Many-Body Boson Systems: Half a Century Later

194 Pages·2011·2.219 MB·English

by  André F. Verbeure (auth.)

Many-Body Boson Systems
Theoretical and Mathematical Physics
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EditorialBoard
W.Beiglboeck,InstituteofAppliedMathematics,UniversityofHeidelberg,Heidelberg,
Germany
P.Chrusciel,GravitationalPhysics,UniversityofVienna,Vienna,Austria
J.-P. Eckmann, Université de Genève, Département de Physique Théorique, Geneva,
Switzerland
H.Grosse,InstituteofTheoreticalPhysics,UniversityofVienna,Vienna,Austria
A.Kupiainen,DepartmentofMathematics,UniversityofHelsinki,Helsinki,Finland
M.Loss,SchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta,USA
H. Löwen, Institute of Theoretical Physics, Heinrich-Heine-University of Duesseldorf,
Duesseldorf,Germany
N.Nekrasov,IHÉS,Bures-sur-Yvette,France
M. Salmhofer, Institute of Theoretical Physics, University of Heidelberg, Heidelberg,
Germany
S.Smirnov,MathematicsSection,UniversityofGeneva,Geneva,Switzerland
L.Takhtajan,DepartmentofMathematics,StonyBrookUniversity,StonyBrook,USA
J.Yngvason,InstituteofTheoreticalPhysics,UniversityofVienna,Vienna,Austria
Forothertitlespublishedinthisseries,goto
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André F. Verbeure
Many-Body
Boson Systems
Half a Century Later
AndréF.Verbeure
K.U.Leuven
InstituteforTheoreticalPhysics
Celestijnenlaan200D
3001Leuven
Belgium
[email protected]
ISSN1864-5879 e-ISSN1864-5887
ISBN978-0-85729-108-0 e-ISBN978-0-85729-109-7
DOI10.1007/978-0-85729-109-7
SpringerLondonDordrechtHeidelbergNewYork
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Mathematics Subject Classification (2010): 22E70, 37A60, 46L57, 46N55, 47D06, 47N50, 81Q80,
81S05,81T05,82B10,82B26,82D50
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ToIvonneandtoourfamily
Preface
The writing of this book has of course been stimulated by the exciting develop-
mentsinthefieldofBose-EinsteinCondensation(BEC)foratomicgasesthathave
manifested since the 1995 experiments. These experiments are showing brand new
featuresneverbeforeobserved.Theirtheoreticalanalysisandunderstandingishow-
everstillbasedonthestandardtheoryofBose-Einsteincondensationdevelopedfor
space-homogeneousboson systems. Just as importantlyare the recent exact results
andnewviewsontheproblemofmany-bodyphysicsdevelopedduringthelastfifty
years.Ofcourse,manyoftheseresultshavetheirownparticularimpactontheprob-
lemofBECforbosonsystems.Moreovermanyofthemseemtobeonlyknownbya
smallnumberofmathematicalphysicistsbutarelessknownbythelargercommunity
ofphysicists.
Faced with this situation, this book is conceived to be an introduction to these
new concepts and results written with considerable attention toward the physical
ideas behind the more technical material. Apart from the study of general and uni-
versalpropertiesoffullyinteractingbosonsystems,numeroushomogeneousboson
modelsareexplicitlytreated.Theapplicationsofthepresentedmaterialtosystems
oftrappedbosonsisonlybrieflydiscussedandtreatedasaposedprobleminChap.4.
Furtherstudyislefttothecareoftheinterestedreader.
Much of the material mentioned in the text was obtained during many years of
collaborations with many colleagues and former students. Warm thanks to all of
them.Wefeelobligedtopointoutonecolleagueinparticular,MarkFannes.Thank
you, Mark, for our long standing collaborations and continuing exchanges about
views, ideas, and techniques along all these years during which we constructed to-
gethermuchofthebackboneofthepresenttext.
2009Leuven
Andre´ F.Verbeure
VII
Contents
1 Introduction................................................. 1
2 Bosesystems ................................................ 7
2.1 Generalities ............................................... 7
2.2 CCRandbosonfields ....................................... 10
2.3 StatesandQuasi-FreeStates ................................. 12
3 EquilibriumStates ........................................... 27
3.1 VariationalPrinciple ........................................ 27
3.2 Energy-EntropyBalanceCriterion ............................ 31
3.3 VariationalPrincipleforSolvableModels ...................... 37
4 BoseEinsteinCondensation(BEC).............................. 43
4.1 IntroductoryRemarks....................................... 43
4.2 FreeBosonGasandBEC.................................... 44
4.2.1 StandardBEC....................................... 45
4.2.2 ThermodynamicLimitandBoundaryConditions.......... 49
4.3 BECinInteractingBosonGases .............................. 52
4.3.1 Mermin-WagnerArgument............................ 53
4.3.2 SpontaneousSymmetryBreaking(SSB)andBEC......... 55
4.3.3 CondensateEquations ................................ 74
4.4 MeanFieldBoseGas ....................................... 81
4.5 Super-radianceandMatterWaveAmplification ................. 85
4.6 TheoryofBogoliubov....................................... 89
4.7 CondensationinTwo-bodyFullyInteractingModels ............. 96
4.8 BECinTraps .............................................. 99
4.8.1 FreeBosonGasinanHarmonicPotential................ 100
4.8.2 InteractingBosonsinTraps............................ 104
IX
X Contents
5 BosonSystemDynamics.......................................109
5.1 ReversibleDynamics ....................................... 110
5.2 IrreversibleDynamics....................................... 118
6 QuantumFluctuationsandBosonization.........................123
6.1 Preliminaries .............................................. 123
6.2 NormalQuantumFluctuations................................ 130
6.3 AbnormalQuantumFluctuations.............................. 140
6.4 Applications............................................... 145
6.4.1 LuttingerModel ..................................... 145
6.4.2 Micro/macro-dynamicsandConservationofEquilibrium... 154
6.4.3 Micro/macro-dynamicsandLinearResponseTheory ...... 156
6.4.4 Micro/macroandSSB ................................ 157
7 Appendix ...................................................165
7.1 DynamicalSystemsandGNSConstruction..................... 165
7.2 DynamicalSemigroups...................................... 172
7.3 CanonicalTransformations .................................. 175
References ......................................................179
Index...........................................................187
1
Introduction
The study of boson models and the problem of their solutions is as old as the free
BosegasmodelforwhichthecelebratedphenomenonofBose-Einsteincondensation
(BEC 1924) has been detected. This phenomenon puts in evidence a macroscopic,
purely quantum phenomenon. Solving boson models means that we are interested
in finding the ground and/or temperature states of the models, or that we are in-
terestedinderivingatleastsomeoftheirpropertiesfromfirstquantum-mechanical
principles.Foralongtimeactivitiesinthisareabelongedtothefieldofmany-body
physics, a field of high activity in theoretical physics. Green’s functions, series ex-
pansions,Feynmangraphsandtheirsummations,andmuchofnumericsarethestan-
dardtechnologies.Theultimateaimis,asalways,tounderstandthephysicalworld
fromthepointofviewofitsbasiclawsandconstituentsand,therefore,toderiveas
manyexactresultsaspossiblehavingalargeoruniversalrangeofvalidityforthese
systems.
From the point of view of exact results in this field of physics, the work of
Araki-Woods [12] greatly boosted the understanding and clear formulation of the
propertiesofthefreeBosegasanditsaccompanyingphenomenonofBose-Einstein
condensation.CarefulstudiesofthethermodynamiclimitforthefreeBosegasalso
inspired efforts toward finding exact results [101]. In both cases these works relate
theproblemoftheequilibriumstatesoftheBosegastoaproblemofrepresentations
of the canonical commutation relations (CCR). This problem is by itself as old as
theearlydaysofquantummechanics.Indeedthebasicproblemassociatedwiththe
foundationsofquantummechanicscenteredontheuniquenessoftheresultsofthe
matrixtheoryofquantummechanics(seeBorn-Jordan-Heisenberg1925-1926).All
this led to the idea that developing quantum mechanics entails the search for new
representations of the canonical commutation relations. Later the famous Gelfand-
Naimark-Segal(GNS)construction[26],relatingeveryrepresentationoftheCCRto
an expectationvalue, or a state of the CCR and vice-versa, openedthe gate for the
stateapproachtothegroundand/orequilibriumstatesofallphysicalmodelsand,in
particular,alsoforbosonsystems.
ThephysicsofBECwasquicklyrealizedtoberelatedtothephenomenonofsu-
perfluidity.Oneunderstoodinshortorderthatsuperfluiditycouldonlybeexplained
A.F.Verbeure,Many-BodyBosonSystems,TheoreticalandMathematicalPhysics, 1
DOI10.1007/978-0-85729-109-7 1,©Springer-VerlagLondonLimited2011
2 1 Introduction
on the basis of an interacting Bose gas [20, 21]. In short, the spectrum of the free
Bosegasdidnotfitwiththepropertyofsuperfluidity.Thefirststeptoovercomethis
difficulty was the introduction of an interaction, the mean field or imperfect Bose
gas,whichconservesneverthelessthesolvabilityofthemodel.However,thismodel,
whose spectrum is identical to that of the free Bose gas, did not produce the right
solution which produces a spectrum giving an explanation for the phenomenon of
superfluidity.
In this context the so-called Bogoliubov model, sometimes called the weakly
interacting Bose gas [169, 22, 169], was conceived. This model takes into account
moreinteractiontermsbutwithoutlosingitsinterestingpropertyofexactsolvability.
The basic ingredients of this model in terms of states on the CCR algebra of the
bosonobservableshavebeenanalyzedrigorouslyin[8,9,7].Latertheboson-pairing
modelwasintroducedasanexercisetowardsafurtherrefinementoftheBogoliubov
model. Theoretical work on this model resulted in some intriguing questions, such
as whether two types of condensation occur simultaneously or not. The question
addressestheco-existenceofaboson-paircondensationandthestandardone-particle
ground state condensation. The other question deals with whether a spectral gap
appearsinthespectrumoftheelementaryexcitations[58,108,90,82,142].
After a number of quiet years came the great and important year, 1995, now
consideredtheyearBECwasexperimentallydiscoveredfortheidealBosegas.We
referofcoursetowhatiscalledBosecondensationfortrappedalkalimetals,research
donebyE.CornellandC.WiemanatJILA,Boulder,R.HuletatRiceUniv.Houston
Texas and W. Kelterle at MIT, Cambridge Mass, and by many people active in the
enormousscientificactivitypresentlytakingplaceinthisfield.
Moreover, many boson models have been heavily studied in the literature, not
onlyinstatisticalmechanicsbutalsoinfieldtheory.Inallaspects,modelstudiesof
boson systems have always represented a large part of the activity in this research
field. Therefore considerable attention is devoted in this book to the discussion of
solvablemodels.
A characteristic feature of all solutions of solvable boson models is that they
shareacommonpropertyfortheequilibriumstatesaswellasforthegroundstates;
namely,theyarecompletelydeterminedbytheirone-andtwo-pointcorrelationfunc-
tions. Higher-order correlation functions are expressed in terms of these one- and
two-point functions. Because of their similarities with the Fock state, the ground
stateofthefreebosongas,suchstatesarecalledgeneralizedfreeorquasi-freestates.
Thisclassofstateswasintensivelystudiedinthe1960sand1970s.Thesequasi-free
statesarenowinadispersedordercommonlyusedastheidealtheoreticallaboratory
in which we can perform tests of all kinds. Although an extensive literature about
thesestatesnowexists,theintenselypuremathematicalanalysisofquasi-freestates
wasapparentlymuchtootechnicalandgeneraltobepracticalforthestudyofboson
systemsbytheoreticalphysicists.
Thesenotesareintendedtoremedytothissituation.Aneffortismadetopresent
a less technical and more accessible presentation of the subset of quasi-free states
withinthesetofallstates.Aproperdefinitionofthenotionofsolvablemodelispro-
vided.Therelationwithquasi-freestatesisalsoclarified.Alsotheirbasicproperties