Quantum Monte Carlo Approaches for Correlated Systems

287 Pages·2017·2.492 MB·English

by  Federico Becca

QUANTUM MONTE CARLO APPROACHES FOR
CORRELATED SYSTEMS
Over the past several decades, computational approaches to studying strongly
interacting systems have become increasingly varied and sophisticated. This
book provides a comprehensive introduction to state-of-the-art quantum Monte
Carlo techniques relevant for applications in correlated systems. Starting with
an overview of variational wave functions, it features a detailed presentation of
stochastic samplings including Markov chains and Langevin dynamics, which are
developed into a discussion of Monte Carlo methods. The variational technique
is also described, from foundations to a detailed description of its algorithms.
Other topics discussed include optimization techniques, real-time dynamics, and
projection methods, including Green’s function, reptation, and auxiliary-field
Monte Carlo, from basic definitions to advanced algorithms for efficient codes,
and concluding with recent developments on the continuum space. This book
providesanextensivereferenceforstudentsandresearchersworkingincondensed
matter theory or those interested in advanced numerical methods for electronic
simulation.
federico becca isaresearcherattheNationalResearchCouncil(CNR)work-
ing in the theoretical group of the Condensed Matter section of the International
School for Advanced Studies (SISSA) in Trieste. His research focuses on differ-
ent aspects of correlated systems on the lattice. His major scientific contributions
include advances in frustrated magnets, superconductivity from strong electronic
correlation, disordered fermionic and bosonic models, and Mott metal-insulator
transitions.
sandro sorella is Professor of Condensed Matter Physics at the Interna-
tionalSchoolofAdvancedStudies(SISSA)inTrieste.Hisfocusisonthestudyof
stronglycorrelatedelectronsystemsbyadvancednumericalsimulationtechniques
based on quantum Monte Carlo. He has developed novel Monte Carlo algorithms
thatarenowwidelyusedandconsideredstate-of-the-artinthefield.

QUANTUM MONTE CARLO
APPROACHES FOR CORRELATED
SYSTEMS
FEDERICO BECCA
NationalResearchCouncil
SANDRO SORELLA
InternationalSchoolforAdvancedStudies,Trieste
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi–110025,India
79AnsonRoad,#06–04/06,Singapore079906
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learning,andresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107129931
DOI:10.1017/9781316417041
©FedericoBeccaandSandroSorella2017
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2017
PrintedintheUnitedKingdombyClays,StIvesplc
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Becca,Federico,1972–author.|Sorella,Sandro,1960–author.
Title:QuantumMonteCarloapproachesforcorrelatedsystems/FedericoBecca
(ScuolaInternazionaleSuperiorediStudiAvanzati),SandroSorella.
Description:Cambridge,UnitedKingdom;NewYork,NY:CambridgeUniversity
Press,2017.|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2017025043|ISBN9781107129931(hardback;alk.paper)|
ISBN1107129931(hardback;alk.paper)
Subjects:LCSH:MonteCarlomethod.|Variationalprinciples.
Classification:LCCQC174.85.M64B432017|DDC530.1201/518282–dc23
LCrecordavailableathttps://lccn.loc.gov/2017025043
ISBN978-1-107-12993-1Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
Contents
Preface pageix
Acknowledgements xii
PartI Introduction 1
1 CorrelatedModelsandWaveFunctions 3
1.1 Introduction 3
1.2 TheMatrixFormulation 6
1.3 EffectiveLatticeModels 7
1.4 TheVariationalPrinciple 13
1.5 VariationalWaveFunctions 14
1.6 SizeExtensivity 31
1.7 ProjectionTechniques 34
PartII ProbabilityandSampling 37
2 ProbabilityTheory 39
2.1 Introduction 39
2.2 EventsandProbability 41
2.3 MomentsoftheDistribution:MeanValueandVariance 44
2.4 ChangingRandomVariables 47
2.5 TheChebyshev’sInequality 48
2.6 SummingIndependentRandomVariables 48
2.7 TheCentralLimitTheorem 53
3 MonteCarloSamplingandMarkovChains 56
3.1 Introduction 56
3.2 ReweightingTechniqueandCorrelatedSampling 59
3.3 DirectSampling 60
3.4 ImportanceSampling 61
v
vi Contents
3.5 SamplingaDiscreteDistributionProbability 62
3.6 SamplingaContinuousDensityProbability 64
3.7 MarkovChains 66
3.8 DetailedBalanceandApproachtoEquilibrium 69
3.9 MetropolisAlgorithm 74
3.10 HowtoEstimateErrorbars 76
3.11 ErrorbarsinCorrelatedSamplings 82
4 LangevinMolecularDynamics 85
4.1 Introduction 85
4.2 Discrete-TimeLangevinDynamics 87
4.3 FromtheLangevintotheFokker-PlanckEquation 89
4.4 Fokker-PlanckEquationandQuantumMechanics 91
4.5 AcceleratedLangevinDynamics 96
PartIII VariationalMonteCarlo 101
5 VariationalMonteCarlo 103
5.1 QuantumAveragesandStatisticalSamplings 103
5.2 TheZero-VarianceProperty 105
5.3 JastrowandJastrow-SlaterWaveFunctions 106
5.4 TheChoiceoftheBasisSets 108
5.5 BosonicSystems 109
5.6 FermionicSystemswithDeterminants 112
5.7 FermionicSystemswithPfaffians 123
5.8 EnergyandCorrelationFunctions 129
5.9 PracticalImplementation 129
6 OptimizationofVariationalWaveFunctions 131
6.1 Introduction 131
6.2 ReweightingTechniquesfortheOptimizationofWaveFunctions 132
6.3 EnergyDerivatives 134
6.4 TheStochasticReconfiguration 137
6.5 StochasticReconfigurationasaProjectionTechnique 146
6.6 TheLinearMethod 147
6.7 CalculationsofDerivativesintheJastrow-SlaterCase 152
7 Time-DependentVariationalMonteCarlo 156
7.1 Introduction 156
7.2 Real-TimeEvolutionoftheVariationalParameters 157
7.3 AnExamplefortheQuantumQuenchinOneDimension 161
Contents vii
PartIV ProjectionTechniques 165
8 Green’sFunctionMonteCarlo 167
8.1 BasicNotionsandFormalDerivations 167
8.2 SingleWalkerTechnique 170
8.3 ImportanceSampling 173
8.4 TheContinuous-TimeLimit 178
8.5 ManyWalkersFormulation 180
8.6 PracticalImplementation 188
9 ReptationQuantumMonteCarlo 189
9.1 ASimplePathIntegralTechnique 189
9.2 ASimpleWaytoSampleConfigurations 191
9.3 TheBounceAlgorithm 195
9.4 TheContinuous-TimeLimit 196
9.5 PracticalImplementation 197
10 Fixed-NodeApproximation 199
10.1 TheSignProblem 199
10.2 ASimpleExampleontheContinuum 204
10.3 ASimpleExampleontheLattice 208
10.4 TheFixed-NodeApproximationontheLattice 209
10.5 PracticalImplementation 213
11 AuxiliaryFieldQuantumMonteCarlo 214
11.1 Introduction 214
11.2 TrotterApproximation 216
11.3 Hubbard-StratonovichTransformation 217
11.4 ThePath-IntegralRepresentation 219
11.5 SequentialUpdates 223
11.6 Ground-StateEnergyandCorrelationFunctions 228
11.7 SimpleCaseswithoutSignProblem 228
11.8 PracticalImplementation 231
PartV AdvancedTopics 233
12 RealisticSimulationsontheContinuum 235
12.1 IntroductionandMotivations 235
12.2 VariationalWaveFunctionwithLocalizedOrbitals 237
12.3 SizeConsistencyoftheVariationalWaveFunctions 244
viii Contents
12.4 OptimizationoftheVariationalWaveFunctions 245
12.5 Lattice-RegularizedDiffusionMonteCarlo 252
12.6 AnImprovedSchemefortheLatticeRegularization 256
Appendix Pseudo-RandomNumbersGeneratedbyComputers 261
References 264
Index 272
Preface
This book originates from the lecture notes of the Ph.D. course on numerical sim-
ulations for strongly-correlated systems at the International School for Advanced
Studies (S.I.S.S.A.) in Trieste. Even though the backbone of this book has been
formedoverthelastfifteenyears,almostallchaptershavebeencompletelyrewrit-
ten,reshaped,orheavilymodifiedinordertoimprovetheclarityofthepresentation.
Inaddition,thesecondpartofthisbook(i.e.,Chapters2,3,and4)hasbeentaught
during the C.E.C.A.M. Summer School on “Atomistic Simulation Techniques for
Material Science, Nanotechnology, and Biophysics”, held in Trieste from 2011
to 2016. This school was addressed to graduate and undergraduate students who
had little experience in numerical simulations and part of the actual presentation
has benefited from their suggestions. The book is envisaged for students and
young researches, who do not know much on Monte Carlo methods and want to
understandhowtoimplementground-statealgorithmsoninteractingmodels.After
a first chapter dealing with a bird’s-eye review on correlated wave functions that
havebeenusedinthepast,thesecondpartisintendedtogivearatherpedagogical
introduction to simple probability theory (just the concepts that are necessary for
the further developments) and methods for statistical samplings, not only based
on Monte Carlo techniques but also including molecular-dynamics approaches.
The central part of the book deals with variational and projection Monte Carlo
approaches that are well established and widely used to treat lattice models
(for both fermions and bosons). Finally, the last part provides an introduction
toMonteCarlomethodsthathavebeenrecentlydevelopedforelectronsystemson
thecontinuum.
Our main motivation to convert the preliminary lecture notes into a structured
bookistoputinasequentialorderalltheconceptsandmachineriesthatareneeded
forwritingefficientcodesbaseduponvariationalandprojectionMonteCarlotech-
niques.Moreover,amoderntextbookdescribinghowtotreatwavefunctionswithin
quantum Monte Carlo algorithms was lacking in the portfolio of books treating
ix
Downloaded from https://www.cambridge.org/core. Teachers College Library - Columbia University, on 04 Dec 2017 at 12:07:20, subject to the Cambridge Core
terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781316417041.001