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Phasesandphasetransitionsin
interactingBosegas

ZurErlangungdesakademischenGradeseines
DoktorsderNaturwissenschaften
derMathematisch–NaturwissenschaftlichenFakulta¨t
derUniversita¨tAugsburgvorgelegte

Dissertation

nov

ChristopherMoseley

am21.12.2006

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Erstgutachter:

Zweitgutachter:

Tagdermu¨ndlichen

Pru¨fung:

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P.vir

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sualK

Ziegler

Doz.Dr.Sigmund

12.Februar2007

Kohler

Meiner

Mutter

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gewidmet

Contents

1Introduction9
2OverviewofBose-Einsteincondensation13
2.1Denitionofthecondensatedensity.....................13
2.2DiluteBosegas.................................14
2.3TrappedBosegas................................16
2.4Lightscatteringandstructurefactor.....................18
2.5Multi-componentandfermioniccondensates.................20
2.6Opticallattices.................................21
3Functionalintegralrepresentation27
3.1Grandcanonicalpartitionfunctionasfunctionalintegral..........27
3.2Correlationfunctions..............................29
4IdealBosegas33
4.1TheHamiltonian................................33
4.2Green’sfunctionandpartitionfunction...................34
4.3One-particlecorrelationfunction.......................36
4.4Structurefactor.................................38
4.5Randomwalkexpansionandworld-lines...................41
5Bogoliubovtheory45
5.1Bogoliubovtransformation...........................45
5.2Bogoliubovgroundstate............................48
5.3Thermalexcitations..............................50
5.4Staticstructurefactor.............................51
5.5Derivationfromsaddlepointapproximation.................53
5.6Partitionfunctionandcorrelationfunctions.................56
6Hard-coreBosegasinonedimension61
6.1Generalremarks................................61
6.2World-linemodel................................62
6.3Particledensityandphasediagram......................65
6.4Densitycorrelationsandstaticstructurefactor...............68

5

Contents
6.5Externaltrappotential.............................71
7Hard-coreBosemodelinmorethanonedimension75
7.1Nilpotentalgebra................................75
7.2Hubbard-Stratonovichdecoupling.......................77
7.3Saddlepointapproximation..........................80
7.4Resultsforthehard-coreBosemodel.....................82
7.5TheN-componentmodel...........................86
8Paired-fermionmodel91
8.1Bosonicmoleculesofspin-1/2fermions....................91
8.2Hubbard-Stratonovichdecoupling.......................93
8.3Saddlepointexpansion.............................94
8.4Resultsforthepaired-fermionmodel.....................96
9Slavebosonmodel101
9.1Hamiltonianandfunctionalintegral.....................101
9.2Two-uidtheoryinclassicalapproximation.................103
9.3Mean-eldtheory................................105
9.4Quasiparticlespectrum.............................107
9.5RenormalisedGross-Pitaevskiiequation...................109
9.6ApplicationtoatrappedBose-Einsteincondensate.............112
10Discussion117
10.1Comparisonoftheresults...........................117
10.2Excitationspectruminthelarge-UoftheBose-Hubbardmodel......120
11Conclusion123
AFinitesummationsandproducts125
A.1Finiteproducts.................................125
A.2Finitesums...................................125
BGaussianintegralsandexpectationvalues129
B.1Gaussianintegrals...............................129
B.2ExpectationvaluesandWick’stheorem...................130
CCoherentstatesforbosons,fermions,andhard-corebosons131
DCorrelationfunctionsforthehard-coreBosemodel135
D.1Decayofthedensity-densityCF.......................137
6

ECalculationstotheslave-bosonmodel
E.1Integrationoftheconstraint..........
E.2Condensatedensityandtotalparticledensity
E.3Zerotemperaturelimit.............

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Contents

8

1Introduction

Thequantumstatisticsofnon-interactingparticleswasestablishedbyS.N.Bosein
1924[1].BosewasabletodeducePlanck’sradiationlawontheassumptionthateach
quantumstatecanbeoccupiedbyanarbitrarynumberofindistinguishablephotons.
ByapplyingthisideatothequantumstatisticsofanidealgasofNatomsenclosedina
volumeV,A.Einsteinpredictedtheoccurrenceofaphasetransition[2]:Belowacritical
temperatureTc,acertainfractionofatomswould“condense”inthegroundstateofthe
system.ThisphenomenoniscalledBose-Einsteincondensation(BEC).
ThepossibilitytostudyquantumphasetransitionsininteractingBosegasesexper-
imentallywithhighaccuracymotivatesthedevelopmentofastatisticaltheory,which
isnotonlyvalidinweaklyinteractingbutalsoinstronglyinteractingsystems.The
stronglyinteractingregimehasbeenrealisedbytheapplicationofFeshbachresonances
[3].Inthisthesis,specialattentionwillbedrawntoBosesystemsinopticallattices,
whereaquantumphasetransitionofaBECtoaMott-insulator(MI)hasbeendescribed
theoretically[4,5,6]bymeansoftheso-calledBose-Hubbardmodel,aswellasobserved
experimentally[7].
Inthisthesis,asystemofhard-corebosonswillbeusedasafoundationforthetheo-
reticalanalysisofthesephasetransitions.Thissystemischaracterisedbytherestriction
thateachlatticesiteiseitherempty,oroccupiedbyonesingleboson.Itdiersfrom
theBose-Hubbardmodelinthewaythatthelatterallowsmultipleoccupationoflattice
sitesbythecostofarepulsiveinteractionenergyU.Thusthemodelofhard-corebosons
requiresoneparameterlessthantheBose-Hubbardmodel.Dierentapproachesshall
bepresentedtomakepredictionstoexperimentallyrelevantphysicalquantitiesandthe
qualitativebehaviourofthesystematzerotemperatureaswellasnon-zerotempera-
tures.Theobjectsofinterestarethephasediagram,theparticledensityandcondensate
density,thespectrumofquasiparticleexcitations,andthestaticstructurefactorwhich
canbemeasuredbymeansoflightscatteringexperiments.
Thefunctionalintegralformalismprovidesanadequateapproachtothisproblemof
manyparticlephysics[8,9].Withinthisformalismthedeterminationofthequantities
mentionedabovereducestothecalculationofcorrelationfunctions,whichcanbeeval-
uatedbymeansofappropriateapproximations.Fourdierentmodelstodescribeagas
ofhard-corebosonsinanopticallatticewillbediscussed:Aone-dimensionalmodel,a
hard-coreBosemodelconstructedbynilpotenteldvariables,amodeldescribingpaired
fermions,andamodelwhichisbasedontheslave-bosonapproach.
Thethesisisorganisedasfollows:Inchapter2anoutlineofthetheoreticaland

9

1Introduction

experimentaldevelopmentsintheeldofBose-Einsteincondensation,whicharerelevant
forthetopic,isgiven.Inchapter3thefunctionalintegralrepresentationisintroduced
intheformasitisappliedtothemodels.Itisshownthatallphysicalquantitiescan
bedrawnoutofthefunctionalintegralrepresentationofthegrandcanonicalpartition
function.Inchapter4itisappliedtoanidealBosegasinanopticallattice.The
procedureofcalculatingphysicalquantitiesfromcorrelationfunctionsisdemonstrated.
Inthelastsectionarandom-walkexpansionisperformedinordertoshowthatbosons
canbeillustratedbyworld-linesalongtheimaginarytimecoordinate.Thisworld-line
picturewillalsobeappliedtomotivatetheone-dimensionalmodel,thehard-coreBose
modelandthepaired-fermionmodel.
Theaimofchapter5istodemonstratetheprincipleofthesaddle-pointapproxi-
mation,whichwillbeappliedtoallmodelsthatwillbediscussedlater,exceptthe
one-dimensionalmodel.Therstfoursectionsofthechaptergiveanintroductionto
Bogoliubovtheory.Thenitisshown,thatthesameresultsarefoundbyapplyingasad-
dlepointapproximationtotheactionofaninteractingBosegaswithweaktwo-particle
interaction:Onthemean-eldlevel,whichisderivedbyminimisingtheactiondueto
thevariationalprinciple,theGross-Pitaevskiiequation[10]isfound.Onthelevelof
Gaussianuctuationsaroundthemean-eldresult,allresultsofBogoliubovtheoryare
found.
Inchapter6theone-dimensionalsystemisdiscussed.Basedonthewell-knownfact
thatasystemofone-dimensionalimpenetrablebosonscanbemappedtoidealfermions,
wedenethemodelbyaspecialconstructionofabipartitelatticewhereitisassuredthat
twoparticlescannotinterchangetheirposition.Thisallowstocalculatesomephysical
quantitiesexactly.
Chapter7isdevotedtothemodelwhichwillbereferredtoashard-coreBosemodel.
Thefunctionalintegralrepresentationofitsgrandcanonicalpartitionfunctioniscon-
structedbyanalgebraofnilpotentcommutatingeldvariables.AHubbard-Stratonovich
transformationallowsamappingtoarepresentationwithtwocomplex

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