Multi-scale Analysis for Random Quantum Systems with Interaction PDF
Preview Multi-scale Analysis for Random Quantum Systems with Interaction
Progress in Mathematical Physics 65 Victor Chulaevsky Yuri Suhov Multi-scale Analysis for Random Quantum Systems with Interaction Progress in Mathematical Physics Volume65 Editors-in-Chief AnneBoutetdeMonvel,UniversitéParisVIIDenisDiderot,France GeraldKaiser,CenterforSignalsandWaves,Austin,TX,USA EditorialBoard C.Berenstein,UniversityofMaryland,CollegePark,USA SirM.Berry,UniversityofBristol,UK P.Blanchard,UniversityofBielefeld,Germany M.Eastwood,UiversityofAdelaide,Australia A.S.Fokas,UniversityofCambridge,UK F.W.Hehl,UniversityofCologne,Germany andUniversityofMissouri-Columbia,USA D.Sternheimer,UniversitédeBourgogne,Dijon,France C.Tracy,UniversityofCalifornia,Davis,USA Forfurthervolumes: http://www.springer.com/series/4813 Victor Chulaevsky • Yuri Suhov Multi-scale Analysis for Random Quantum Systems with Interaction VictorChulaevsky YuriSuhov DépartementdeMathématiques StatisticalLaboratory UniversitédeReimsChampagne-Ardenne UniversityofCambridge Reims,France Cambridge,UK ISSN1544-9998 ISSN2197-1846(electronic) ISBN978-1-4614-8225-3 ISBN978-1-4614-8226-0(eBook) DOI10.1007/978-1-4614-8226-0 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2013950173 MathematicsSubjectClassification(2010):47B80,47A75,81Q10,60H25,35P10 ©SpringerScience+BusinessMediaNewYork2014 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.birkhauser-science.com) Preface This book is about rigorous results on localization in multi-particle interactive Anderson models, mostly on a cubic lattice Zd of dimension d (cid:2) 1. The book emerged in the wake of recent publications on the mathematical theory of localization in such models (also called multi-particle Anderson tight-binding models).Theaim of the book,asthe authorssee it, is to introducethe readerto a recentprogressinthisfieldandtoattractattentiontopossibledirectionsforfuture research. Thetermmulti-particleaddressesherea modelwithN particleswhereN > 1 isanarbitrarygivennumber.Thisisanaturalgeneralizationofthemodelwithone particle, andit is still a far cry from the ultimate goalto constructa theorywhere particlesare distributedwith a positivespatialdensity.However,eventhis modest steprequired(andnodoubtwillfurtherrequire)somenewideasandtechnicaltools. Itisthepresenceofinteractionandaspecificformoftheexternalrandompotential field which makes the principal difference between the model with one particle andthat with manyparticles. More precisely,the structureof the externalrandom term in the multi-particle Hamiltonian prevents one from applying single-particle techniques in a direct fashion. Compared with its single-particle counterpart, the multi-particlelocalizationtheoryisstillataveryearlystageofits development;a numberofquestionshereremainopen,whereasthesingle-particleversionsofthese questionshavereceivedanswers.Wehopethatourbookwillhelpthemulti-particle localizationtheorytotakeoffstrongly;aswesaidabove,anultimategoalwouldbe toattackthecaseofinfinitelymanyparticleswithapositivespatialdensitymoving inarandomenvironment. On the other hand, this book was written with a degree of uncertitude, and we do not make an attempt to conceal it. The fact is that at present, the single- particleAndersontheorydoesnotoffermuchofaclueastowhatoneshouldexpect whendealingwithmulti-particlesystemswithinteraction.Moreprecisely,working on the rigorous multi-particle Anderson theory requires a careful review of all technicalaspectsofthesingle-particletheoryincombinationwithawell-developed intuition.AllthisdemandstheuseofProbability,FunctionalAnalysis,Dynamical Systems, and Quantum Theory, including Quantum Statistical Mechanics, quite v vi Preface a formidable band of reputable mathematical disciplines. It is conceivable that the crucial progress in the development of the rigorous theory of multi-particle Anderson systems can only be achieved when a serious breakthrough (or rather a series of breakthroughs) is made, perhaps involving the above mathematical disciplinesinthefirstplace. Methodologically,ourbookfocusesmainlyontheso-calledMulti-ScaleAnalysis (MSA). In our context, the MSA represents a method that emerged in the single- particle Anderson theory after the 1983 paper by Fröhlich and Spencer [95] (for a brief and by no means complete historical outline, see Sect.1.1). We believe that possibilities offered by this method are far from being exhausted, and the currentand future generationsof researcherswould greatly benefitfrom a unified presentationwherestrongandweak pointsofthe MSA methodarediscussed in a contextofpossibleapplicationstocomplexsystems.Wedecidedtorestrictthebook to tight-bindingAnderson models on a cubic lattice with independent,identically distributed randomamplitudesof the externalpotential, as it allows us to free the presentation from a number of technical complications. (Models in a continuous space are mentioned in passing.) Multi-particle localization results and findings based on the MSA and related or applicable to the tight-bindingAnderson model have been published in [58,69–71], and, in part, in [49,50,72,148]. However, it wouldbefairtosaythattheentireprogressinthemathematicaltheoryofdisorder in random quantum systems contributed to the development of the multi-particle MSA. WealsomentioninvariousplacesofthebookthealternativeFractionalMoment Method (FMM) proposed by Aizenman and Molchanov, in the set-up of single- particle systems, in 1993 (cf. [7]) and modified for multi-particle models in 2009 (thefirstpreprintappearedin2008);see[9,10]. Sofar,progressinmulti-particletheoryhasbeenachievedbyusingmodifications ofoneoranotherofthesetwomethods,allowingthepassagefromasingleparticle to N particles; as was said earlier, it requires a thorough scrutiny of all technical stepsfromtheoriginalmethod(s). In the context of single-particle Anderson localization, various aspects of the MSA and FMM (as well as other technical means) have been discussed at the mathematicallyrigorouslevelinanumberofpreviouslywrittenbooksanddetailed reviews; cf. [39,52,53,77,113,118,144,155,158]. Section 1.2 of our book also gives a short version of such a review, with a view of possible extensions from one-particle systems to systems with several interacting particles, although such extensionsinsomeinstancescanbequitechallenging.Itisworthnotingthateven inthecaseofasingle-particletight-bindingmodel,theMSArequiressomerather involvedtechnicaltools(albeitbased,asarule,onquitesimpleobservations). Before we proceed further, we would like to point out that localization can manifestitselfinvariousways,althoughinthephysicscommunityitiscustomary toconsiderthesemanifestationsasdifferent“signatures”ofasinglephenomenon. Untilthelate1990s,rigorousresultsonsingle-particlelocalizationhavebeenabout exponentialspectrallocalizationwherethegoalwastoestablishthat,onthewhole lattice, the spectrum (more precisely, the spectral measure) of a given (random) Preface vii Hamiltonian(wherethekineticenergypartisrepresentedbyadiscreteLaplacian)is purepointanditseigenfunctionsdecayexponentiallyfastatinfinity.Inthisregard, westressthatwhenwespeakinthisbookoflocalizedeigenfunctions(ortalkabout localizationloosely),wemeanexponentiallocalization. However,inlinewithtechnicalprogress,thereappearedpossibilitiestoanalyze dynamicallocalization,i.e.,toassessthespreadofaninitialwavefunction(say,with a finite-point support on the lattice) in the course of the time evolution generated bytheHamiltonian.Dynamicallocalization,besidesitspurelytheoreticalvalue,is importantinexperimentsaswell.Thelinkbetweenthetwoformsoflocalizationis providedbythe so-calledRAGE theorems(see [77];theoriginalpublicationsthat gaverisetotheacronymare[19,88,147])allowingtodeducespectrallocalization (more precisely, the pure-point nature of the spectral measure) from dynamical localization. However, the method developed in this book derives both forms of localizationfromthe same (probabilistic)estimates, alongthe same path as in the single-particle theory. In our view, this fact puts these two manifestations of the localization,essentially,onequalfootage. By the time of the book’s completion, some new results appeared or were expectedtoappear.Welistedthem(withvariousdegreesofdetail)intheconcluding Sect.4.7ofChap.4. A couple of words about the style of the book: Due to a special character of thematerial(andrestrictionsuponthelengthofthebook),wewerenotinposition to deriveall necessaryconceptsandfactsfromthe first principles.However,most of the arguments used in the book are straightforward and, taken in isolation, should not be overtly difficult to follow. It is the combination of these arguments that makes the presentation technically involved. In this situation, we decided to follow the principle reflected in a popular Russian saying “Repetition is the mother of learning,” which is actively used in various educational systems. (In our own schooling experience, teachers sometimes abused this principle, which caused lack of enthusiasm among pupils, but the maxim itself should not have beenblamed.)Asfarasthisbookisconcerned,wesystematicallyrepeatformulas, equations,assertions,andcomments.However,theauthorshopethattheyhavenot exceededreasonablelevelsofrepetitiveness,beyondwhichitbecomesirritatingand counterproductive. Thebookcontainsfourchapters.Chapters1and2formPartI,onsingle-particle systems, while Chaps.3 and 4 constitute Part II, about multi-particle systems. Chapter 1 is introductory and contains comments on a number of crucial results aboutlocalizationanddelocalizationforvarioussingle-particlemodelsofquantum disorder.Chapter2developsthetheoryofsingle-particleAndersonlocalizationvia the MSA. We pay a particular attention to key parts of the single-particle MSA andpresentthemfromthepointof viewoffutureextensionoftheMSA to multi- particleAndersonmodels.Chapters3and4aredevotedtothemulti-particleMSA, culminatingintheproofofspectralanddynamicallocalizationforlargeamplitudes oftheexternalrandompotentialfield. Takingintoaccountanabundanceofvariousconstantsandotherminorprotago- nistsinthepresentation(afeaturethatisdifficulttoavoidwhenyouspeakaboutan viii Preface asymptoticaltheory),wechosetovarythenumerationofconstantsfromonesection toanother.Besides,wedecidedtofollowanOrwellianprinciplethat“allconstants are constantbut some are more constant than others.” The lower caste is formed bythosefacelessvaluesthatappearinproofsofvarioustechnicalstatementsorare mentionedinpassing:ThesearesimplymarkedasConst.Theuppercasteisformed byconstantsdeservinganindividualnotationoraspecificcomment.Forthesewe employavarietyofnotationasweseefitinaparticularcontext;wearesurethatour system(ifindeeditcanbecalledasystematall)maybesignificantlyimproved.But, rephrasingHitchcock1:“Allofthemareonlyconstants...”(andinmanyinstances theyhavebeenidentifiednumericallyorintermsofbasicquantities). The authors would like to acknowledge the hospitality, including financial support, at a number of universities and institutes they visited, individually or jointly, in recent years, where countless discussions took place with colleagues workingon related subjects and with membersof wider mathematicsand physics groups.TheseincludetheUniversityofCambridge(particularly,theIsaacNewton Institute), Université de Reims Champagne-Ardenne, IHES (Bures-sur-Yvette), IAS (Princeton) and Princeton University, University of Alabama (Birmingham), University of California (campuses at Berkeley, Davis, Irvine, and Los Angeles), UniversityofToronto,UniversitéParis6–7,Technion(Haifa),ETH(Zürich),EPF (Lausanne),DIAS (Dublin),IMPA (Rio de Janeiro),and Universityof Sao Paulo, the latter under the financial support from FAPESP. It would take up a lot of space to name all the individuals who generously shared their thoughts with us, eitherprivatelyorinpublic,orsimplylistenedpatientlytowhatwewantedtosay (sometimesinaratherincoherentfashion).However,weshouldparticularlythank Michael Aizenman, Anne Boutet de Monvel (who made a considerable number of comments and suggestions which contributed to improving the quality of the manuscript), David Damanik, Efim I. Dinaburg, Shmuel Fishman, Jürg Fröhlich, YanFyodorov,FrançoisGerminet,Ilya Goldsheid,IgorGornyy,Misha Goldstein, Gian Michele Graf, Lana Jitomirskaya, Yulia Karpeshina, Dmitrii Khmelniskii, Werner Kirsch, Abel Klein, Frédéric Klopp, Shinichi Kotani, Alexander Mirlin, Stanislav A. Molchanov, Peter Müller, Fumihiko Nakano, Leonid A. Pastur, Jeff Schenker, Boris Shapiro, Dima Shepelyansky, Hermann Schulz-Baldes, Sasha Sodin, Tom Spencer, Roman Shterenberg, Peter Stollmann, Günter Stolz, Ivan Veselic´,SimoneWarzel,andMartinZirnbauer. Our special thanks go to our teacher Yakov Grigorievich Sinai: one of the traits of his inimitable tutoring style, which helped with this project, was that he instilledinusacompletedisregardforformalboundariesbetweendifferentareasof mathematics. Cambridge, UK-Reims, France-SaoPaolo, Brazil VictorChulaevsky Cambridge,UK YuriSuhov 1AlfredJosephHitchcock(1899–1980)wasanEnglishfilmdirectorandproducerwhoinvented noveltechniquesofsuspenseinthegenreofapsychologicalthriller.“It’sOnlyaMovie”isthetitle ofhisbiography(writtenbyCharlotteChandler). Contents PartI Single-ParticleLocalization 1 ABriefHistoryofAndersonLocalization................................. 3 1.1 AndersonLocalizationinTheoreticalPhysics......................... 3 1.2 LocalizationinanIIDExternalPotential.............................. 5 1.3 LocalizationVersusDelocalizationinaQuasiperiodic ExternalPotential....................................................... 8 1.4 SpectralandDynamicalManifestationsofAnderson Localization ............................................................. 18 1.4.1 SpectralLocalization........................................... 18 1.4.2 DynamicalLocalization........................................ 18 1.5 TheN-ParticleModelinaRandomEnvironment .................... 20 1.5.1 TheHamiltonianoftheN-ParticleSysteminZd............. 21 1.5.2 TheTwo-ParticleCase.......................................... 23 1.5.3 SystemsofPositiveSpatialDensities.......................... 24 2 Single-ParticleMSATechniques ........................................... 27 2.1 AnInitiationintotheSingle-ParticleMSA............................ 28 2.1.1 TechnicalRequisites............................................ 28 2.1.2 TheMSAInduction............................................. 32 2.1.3 Fixed-Energyand Variable-EnergyMSA: InformalDiscussion ............................................ 33 2.1.4 Fixed-EnergyMSA............................................. 34 2.1.5 Variable-EnergyMSA.......................................... 37 2.1.6 Single-ParticleLocalizationResults........................... 39 2.2 EigenvalueConcentrationBounds ..................................... 43 2.2.1 Wegner’sBounds ............................................... 43 2.2.2 Stollmann’sProduct-MeasureLemma......................... 44 2.2.3 Stollmann’sEVCBound ....................................... 47 2.2.4 TheDensityofStatesandHigher-OrderEVCBounds....... 50 ix
The list of books you might like
