Introduction to Superfluidity: Field-theoretical Approach and Applications

160 Pages·2015·2.03 MB·English

by  Andreas Schmitt (auth.)

Lecture Notes in Physics 888
Andreas Schmitt
Introduction to
Superfl uidity
Field-theoretical Approach and
Applications
Lecture Notes in Physics
Volume 888
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Andreas Schmitt
Introduction to Superfluidity
Field-theoretical Approach and Applications
123
AndreasSchmitt
InstitutfuRrTheoretischePhysik
TechnischeUniversitätWien
Wien
Austria
ISSN0075-8450 ISSN1616-6361(electronic)
LectureNotesinPhysics
ISBN978-3-319-07946-2 ISBN978-3-319-07947-9(eBook)
DOI10.1007/978-3-319-07947-9
SpringerChamHeidelbergNewYorkDordrechtLondon
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Preface
Thiscourseisaboutthetheoryoflow-energyandhigh-energy,non-relativisticand
relativistic, bosonic and fermionic superfluidity and superconductivity. Does that
sound too much? Well, one important point of the course will be to show that
these things are not as diverse as they might seem: the mechanism behind and the
basicphenomenologicalpropertiesofsuperfluidityarethesamewhetherappliedto
“ordinary” low-energy superfluids or to more “exotic” superfluids in high-energy
physics;non-relativisticandrelativistictreatmentsmaylookquitedifferentatfirst
sight,butofcoursetheformerisonlyalimitcaseofthelatter;bosonicandfermionic
superfluidscanbecontinuouslyconnectedinsomesense;and,onceyouunderstand
whatasuperfluidis,itisveryeasytounderstandwhatasuperconductorisandvice
versa.
The motivation for this course arose from my own research in high-energy
physics where certain kinds of superfluids and superconductors are predicted in
ultra-dense nuclear and quark matter. These are “stellar superfluids”, since they
are likely to occur in the interior of compact stars. Working on stellar superfluids,
it was natural to learn about more down-to-earth superfluids which are firmly
established experimentally. Therefore, this course is interesting for researchers
who are in a similar situation like myself, who have some background in high-
energyphysicsandwanttolearnaboutsuperfluidity,explainedinafield-theoretical
languagetheyareusedto.Ibelievethatthecourseisalsoinsightfulforresearchers
with a background in condensed matter physics who are interested in high-energy
applicationsoftheirfieldandarelativisticfield-theoreticalformalismtheyusually
do not employ. And, most importantly, this course is intended for advanced
undergraduate students, graduate students, and researchers who simply want to
understandwhatsuperfluidityisandwhatitsapplicationsinmodernphysicsare.
Readers unfamiliar with quantum field theory might find some of the chapters
challenging, even though I have tried to present most of the calculations in a self-
contained way. When this was not possible, I have mentioned suitable references
where the necessary elements of field theory are explained. However, not all of
the chapters rely on field-theoretical methods. For instance, the course starts with
an introduction to superfluid helium that can easily be understood with basic
v
vi Preface
knowledge of statistical physics and thermodynamics. Most of the chapters that
do employ quantum field theory aim at a microscopic description of superfluids,
i.e., the degrees of freedom of the theory are the bosons that condense or the
fermionsthatformCooperpairs.Inthissense,thecourseisinlargepartsaboutthe
fundamentalmechanismsbehindsuperfluidity.But,Iwillemphasizetheconnection
to phenomenology throughout the course and do not want the reader to get lost in
technical details. For instance, I will show in a simple setting how a microscopic
quantumfieldtheorycanbeconnectedtothephenomenologicaltwo-fluidmodelof
asuperfluid.
Despitethepompousannouncementinthefirstsentence,thisisacoursethatcan
be taught in about one semester. Therefore, it can only deal with a few selected
aspects of superfluidity. This selection has been based on the aim to convey the
underlyingmicroscopicphysicsofsuperfluidity,onpedagogicalconsiderations,and
of course is also, to some extent, a matter of taste. As a result of this subjective
selection, there are many important aspects that I will not, or only marginally,
discuss, such as vortices in a rotating superfluid, dissipative effects, or observable
signaturesofstellarsuperfluids.Literaturethatcanbeconsultedforsuchtopicsand
forfurtherreadingingeneralisgivenattheendoftheintroductionandthroughout
thetext.
TheselecturenotesarebasedonacoursethatItaughtattheViennaUniversityof
Technology in the winter semester 2011/2012 and in the summer semester 2013.
I would like to thank all participants for numerous questions and many lively
discussions that have improved my understanding of superfluidity. I am grateful
to Mark Alford, Karl Landsteiner, S. Kumar Mallavarapu, David Müller, Denis
Parganlija, Florian Preis, Anton Rebhan, and Stephan Stetina for many helpful
comments and discussions. This work has been supported by the Austrian science
foundationFWFunderprojectno.P23536-N16andbytheNewCompStarnetwork,
COSTActionMP1304.
Vienna,Austria AndreasSchmitt
April2014
Contents
1 Introduction .................................................................. 1
1.1 SettingtheStage:WhatIsaSuperfluid?............................... 1
1.2 PlanoftheCourseandFurtherReading............................... 4
References..................................................................... 5
2 SuperfluidHelium ........................................................... 7
2.1 Landau’sCriticalVelocity.............................................. 8
2.2 ThermodynamicsofSuperfluidHelium................................ 10
2.3 Two-FluidModel........................................................ 13
2.4 FirstandSecondSound................................................. 17
2.4.1 Single-FluidHydrodynamics................................... 17
2.4.2 Two-FluidHydrodynamics..................................... 22
2.4.3 SoundModes ................................................... 25
References..................................................................... 30
3 SuperfluidityinQuantumFieldTheory .................................. 33
3.1 LagrangianandConservedCharge..................................... 34
3.2 SpontaneousSymmetryBreaking...................................... 37
3.3 SuperfluidVelocity...................................................... 40
3.4 GoldstoneMode......................................................... 43
3.5 SymmetryRestorationattheCriticalTemperature.................... 50
References..................................................................... 52
4 RelativisticTwo-FluidFormalism ......................................... 53
4.1 CovariantFormulation.................................................. 53
4.2 RelationtotheOriginalTwo-FluidFormalism........................ 57
4.3 ConnectingFieldTheorywiththeTwo-FluidFormalism............. 59
4.3.1 GoldstoneModeandSuperfluidDensity...................... 61
4.3.2 GeneralizedPressureandSonicMetric........................ 64
References..................................................................... 66
vii
viii Contents
5 FermionicSuperfluidity:CooperPairing ................................. 67
5.1 DerivationoftheGapEquation ........................................ 69
5.1.1 Lagrangian ...................................................... 70
5.1.2 Mean-FieldApproximation .................................... 72
5.1.3 Nambu-GorkovSpace.......................................... 74
5.1.4 GapEquation.................................................... 77
5.2 QuasiparticleExcitations ............................................... 79
5.3 SolvingtheGapEquation .............................................. 84
5.4 Examples ................................................................ 87
5.4.1 ElectronicSuperconductor ..................................... 87
5.4.2 AnisotropicSuperfluid ......................................... 88
5.4.3 ColorSuperconductor .......................................... 90
References..................................................................... 92
6 MeissnerEffectinaSuperconductor ..................................... 93
6.1 MassiveGaugeBoson .................................................. 94
6.2 MeissnerMassfromtheOne-LoopPolarizationTensor.............. 96
6.2.1 GaugeBosonPropagatorandScreeningMasses.............. 96
6.2.2 CalculationoftheMeissnerMass.............................. 99
References..................................................................... 103
7 BCS-BECCrossover ........................................................ 105
7.1 Ultra-ColdAtomicGases............................................... 106
7.2 CrossoverintheMean-FieldApproximation.......................... 110
References..................................................................... 119
8 Low-EnergyExcitationsinaFermionicSuperfluid ..................... 121
8.1 FluctuationsAroundtheMean-FieldBackground .................... 122
8.2 ExpandingintheFluctuations.......................................... 124
8.3 GoldstoneModeandLow-EnergyExpansion......................... 129
References..................................................................... 135
9 CooperPairingwithMismatchedFermiMomenta ..................... 137
9.1 QuasiparticleExcitations ............................................... 138
9.2 FreeEnergy.............................................................. 141
9.2.1 Chandrasekhar-ClogstonLimit ................................ 144
9.3 SuperfluidswithMismatchedChargeDensities....................... 149
References..................................................................... 154
Chapter 1
Introduction
1.1 Setting the Stage:WhatIs a Superfluid?
Superfluiditywasfirstobservedinliquidhelium.Thekeyexperimentwasthestudy
of flow through a thin capillary, and the key observation was that the fluid flows
without friction. Hence the name superfluid. What is behind this phenomenon?
Does it only occur in liquid helium? If not, where else? To generalize the specific
observationoffrictionlessflow,wenoticethatinordertoobserveaflow,something
is transported through the capillary. In liquid helium, we can say that mass is
transported.Wemayalsosaythatheliumatomsaretransported.Thisdoesnotmake
adifference,neitherthetotalmassoftheliquidnorthetotalnumberofheliumatoms
is changed during the experiment. Both are conserved quantities. In relativistic
systems, mass is not a conserved quantity in general. So, if we call the mass,
or better, the number of helium atoms, a “charge”, we can say that superfluidity
is frictionless transport of a conserved charge. Formulated in this way, we can
ask whether there are other systems where some other conserved charges show a
dissipationlessflow.
Before we do so, let us stay with superfluid helium for a moment. It turns out
thatthefrictionlessflowisnotitsonlyspectacularproperty.Forinstance,ifwetry
to rotate it, it will develop vortices, quasi-one-dimensional strings whose number
is proportional to the externally imposed angular momentum. The existence of
vortices is, besides the frictionless flow, another clear signature of superfluidity.
Furthermore,onefindsthatthespecificheatshowsapeculiarbehavioratacertain
temperature. This is the temperature below which helium becomes superfluid and
above which it behaves like a normal fluid. Therefore, a superfluid is a phase of
a given system below a certain critical temperature at which a phase transition
happens. What is the nature of this phase transition and how can we describe it
theoretically? For the case of liquid helium, more precisely for liquid 4He, the
answer is Bose-Einstein condensation, where the helium atoms occupy a single
quantumstate,forminga“condensate”.Thisphasetransitioncanbecharacterizedin
termsofsymmetriesofthesystem,andwecanmaketheconnectiontotheconserved
A.Schmitt,IntroductiontoSuperfluidity,LectureNotesinPhysics888, 1
DOI10.1007/978-3-319-07947-9__1,
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