Renormalization group approach to superfluid neutron matter [Elektronische Ressource] / von Kai Hebeler
145 pages
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Renormalization group approach to superfluid neutron matter [Elektronische Ressource] / von Kai Hebeler

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145 pages

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Renormalization Group Approach toSuperfluid Neutron MatterVom Fachbereich Physikder Technischen Universit¨at Darmstadtzur Erlangung des Gradeseines Doktors der Naturwissenschaften(Dr. rer. nat.)genehmigte Dissertation vonDipl.-Phys. Kai Hebeleraus Bad HersfeldDarmstadt 2007D17Referent: Prof. Dr. Bengt FrimanKoreferent: Prof. Dr. Jochen WambachTag der Einreichung: 23. Januar 2007Tag der Pru¨fung: 12. Februar 2007Meinen Eltern.The most incomprehensible thing about the worldis that it is at all comprehensible.Albert EinsteinIIContents IIIContentsList of Figures VIntroduction and Overview i1. Superfluid Nuclear systems 11.1. History of Superconductivity . . . . . . . . . . . . . . . . . . . . . 11.2. The Cooper Phenomenon . . . . . . . . . . . . . . . . . . . . . . 31.3. The Superconducting Ground State . . . . . . . . . . . . . . . . . 61.4. From Superconductivity to Superfluidity . . . . . . . . . . . . . . 111.5. Superfluidity in Neutron Matter . . . . . . . . . . . . . . . . . . . 131.6. Nambu-Gorkov Propagators . . . . . . . . . . . . . . . . . . . . . 151.7. Scattering Equation in a Superfluid . . . . . . . . . . . . . . . . . 211.8. Separable Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.9. Relation between Gap Equation and Scattering Equation . . . . . 361.10.Problems and Limitations of the Inversion Approach . . . . . . . 382. RG approach to Superfluid many-body systems 412.1. Basic Ideas . . . . . . . . . . . . . . . .

Sujets

Informations

Publié par
Publié le 01 janvier 2007
Nombre de lectures 18
Poids de l'ouvrage 1 Mo

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RenormalizationGroupApproach
SuperuidNeutronMatter

VomFachbereichPhysik
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Contents

Contents

III

ListofFiguresV
IntroductionandOverviewi
1.SuperuidNuclearsystems1
1.1.HistoryofSuperconductivity.....................1
1.2.TheCooperPhenomenon......................3
1.3.TheSuperconductingGroundState.................6
1.4.FromSuperconductivitytoSuperuidity..............11
1.5.SuperuidityinNeutronMatter...................13
1.6.Nambu-GorkovPropagators.....................15
1.7.ScatteringEquationinaSuperuid.................21
1.8.SeparableModel...........................29
1.9.RelationbetweenGapEquationandScatteringEquation.....36
1.10.ProblemsandLimitationsoftheInversionApproach.......38
2.RGapproachtoSuperuidmany-bodysystems41
2.1.BasicIdeas..............................41
2.2.RGApproachtoMany-FermionSystems..............43
2.3.SolvingMany-BodyProblemsbyusingtheRG...........46
2.4.InteractionDependenceofthePairingGap.............51
2.5.RGEvolutionfromtheNormalintotheSuperuidPhase.....57
2.6.ResultsforRealisticInteractions...................68
3.BeyondtheBCSapproximation75
3.1.Overview................................75
3.2.RGApproachtotheParquetEquations..............80
3.3.Self-EnergyCorrections........................83
3.4.GeneralizedGapEquation......................86
3.5.ComparisontotheExactRGEquations..............93
3.6.ParquetEquationsinaSuperuid..................98
3.7.PartialWaveCoupling........................104
4.ConclusionsandOutlook107

VI

A.Appendix
A.1.NumericalEvaluationoftheFlowEquation
A.2.Particle-HoleInteraction...........
A.3.FlowEquationsinVacuum..........

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ListofFigures

ListofFigures

V

1.1.Heatcapacityofasuperconductor..................2
1.2.Thethreemany-bodychannels....................3
1.3.AnalyticstructureofthevertexfunctionintheCooperproblem..5
1.4.FinalstatephasespaceintheCooperproblem...........6
1.5.Distributionfunctionsandexcitationspectruminsuperuidsystems.9
1.6.Excitationspectrumofliquid4He...................12
1.7.1S0and3P23F2paringgapinneutronmatterintheBCSap-
proximation..............................14
1.8.One-bodyNambu-GorkovpropagatorintheBCSapproximation.16
1.9.GapfunctionandgapequationintheBCSapproximation....17
1.10.Two-bodyNambu-GorkovpropagatorintheBCSapprox.....20
1.11.Illustrationofthepossibleprocessesinasuperuidsystem.....22
1.12.Antisymmetrizedbaretwo-particleinteraction............23
1.13.Typicaldiagramsofthetwo-particleNambu-Gorkovpropagatorin
second-orderperturbationtheory...................25
1.14.Thenon-superuidlimitoftheBCS-channelscatteringequation..26
1.15.ThethreeBethe-Salpeterequationsinanormalsystem.......27
1.16.Analyticstructureofthevertexfunctioninthesuperconducting
groundstate..............................32
1.17.Paringgapasafunctionofdensityforaseparablemodel......33
1.18.DispersionrelationoftheGoldstoneboson..............34
1.19.Imaginarypartofthevertexfunctionasafunctionofenergyfor
aseparablemodel...........................34
1.20.Realpartofthevertexfunctionasafunctionofenergyforasep-
arablemodel..............................35
2.1.Comparisonofthelowmomentumone-bodyHilbertspaceinvac-
cumandmedium............................43
2.2.Kinematicsofinteractinglowlyingexcitationsinamany-body
system..................................44
2.3.Marginalinteractionprocessesinamany-bodysystemattreelevel.46
2.4.Spuriouspolesoftheon-shellT-matrixelements..........52
2.5.Neutron-neutronpairinggapsasafunctionofdensityforlowmo-
mentuminteractions..........................54
2.6.Chargedependenceofthepairinggap................55

IV

ListofFigures

2.7.Pairinggapsatxeddensitiesasafunctionofthecutofordif-
ferentwidths..............................56
2.8.Pairinggapsatxeddensitiesasafunctionofthecutofordif-
ferentinteractions...........................57
2.9.Dierentcutofunctions.......................61
2.10.Formationofthegapasafunctionofthecuto...........62
2.11.Snapshotsofthevertexfunctionatdierentcutoscales......63
2.12.ResultsoftheRGalgorithmfortherealpartofvertexfunctionas
afunctionofenergy..........................64
2.13.ResultsoftheRGalgorithmfortheimaginarypartofvertexfunc-
tionasafunctionofenergy......................65
2.14.Illustrationofowintothebrokenphase...............67
2.15.Realpartofthevertexfunctionasafunctionofmomentumat
xedenergiesforrealisticinteractions................68
2.16.ResultsfortherealpartofthevertexfunctionontheFermisurface
forrealisticinteractions........................69
2.17.ResultsfortheimaginarypartofthevertexfunctionontheFermi
surfaceforrealisticinteractions....................70
2.18.Realpartofthevertexfunctionforrealisticinteractionsonthe
FermisurfacefordierentCMmomenta...............72
2.19.ImaginarypartofthevertexfunctionontheFermisurfacefor
dierentCMmomenta.........................73
3.1.Self-energyeectsinthegapfunctioninquasiparticleapprox...76
3.2.Deningsystemfortheinducedinteraction.............78
3.3.Monte-Carlosimulationresultsforthegapfunction.........79
3.4.Resultsforthe1S0gapfunctionindierentapproaches.......80
3.5.Theone-loopparticle-holeparquetowequation..........81
3.6.1S0pairinggapintheRGapproachtotheparticle-holeparquet..82
3.7.Dysonequationsforthesingle-particleGorkovpropagators.....84
3.8.Thetwo-particleirreduciblevertexfunctions.............90
3.9.Representationsofthegapfunction..................90
3.10.Thegeneralizedgapequation.....................91
3.11.TheexactRGequationsforthefourpointvertex..........95
3.12.Tworepresentationsoftheowequationforthetwo-pointfunction.97
3.13.Bethe-Salpeterequationsinasuperuid...............99
3.14.Parquetequationsinasuperuid...................100
3.15.Comparisonoftheexplicitanditerativesolution..........103
3.16.Illustrationofthepropertiesofwithpartialwavecoupling....106
3.17.Collectiveexcitationspectruminanattractivepartialwave....106

IntroductionandOverview

i

Thepairingphenomenonplaysacentralroleincoldfermionicmany-bodysys-
tems.Fromthepresenttheoreticalunderstandingpairingshouldappearinalmost
everymany-fermionsystematsucientlylowtemperatures.Indeed,duringthe
lastcenturyithasbeenobservedinmanydierentsystemslikemetals[1],atomic
nuclei[2]andliquid3He[3].Thebasicunderlyingmicroscopicmechanismwas
discoveredinthepioneeringworkofBardeen,SchrieerandCooper[4]in1957,
todaywellknownastheBCStheoryofsuperconductivity.Basedonthistheory
theoccurrenceofsuperconductivityandsuperuiditycanbeunderstoodquite
intuitivelyonthebasisofsimplephasespacearguments.
Ingeneraltwoparticularscatteringprocessesplayaspecialroleinisotropic
many-fermionsystemsatlowenergies.Therstcorrespondstoforwardscat-
teringprocesses.TheseformthebasisofLandau’stheoryofFermiLiquidsand
describethedominantinteractionsoflowenergyelementaryexcitationsinnormal
Fermisystems.Thesecondcorrespondstotwobodyprocessesatzerocenterof
mass(CM)momentum.Inthecaseofattractiveinterparticleinteractionsthese
processesleadnaturallytoaninstabilityofthesystemtowardstheformationof
two-particleboundstatescalledCooperpairs.Thecharacteristicphenomenology
ofsuperuidandsuperconductingsystemscanbequalitativelyunderstoodsolely
bythepresenceofthecondensate,amacroscopicallypopulatedcollectivestate
consistingofCooperpairs.Inthissensetheoccurrenceofpairingandsuperu-
idityinFermisystemsisanaturalconsequenceoftheexclusionprincipleand
consequentlythefundamentalpropertiesofsuperuidsystemscanqualitatively
beunderstoodfromrstprinciples.
However,aquantitativeunderstandingofthebulkpropertiesofthesesystems
ismuchmoreinvolved.Ingeneralmany-bodycorrelations,whichareneglected
intheoriginalBCStheory,willchangethevalueofthecondensateconsiderably.
Especiallyinstronglycorrelatedsystemslikenuclearsystems,itisaverycomplex
tasktotreatallinterparticlecorrelationsatdierentlengthscalesinasystematic
way.Ontheotherhandmanymacroscopicpropertiesdependverysensitivelyon
theparticularvalueofthecondensate.Thusforasystematicquantitativeunder-
standingofthesesystemstheapplicationofcontrolledmicroscopictechniquesis
necessary.
Inthecaseofnuclearsystemsthebasicingredientofsuchmicroscopicap-
proachesisthetwo-bodynucleon-nucleon(NN)interactionandpossiblethree-
body(3N)interactions,determinedfromvacuumscatteringexperimentsandthe
propertiesofthedeuteron.TheNN-interactionhasbeenstudiedingreatde-

ii

IntroductionandOverview

tailduringthelastdecades.Atpresentthereexistdierentmodelswhichallt
thebasicobservablestoverygoodaccuracy.Allthesemodelssharesomebasic
features.
•Almostallpotentialsparametrizetheinteractiononlyforenergiesbelow
thepionproductionthreshold.Beyondthisscalethepotentialdescription
oftheinteractionbreaksdownandthenon-relativisticquantummechanical
descriptionmustbereplacedbyaneectivequantumeldtheory.Dueto
thisprincipallimitation,onlyneutronsystemsuptoamaximaldensityof
abouttwicenucleardensityncanbeproperlydescribedmicroscopically
startingfromtheseNNinteractions.
•Thepotentialincludesstrongrepulsionatsmalldistancesandanattractive
longrangepart.Thesepropertiesleadnaturallytothesaturationofnuclear
systems.
•Scatteringexperimentsshowalargenegativescatteringlengthinthe1S0
partialwavechannelsignallingthepresenceofanalmostbound(”anti-
bound”)state.Incontrast,duetothepresenceofthetensorforcecompo-
nentinthespin-triplet3S1-3D1channel,theneutron-protoninteractioncan
supportaboundstate,thedeuteron,whereasthecorrespondingneutron-
neutronstateinthesamepartialwavechannelremainsunbound.Due
tothelargescatteringlengtha,lowdensityapproximations,basedonan
expansioninthesmallparameterakF,alreadybreakdownatquitesmall
FermimomentakF.
Startingfromthis”bare”interactionallmany-bodycorrelationeectsleadingto
theformationofmany-bodyboundstates,collectiveexcitationmodesetc.are
generateddynamicallybysolvingthemany-bodyprobleminacertainapproxi-
mation.
Innature,thesizeoftheCooperPairwavefunctionintypicalsuperuidscovers
manyotherparticles.Duetothepresenceoftheseparticlesthebareinteraction
getsmodiedsubstantiallybymany-bodyeects.Besidethesepolarizationef-
fects,thepresenceoftheotherparticleswillaectthesingle-particleproperties.
Ideallythesetwoeectshavetobetakenintoaccountatthesametimeandon
equalfooting.
ApromisingapproachtothiscomplexproblemistheRenormalizationGroup
(RG):insteadoftryingtosolvetheequationsinonestepforallmomentum
scales,onedividestheproblemintosmallerandeasierstepsbyintroducinga
cutoscale.Thisscalerepresentsthedividinglinebetweentheslowandthe
fastmodesofthesystem.Theslowmodesrepresentthemodesoftheeective
Hilbertspaceatthecurrentscale,whereasthefastmodeshavealreadybeen
integratedoutintotheeectiveoperatorsoftheHamiltonianatthisscale.The
RGowequationdescribeshowtheeectiveHamiltonianismodi