The one-dimensional Hubbard model

692 Pages·2005·3.55 MB·English

by  Fabian H. L. Essler, Holger Frahm, Frank Göhmann, Andreas Klümper, Vladimir E. Korepin

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THE ONE-DIMENSIONAL HUBBARD MODEL
The description of a solid at a microscopic level is complex, involving the interaction of
a huge number of its constituents, such as ions or electrons. It is impossible to solve the
correspondingmany-bodyproblemsanalyticallyornumerically,althoughmuchinsightcan
be gained from the analysis of simplified models. An important example is the Hubbard
model, which describes interacting electrons in narrow energy bands, and which has been
appliedtoproblemsasdiverseashigh-T superconductivity,bandmagnetismandthemetal-
c
insulatortransition.
Remarkably,theone-dimensionalHubbardmodelcanbesolvedexactlyusingtheBethe
ansatz method. The resulting solution has become a laboratory for theoretical studies of
non-perturbativeeffectsinstronglycorrelatedelectronsystems.Manymethodsdevisedto
analysesucheffectshavebeenappliedtothismodel,bothtoprovidecomplementaryinsight
intowhatisknownfromtheexactsolutionandasanultimatetestoftheirquality.
Thisbookpresentsacoherent,self-containedaccountoftheexactsolutionoftheHubbard
modelinonedimension.Theearlychaptersdevelopaself-containedintroductiontoBethe’s
ansatzanditsapplicationtotheone-dimensionalHubbardmodel,andwillbeaccessibleto
beginninggraduatestudentswithabasicknowledgeofquantummechanicsandstatistical
mechanics. The later chapters address more advanced topics, and are intended as a guide
forresearcherstosomeofthemorerecentscientificresultsinthefieldofintegrablemodels.
The authors are distinguished researchers in the field of condensed matter physics and
integrable systems, and have contributed significantly to the present understanding of the
one-dimensionalHubbardmodel. Fabian Essler isaUniversityLecturerinCondensed
MatterTheoryatOxfordUniversity. Holger Frahm isProfessorofTheoreticalPhysics
at the University of Hannover. Frank Go¨hmann is a Lecturer at Wuppertal University,
Germany. Andreas Klu¨mper isProfessorofTheoreticalPhysicsatWuppertalUniver-
sity. Vladimir Korepin isProfessorattheYangInstituteforTheoreticalPhysics,State
UniversityofNewYorkatStonyBrook,andauthorofQuantumInverseScatteringMethod
andCorrelationFunctions(Cambridge,1993).

THE ONE-DIMENSIONAL
HUBBARD MODEL
FABIAN H. L. ESSLER
OxfordUniversity
HOLGER FRAHM
UniversityofHannover
FRANK GO¨ HMANN
WuppertalUniversity
ANDREAS KLU¨ MPER
WuppertalUniversity
VLADIMIR E. KOREPIN
StateUniversityofNewYorkatStonyBrook
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Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
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© F. H. L. Essler, H. Frahm, F. Göhmann, A. Klümper and V. Korepin 2005
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Contents
Preface pagexi
1 Introduction 1
1.1 OntheoriginoftheHubbardmodel 1
1.2 TheHubbardmodel–aparadigmincondensedmatterphysics 5
1.3 Externalfields 11
1.4 Conclusions 14
AppendicestoChapter1 15
1.A Responsetoexternalfields 15
2 TheHubbardHamiltoniananditssymmetries 20
2.1 TheHamiltonian 20
2.2 Symmetries 25
2.3 Conclusions 35
AppendicestoChapter2 36
2.A Thestrongcouplinglimit 36
2.B Continuumlimits 45
3 TheBetheansatzsolution 50
3.1 TheHamiltonianinfirstquantization 51
3.2 Solutionofthetwo-particleproblem 54
3.3 Many-particlewavefunctionsandLieb-Wuequations 64
3.4 Symmetrypropertiesofwavefunctionsandstates 67
3.5 Thenormoftheeigenfunctions 68
3.6 Conclusions 72
AppendicestoChapter3 73
3.A Scalarproductsandprojectionoperators 73
3.B DerivationofBetheansatzwavefunctionsandLieb-Wuequations 76
3.C Sometechnicaldetails 94
3.D HighestweightpropertyoftheBetheansatzstateswithrespectto
totalspin 96
3.E ExplicitexpressionsfortheamplitudesintheBetheansatzwave
functions 101
v
vi Contents
3.F Lowestweighttheoremfortheη-pairingsymmetry 105
3.G LimitingcasesoftheBetheansatzsolution 112
4 Stringhypothesis 120
4.1 Stringconfigurations 121
4.2 Stringsolutionsasboundstates 125
4.3 Takahashi’sequations 128
4.4 CompletenessoftheBetheansatz 131
4.5 Higher-levelBetheansatz 133
AppendicestoChapter4 134
4.A Ondeviationsfromthestringhypothesis 134
4.B Detailsabouttheenumerationofeigenstates 137
5 ThermodynamicsintheYang-Yangapproach 149
5.1 Apointofreference:noninteractingelectrons 149
5.2 ThermodynamicBetheAnsatz(TBA)equations 153
5.3 Thermodynamics 161
5.4 Infinitetemperaturelimit 162
5.5 Zerotemperaturelimit 163
AppendicestoChapter5 168
5.A Zerotemperaturelimitforε(cid:2)((cid:3)) 168
1
5.B Propertiesoftheintegralequationsat T = 0 168
6 Groundstatepropertiesinthethermodynamiclimit 175
6.1 Apointofreference:noninteractingelectrons 175
6.2 Definingequations 177
6.3 Groundstatephasediagram 178
6.4 Densityandmagnetization 184
6.5 Spinandchargevelocities 187
6.6 Susceptibilities 188
6.7 Groundstateenergy 193
AppendicestoChapter6 195
6.A Numericalsolutionofintegralequations 195
6.B Groundstatepropertiesinzeromagneticfield 197
6.C Smallmagneticfieldsathalffilling:applicationofthe
Wiener-Hopfmethod 202
7 Excitedstatesatzerotemperature 209
7.1 Apointofreference:noninteractingelectrons 210
7.2 Zeromagneticfieldandhalf-filledband 211
7.3 Rootdensityformalism 225
7.4 Scatteringmatrix 236
7.5 ‘Physical’Betheansatzequations 242
7.6 Finitemagneticfieldandhalf-filledband 244
7.7 Zeromagneticfieldandlessthanhalf-filledband 253
7.8 Finitemagneticfieldandlessthanhalf-filledband 261
7.9 Emptybandintheinfinitevolume 262
Contents vii
AppendicestoChapter7 265
7.A Relatingroot-densityanddressed-energyformalisms 265
7.B Lowerboundsforε (0),n ≥ 2athalffillinginafinitemagneticfield 267
n
8 Finitesizecorrectionsatzerotemperature 268
8.1 Genericcase–therepulsiveHubbardmodelin
amagneticfield 268
8.2 Specialcases 276
8.3 FinitesizespectrumoftheopenHubbardchain 283
8.4 Relationofthedressedchargematrixtoobservables 290
AppendicestoChapter8 294
8.A WienerHopfcalculationofthedressedcharge 294
9 Asymptoticsofcorrelationfunctions 297
9.1 Lowenergyeffectivefieldtheoryatweakcoupling 297
9.2 Conformalfieldtheoryandfinitesizescaling 303
9.3 Correlationfunctionsoftheone-dimensionalHubbardmodel 308
9.4 Correlationfunctionsinmomentumspace 320
9.5 CorrelationfunctionsintheopenboundaryHubbardchain 324
AppendicestoChapter9 331
9.A Singularbehaviourofmomentum-spacecorrelators 331
10 Scalingandcontinuumlimitsathalf-filling 333
10.1 Constructionofthescalinglimit 333
10.2 TheS-matrixinthescalinglimit 335
10.3 Continuumlimit 337
10.4 Correlationfunctionsinthescalinglimit 344
10.5 Correlationfunctionsinthecontinuumlimit 361
10.6 Finitetemperatures 367
AppendicestoChapter10 369
10.A Currentalgebra 369
10.B Two-particleformfactors 371
10.C CorrelationfunctionsintheGaussianmodel 372
11 Universalcorrelationsatlowdensity 376
11.1 TheHubbardmodelinthegasphase 377
11.2 Correlationfunctionsoftheimpenetrableelectrongas 383
11.3 Conclusions 392
12 ThealgebraicapproachtotheHubbardmodel 393
12.1 Introductiontothequantuminversescatteringmethod 393
12.2 Shastry’sR-matrix 411
12.3 Gradedquantuminversescatteringmethod 425
12.4 TheHubbardmodelasafundamentalgradedmodel 440
12.5 Solutionofthequantuminverseproblem 450
12.6 OnthealgebraicBetheansatzfortheHubbardmodel 452
12.7 Conclusions 470
AppendicestoChapter12 472
viii Contents
12.A AproofthatShastry’sR-matrixsatisfiestheYang-Baxterequation 472
12.B Aproofoftheinversionformula 479
12.C Alistofcommutationrelations 484
12.D Someidentitiesneededintheconstructionofthetwo-particle
algebraicBetheansatz-states 484
12.E AnexplicitexpressionforthefermionicR-operator
oftheHubbardmodel 486
13 Thepathintegralapproachtothermodynamics 488
13.1 Thequantumtransfermatrixandintegrability 489
13.2 TheHeisenbergchain 496
13.3 Shastry’smodelasaclassicalanalogueofthe1dHubbardmodel 509
13.4 Diagonalizationofthequantumtransfermatrix 510
13.5 Associatedauxiliaryproblemofdifferencetype 514
13.6 Derivationofnon-linearintegralequations 519
13.7 Integralexpressionfortheeigenvalue 525
13.8 Numericalresults 536
13.9 Analyticalsolutionstotheintegralequations 547
13.10 Conclusions 555
AppendicestoChapter13 557
13.A DerivationofTBAequationsfromfusionHierarchyanalysis 557
13.B Derivationofsingleintegralequation 560
14 TheYangiansymmetryoftheHubbardmodel 563
14.1 Introduction 563
14.2 Thevariable-range-hoppingHamiltonian 564
14.3 ConstructionoftheYangiangenerators 566
14.4 Specialcases 570
14.5 Conclusions 573
AppendicestoChapter14 575
14.A Yangians 575
15 S-matrixandYangiansymmetryintheinfiniteintervallimit 599
15.1 Preliminaries 599
15.2 Passagetotheinfiniteinterval 600
15.3 Yangiansymmetryandcommutingoperators 605
15.4 ConstructingN-particlestates 607
15.5 EigenvaluesofquantumdeterminantandHamiltonian 617
15.6 Conclusions 617
AppendicestoChapter15 618
15.A Someusefulformulae 618
16 Hubbardmodelintheattractivecase 620
16.1 Half-filledcase 622
16.2 Thegroundstateandlowlyingexcitationsbelowhalffilling 625
16.3 Interactionwithmagneticfield 626