Real-Time Dynamics from Imaginary-Time Quantum Mon

arXiv:cond-mat/9510098v219Oct1995Real-TimeDynamicsfromImaginary-TimeQuantumMonteCarloSimulations:TestsonOscillatorChainsJ.Bonˇca∗andJ.E.GubernatisTheoreticalDivision,LosAlamosNationalLaboratory,LosAlamos,NM87545(February1,2008)WeusedmethodsofBayesianstatisticalinferenceandtheprincipleofmaximumentropytoan-alyticallycontinueimaginary-timeGreen’sfunctiongeneratedinquantumMonteCarlosimulationstoobtainthereal-timeGreen’sfunctions.Fortestproblems,weconsideredchainsofharmonicandanharmonicoscillatorswhosepropertieswesimulatedbyahybridpath-integralquantumMonteCarlomethod.Fromtheimaginary-timedisplacement-displacementGreen’sfunction,wefirstob-taineditsspectraldensity.Forharmonicoscillators,wedemonstratedthepeaksofthisfunctionwereinthecorrectpositionandtheirareasatisfiedasumrule.Additionally,asafunctionofwavenum-ber,thepeakpositionsfollowedthecorrectdispersionrelation.Foradouble-welloscillator,wedemonstratedthepeaklocationcorrectlypredictedthetunnelsplitting.Transformingthespectraldensitiestoreal-timeGreen’sfunctions,weconcludethatwecanpredictthereal-timedynamicsforlengthoftimescorrespondingto5to10timesthenaturalperiodofthemodel.Thelengthoftimewaslimitedbyanoverbroadeningofthepeaksinthespectraldensitycausedbythesimulationalgorithm.02.70.Lq,05.30.-d,02.50.WpI.INTRODUCTIONOneofthegoalsfordoingcomputersimulationsistheproductionofinformationusefulintheinterpretationanddesignofexperiments.NotwithstandingimportantissuesregardingHamiltonianselectionandparameterization,theinterfaceofsimulationswithexperimentisparticularlychallengingforquantumsystems.ThecurrentMonteCarloalgorithms,whethertheyimposequantumparticlestatisticsconstraintsornot,areperformedeitherinreal-timetorinimaginary-time(Euclideantime)τ=−it.Inreal-time,thepropagatorexp(itH)forasystem,describedbyaHamiltonianH,oscillateswildlyatlong-times.Analytically,theserapidoscillationsself-cancel,butaMonteCarloprocess,asitistypicallyused,hasdifficultyachievingthiscancellation.Asaconsequence,modificationsofthebasicalgorithmshavebeenproposedtoextendthesimulationsaslongaspossibleinthereal-timedomain[1].Withthesenewalgorithms,simulationstypicallyproducedynamicsextendingto2to3timesthenaturalperiodsofthesystems.Inimaginary-time,thepropagatorexp(−τH)isdiffusiveandtherapidoscillationsareavoided.CorrelationsfunctionsG(τ),however,arenowafunctionofimaginary-time,andsuchfunctionsdonoteasilyconveytheactualdynamicsofthesystem.Inprinciple,real-timecorrelation(Green’s)functionsˆG(t)canbeobtainedfromtheimaginary-timeonesbytheprocessofanalyticcontinuation.Inpractice,thisprocessisdifficultbecauseitisill-posedandbecausetheMonteCarlodataisincompleteandnoisy.Recently,procedureswereproposedtoperformthisanalyticcontinuation[2].TheseproceduresdrawheavilyuponmethodsofBayesianstatisticalinferenceandtheprincipleofmaximumentropytoinferfromimaginary-timecorrelationfunctionstheirassociatedspectraldensitiesA(ω).Throughlinear-responsetheory,thespectraldensitiesrepresentthespectraassociatedwithnumerousreal-timemeasurementsofcurrent-current,spin-spin,etc.correlationsfunctions.WhatapparentlyhasnotyetbeentriedisperformingtheHilberttransformofthesespectraldensitiestoobtainthefrequency-dependentretardedcorrelationfunctionandthenFouriertransformingthisquantitytoobtainthereal-timecorrelationfunction.Inthispaper,wewillcarryouttheseadditionalsteps.Bydoingthis,wehopetogainagreaterunderstandingofthephysicalcontentpresentinthespectraldensityreturnedbytheBayesianmethods.Weexpectedthattheresultantreal-timeinformationwouldbelimitedbytheapproximateandprobabilisticnatureoftheanalyticcontinuationmethods.Wefound,however,thatthedistanceinreal-timeoverwhichourresultsarevalidislimitedprimarilybytheabilityofthesimulationalgorithmtoproducegooddata.Tointerfaceprofitablywiththenumericalanalyticcontinuation,thesimulationalgorithmhastoproducehighqualitydataconsistentwiththeassumptionsofprocedures.Thealgorithmweusedhadproblemsdoingthis,andwewilldescribethemeasurestakentoreducethisdifficulty.Evenso,inmostcaseswewereabletoextendinreal-timeuptofactorof10naturalperiodsofthephysicalsystems.LongerextensionsarepossibleandrequirelongerMonteCarloruns.Forpresentpurposes,wehadnophysicalmotivationtodoso.InSectionII,wewilldiscussthevariousmodelsstudied.Wesimulatedaparticlemovinginsingleharmonicand1double-wellanharmonicpotentialandacollectionofparticlesmovinginchainsofthesepotentials.Forthesemodelsweknowtheexactsolutions.Bycalculatingtheirpropertiesnumerically,wecanbenchmarkourmethods.Certainpropertiesofasingledouble-wellpotential,likethetunnelsplitting,areeasilyobtainednumerically.Thephasediagramforachainofsuchoscillatorsisalsoknown[3].Thistypeofchaincanexistinasymmetricordisplacivestate.InSectionIII,wesummarizethenumericalanalyticcontinuationprocedureweusedanddiscussoursimulationtechnique.Modifyingthesimulationtechniquetobemorenaturallyergodicandtoproducedatawithshortstatisticalauto-correlationtimeswasthemostdifficultandrestrictivepartofourstudy.WepresentourresultsandconclusionsinSectionsIVandV.II.MODELSWesimulatedfiveHamiltonians.Onewasthatforasingleharmonicosc