Table Of Content

Lecture Notes in Physics 888 Andreas Schmitt Introduction to Superfl uidity Field-theoretical Approach and Applications Lecture Notes in Physics Volume 888 FoundingEditors W.Beiglböck J.Ehlers K.Hepp H.Weidenmüller EditorialBoard B.-G.Englert,Singapore,Singapore P.Hänggi,Augsburg,Germany W.Hillebrandt,Garching,Germany M.Hjorth-Jensen,Oslo,Norway R.A.L.Jones,Sheffield,UK M.Lewenstein,Barcelona,Spain H.vonLöhneysen,Karlsruhe,Germany M.S.Longair,Cambridge,UK J.-F.Pinton,Lyon,France J.-M.Raimond,Paris,France A.Rubio,Donostia,SanSebastian,Spain M.Salmhofer,Heidelberg,Germany S.Theisen,Potsdam,Germany D.Vollhardt,Augsburg,Germany J.D.Wells,Geneva,Switzerland Forfurthervolumes: http://www.springer.com/series/5304 The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new devel- opments in physics research and teaching-quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in anaccessibleway.Bookspublishedinthisseriesareconceivedasbridgingmaterial between advanced graduate textbooks and the forefront of research and to serve threepurposes: (cid:129) to be a compact and modern up-to-date source of reference on a well-defined topic (cid:129) to serve as an accessible introduction to the field to postgraduate students and nonspecialistresearchersfromrelatedareas (cid:129) to be a source of advanced teaching material for specialized seminars, courses andschools Both monographs and multi-author volumes will be considered for publication. Editedvolumesshould,however,consistofaverylimitednumberofcontributions only.ProceedingswillnotbeconsideredforLNP. Volumes published in LNP are disseminated both in print and in electronic for- mats,theelectronicarchivebeingavailableatspringerlink.com.Theseriescontent isindexed,abstractedandreferencedbymanyabstractingandinformationservices, bibliographicnetworks,subscriptionagencies,librarynetworks,andconsortia. Proposals should be sent to a member of the Editorial Board, or directly to the managingeditoratSpringer: ChristianCaron SpringerHeidelberg PhysicsEditorialDepartmentI Tiergartenstrasse17 69121Heidelberg/Germany [email protected] 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 LibraryofCongressControlNumber:2014944556 ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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, ©SpringerInternationalPublishingSwitzerland2015