Supermathematics and its Applications in Statistical Physics: Grassmann Variables and the Method of Supersymmetry PDF
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Lecture Notes in Physics 920 Franz Wegner Supermathematics and its Applications in Statistical Physics Grassmann Variables and the Method of Supersymmetry Lecture Notes in Physics Volume 920 FoundingEditors W.Beiglböck J.Ehlers K.Hepp H.Weidenmüller EditorialBoard M.Bartelmann,Heidelberg,Germany B.-G.Englert,Singapore,Singapore P.HaRnggi,Augsburg,Germany M.Hjorth-Jensen,Oslo,Norway R.A.L.Jones,Sheffield,UK M.Lewenstein,Barcelona,Spain H.vonLoRhneysen,Karlsruhe,Germany 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,AnnArbor,USA G.P.Zank,Huntsville,USA The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new devel- opmentsin physicsresearch and teaching-quicklyand informally,but with a high qualityand the explicitaim to summarizeand communicatecurrentknowledgein 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 Bothmonographsandmulti-authorvolumeswillbeconsideredforpublication. Editedvolumesshould,however,consistofaverylimitednumberofcontributions only.ProceedingswillnotbeconsideredforLNP. VolumespublishedinLNParedisseminatedbothinprintandinelectronicfor- mats,theelectronicarchivebeingavailableatspringerlink.com.Theseriescontent isindexed,abstractedandreferencedbymanyabstractingandinformationservices, bibliographicnetworks,subscriptionagencies,librarynetworks,andconsortia. Proposalsshouldbe sent to a memberof the EditorialBoard, ordirectly to the managingeditoratSpringer: ChristianCaron SpringerHeidelberg PhysicsEditorialDepartmentI Tiergartenstrasse17 69121Heidelberg/Germany [email protected] Moreinformationaboutthisseriesathttp://www.springer.com/series/5304 Franz Wegner Supermathematics and its Applications in Statistical Physics Grassmann Variables and the Method of Supersymmetry 123 FranzWegner InstitutfuRrTheoretischePhysik UniversitaRtHeidelberg Heidelberg,Germany ISSN0075-8450 ISSN1616-6361 (electronic) LectureNotesinPhysics ISBN978-3-662-49168-3 ISBN978-3-662-49170-6 (eBook) DOI10.1007/978-3-662-49170-6 LibraryofCongressControlNumber:2016931278 SpringerHeidelbergNewYorkDordrechtLondon ©Springer-VerlagBerlinHeidelberg2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) To Anne-Gret, Annette, andChristian Preface This book arose from my interest in disordered systems. It was known, for some time,thatdisorderinaone-particleHamiltonianusuallyleadstolocalizedstatesin one-dimensionalchains.Andersonhadarguedthatinhigher-dimensionalsystems, theremayberegionsoflocalizedandextendedstates,separatedbyamobilityedge. In1979and1980,itbecameclearthatthisAndersontransitioncouldbedescribed intermsofanonlinearsigmamodel.LotharSchäferandmyselfreducedthemodel to one described by interacting matrices by means of the replica trick. Efetov, Larkin,and Khmel’nitskiiperformeda similar calculation. They,however,started fromadescriptionbymeansofanticommutingcomponents.In1982Efetovshowed that a formulation without the replica trick was possible using supervectors and supermatriceswithequalnumberofcommutingandanticommutingcomponents. I hadthe pleasureof givingmanylecturesandseminarson disorderedsystems and critical systems, and also on fermionic systems, where Grassmann variables playanessentialrole.AmongthemwereseminarsintheSonderforschungsbereich (collaborativeresearchcenter)onstochasticmathematicalmodelswithmathemati- cians and physicists and in the Graduiertenkolleg (research training group) on physical systems with many degrees of freedom and seminars with Heinz Horner andChristofWetterich.Inparticular,IrememberaseminarwithGüntherDoschon Grassmannvariablesinstatisticalmechanicsandfieldtheory. Some of the applicationsof Grassmann variablesare presented in this volume. Thebookisintendedforphysicists,whohaveabasicknowledgeoflinearalgebra and the analysis of commuting variables and of quantum mechanics. It is an introductory book into the field of Grassmann variables and its applications in statisticalphysics. The algebra and analysis of Grassmann variables is presented in Part I. The mathematicsofthesevariablesisappliedtoarandommatrixmodel,topathintegrals forfermions(incomparisontothe pathintegralsforbosons)andto dimermodels andtheIsingmodelintwodimensions. Supermathematics, that is, the use of commuting and anticommuting variables on an equal footing, is the subject of Part II. Supervectors and supermatri- ces, which contain both commuting and Grassmann components, are introduced. vii viii Preface In Chaps.10–14, the basic formulae for such matrices and the generalization of symmetric,real, unitary,andorthogonalmatricesto supermatricesare introduced. Chapters 15–17 contain a number of integral theorems and some additional information on supermatrices. In many cases, the invariance of functions under certaingroupsallowsthereductionoftheintegralstothosewherethesamenumber ofcommutingandanticommutingcomponentsiscanceled. In Part III, supersymmetric physical models are considered. Supersymmetry appearedfirstinparticlephysics.Ifthissymmetryexists,thenbosonsandfermions exist with equal masses. So far, they have not been discovered. Thus, either this symmetrydoesnotexistoritisbroken.Theformalintroductionofanticommuting space-timecomponents,however,canalsobeusedinproblemsofstatisticalphysics and yields certain relations or allows the reduction of a disordered system in d dimensionsto a pure system in d (cid:2)2 dimensions. Since supersymmetryconnects stateswithequalenergies,ithasalsofounditswayintoquantummechanics,where pairs of Hamiltonians, Q(cid:2)Q and QQ(cid:2), yield the same excitation spectrum. Such modelsareconsideredinChaps.18–20. In Chap.21, the representation of the random matrix model by the nonlinear sigma model and the determination of the density of states and of the level correlation are given. The diffusive model, that is, the tight-binding model with random on-site and hopping matrix elements, is considered in Chap.22. These models show collective excitations called diffusions and if time-reversal holds, alsocooperons.Chapter23discussesthemobilityedgebehaviorandgivesashort account of the ten symmetry classes of disorder, of two-dimensional disordered models,andofsuperbosonization. I acknowledge useful comments by Alexander Mirlin, Manfred Salmhofer, Michael Schmidt, Dieter Vollhardt, Hans-Arwed Weidenmüller, Kay Wiese, and MartinZirnbauer.VirafMehtakindlymadesomeimprovementstothewording. Heidelberg,Germany FranzWegner September2015 Contents PartI GrassmannVariablesandApplications 1 Introduction................................................................. 3 1.1 History................................................................ 3 1.2 Applications.......................................................... 4 References.................................................................... 5 2 GrassmannAlgebra........................................................ 7 2.1 ElementsoftheAlgebra ............................................. 7 2.2 EvenandOddElements,GradedAlgebra .......................... 8 2.3 BodyandSoul,Functions............................................ 10 2.4 ExteriorAlgebraI.................................................... 10 References.................................................................... 12 3 GrassmannAnalysis........................................................ 13 3.1 Differentiation........................................................ 13 3.2 Integration............................................................ 15 3.3 GaussIntegralsI...................................................... 16 3.4 ExteriorAlgebraII................................................... 21 References.................................................................... 27 4 DisorderedSystems......................................................... 29 4.1 Introduction........................................................... 29 4.2 ReplicaTrick ......................................................... 30 4.2.1 FirstVariant................................................. 30 4.2.2 SecondVariant.............................................. 30 4.3 QuantumMechanicalParticleinaRandomPotential.............. 31 4.4 SemicircleLaw....................................................... 32 References.................................................................... 35 ix
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