoolrequiresthenotesofthelecturestobeavailableatthemeet-
ing. Inthisway,theparticipantshavethechancetoworkthroughthelecturesthoroughly
duringorafterthelectures. Weareverythankfultoallauthorswhoprovidedwrittencon-
tributionstothisbookoflecturenotes.Itisintendedtoserveasafuturestandardreference
totherapidlyevolvingfieldofquantumsimulationsofcomplexmany-bodysystems. The
articlesgivea broadreviewofmoderntime-independentandtime-dependentmethodsas
wellasoftherelevantstate-of-the-artnumericalandparallelcomputationtechniques. In
additiontosuchtraditionaltext-basedproceedings,audio-visualproceedingswillbepro-
duced.Alllectureswillberecordedonvideo.AftertheSchooltheserecordingscombined
withtheslideswillbemadeavailabletotheparticipantsonDVDandtothegeneralscien-
tific communityinthe internetathttp://www.fz-juelich.de/nic-series/volume10,the same
placewherethebookoflecturenotesispublished.
Ju¨lich,Bochum,andStuttgart
February2002
JohannesGrotendorst
DominikMarx
AlejandroMuramatsu
Contents
Time-Independent Quantum Simulation Methods
MonteCarloMethods: OverviewandBasics
MariusLewerenz 1
1 Introduction 1
2 ReviewofProbabilityandStatistics 6
3 SourcesofRandomness 14
4 MonteCarloIntegration 20
DiffusionandGreen’sFunctionQuantumMonteCarloMethods
JamesB.Anderson 25
1 Introduction 25
2 HistoryandOverview 26
3 VariationalQuantumMonteCarlo 28
4 DiffusionQuantumMonteCarlo 29
5 Green’sFunctionQuantumMonteCarlo 33
6 NodeStructure 34
7 ImportanceSampling 35
8 TrialWavefunctions 37
9 Fixed-NodeCalculations 39
10 ExactCancellationMethod 40
11 DifferenceSchemes 43
12 ExcitedStates 45
13 UseofPseudopotentials 45
PathIntegralMonteCarlo
BernardBernu,DavidM.Ceperley 51
1 Introduction 51
2 MappingoftheQuantumtoaClassicalProblem 52
3 BoseSymmetry 57
4 Applications 58
Exchange Frequencies in 2D Solids: Example of Helium 3 Adsorbed on
GraphiteandtheWignerCrystal
BernardBernu,LadirCaˆndido,DavidM.Ceperley 63
1 Introduction 63
2 PIMCMethod 65
3 ReactionCoordinate 67
4 AOneDimensionalToyModel:AParticleinaSymmetricalDoubleWell 68
5 ResultsandMagneticPhaseDiagram 70
6 Conclusion 73
ReptationQuantumMonteCarlo
StefanoBaroni,SaverioMoroni 75
1 Introduction 75
2 FromClassicalDiffusiontoQuantumMechanics 76
3 FromQuantumMechanicsBacktoClassicalDiffusion 81
4 TheAlgorithm 86
5 ACaseStudyof He 88
4
6 Conclusions 96
QuantumMonteCarloMethodsonLattices:TheDeterminantalApproach
FakherF.Assaad 99
1 Introduction 99
2 TheWorldLineApproachfortheXXZModelandRelationtothe6-Vertex
Model 101
3 AuxiliaryFieldQuantumMonteCarloAlgorithms 107
4 ApplicationoftheAuxiliaryFieldQMCtoSpecificHamiltonians 129
5 TheHirsch-FyeImpurityAlgorithm 144
6 Conclusion 147
EffectiveHamiltonianApproachforStronglyCorrelatedLatticeModels
SandroSorella 157
1 Introduction 157
2 TheLanczosTechnique 159
3 TheEffectiveHamiltonianApproach 161
4 TheGeneralizedLanczos 163
5 Resultsonthet-JModel 168
6 Conclusions 172
TheLDA+DMFTApproachtoMaterialswithStrongElectronicCorrelations
Karsten Held, Igor A. Nekrasov, Georg Keller, Volker Eyert, Nils Blu¨mer,
AndrewK.McMahan,RichardT.Scalettar,ThomasPruschke,VladimirI.Anisimov,
DieterVollhardt 175
1 Introduction 175
2 TheLDA+DMFTApproach 177
3 Comparison of Different Methods to Solve DMFT: The Model System
La Sr TiO 191
4 Mo1t(cid:0)t-xHuxbbard3Metal-InsulatorTransitioninV O 194
5 TheCeriumVolumeCollapse:AnExamplefo2ra3 -ElectronSystem 198
4f
6 ConclusionandOutlook 203
ii
Time-Dependent Quantum Simulation Methods
ClassicalMolecularDynamics
GodehardSutmann 211
1 Introduction 211
2 ModelsforParticleInteractions 215
3 TheIntegrator 221
4 SimulatinginDifferentEnsembles 229
5 ParallelMolecularDynamics 235
StaticandTime-DependentMany-BodyEffectsviaDensity-FunctionalTheory
HeikoAppel,EberhardK.U.Gross 255
1 Introduction 255
2 BasicConceptsofDFT 256
3 PropagationMethodsfortheTDKSEquations 259
4 ExamplesfortheSolutionoftheTDKSEquations 263
PathIntegrationviaMolecularDynamics
MarkE.Tuckerman 269
1 Introduction 269
2 TheDensityMatrixandQuantumStatisticalMechanics 270
3 Path Integral Formulation of the Canonical Density Matrix and Partition
Function 272
4 TheContinuousLimit 275
5 ThermodynamicsandExpectationValuesinTermsofPathIntegrals 281
6 PathIntegralMolecularDynamics 283
7 Many-BodyPathIntegrals 294
8 Summary 297
AbInitioMolecularDynamicsandAbInitioPathIntegrals
MarkE.Tuckerman 299
1 Introduction 299
2 TheBorn-OppenheimerApproximationandAbInitioMolecularDynamics 300
3 PlaneWaveBasisSets 305
4 The Path Integral Born-Oppenheimer Approximation and Ab Initio Path
IntegralMolecularDynamics 311
5 IllustrativeApplications 314
DynamicPropertiesviaFixedCentroidPathIntegrals
RafaelRam´ırez,TelesforoLo´pez-Ciudad 325
1 Introduction 325
2 DefinitionofAuxiliaryQuantities 327
3 DefinitionofFixedCentroidPathIntegrals 328
4 TheSchro¨dingerFormulationofFixedCentroidPathIntegrals 330
5 ConstrainedTimeEvolutionoftheOperator 337
(cid:27)^~(X;P)
6 NumericalTestonModelSystems 344
iii
7 AReviewofCMDApplications 351
8 OpenProblems 355
9 Conclusions 356
QuantumMolecularDynamicswithWavePackets
UweManthe 361
1 Introduction 361
2 SpatialRepresentationofWavefunctions 362
3 PropagationofWavePackets 366
4 IterativeDiagonalization 369
5 FilterDiagonalization 370
6 MCTDH 371
NonadiabaticDynamics:Mean-FieldandSurfaceHopping
NikosL.Doltsinis 377
1 Introduction 377
2 Born-OppenheimerApproximation 378
3 SemiclassicalApproach 380
4 ApproachestoNonadiabaticDynamics 382
RelievingtheFermionicandtheDynamicalSignProblem:MultilevelBlocking
MonteCarloSimulations
ReinholdEgger,ChiH.Mak 399
1 Introduction:TheSignProblem 399
2 MultilevelBlocking(MLB)Approach 401
3 Applications 412
4 ConcludingRemarks 421
Numerical Methods and Parallel Computing
StatisticalAnalysisofSimulations:DataCorrelationsandErrorEstimation
WolfhardJanke 423
1 Introduction 423
2 ModelSystemsandPhaseTransitions 424
3 Estimators,AutocorrelationTimes,BiasandResampling 430
4 ASimplifiedModel 435
5 ARealisticExample 439
6 ErrorPropagationinMulticanonicalSimulations 440
7 Summary 443
PseudoRandomNumbers:GenerationandQualityChecks
WolfhardJanke 447
1 Introduction 447
2 PseudoRandomNumberGenerators 447
3 QualityChecks 451
iv