Anyons in Infinite Quantum Systems PDF
Preview Anyons in Infinite Quantum Systems
Anyons in Infinite Quantum Systems ˘ ¯ QFT in d 2 1 and the Toric Code PROEFSCHRIFT terverkrijgingvandegraadvandoctor aandeRadboudUniversiteitNijmegen opgezagvanderectormagnificusprof. mr. S.C.J.J.Kortmann volgensbesluitvanhetcollegevandecanen inhetopenbaarteverdedigenopdinsdag15mei2012 om13.30uurprecies door PieterNaaijkens geborenop12september1982 teTilburg Promotor: Prof.dr.N.P.Landsman Copromotor: Dr.M.Müger Manuscriptcommissie: Prof.dr.M.Fannes(KULeuven) Prof.dr.M.I.Katsnelson Dr.J.D.M.Maassen Prof.dr.K.-H.Rehren(Georg-August-UniversitätGöttingen) Prof.dr.R.F.Werner(LeibnizUniversitätHannover) This research was supported by the Netherlands Organisation for Scientific Re- search(NWO)underprojectno.613.000.608. MathematicsSubjectClassification(MSC)2010:81T05,18D10,46L60 GedruktdoorIpskampDrukkers,Enschede Contents Preface vii Introduction ix Localquantumphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Mainresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Organisationofthethesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi I Preliminaries 1 1 Operatoralgebras 5 1.1 Basictheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 vonNeumannalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Inductivelimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 HilbertspacesinvonNeumannalgebras . . . . . . . . . . . . . . . 16 2 Tensorcategories 19 2.1 Categorytheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Tensorcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Symmetryandbraiding . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Linearstructureandfusion . . . . . . . . . . . . . . . . . . . . . . . 27 2.5 Dualsanddimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Modulartensorcategories . . . . . . . . . . . . . . . . . . . . . . . . 34 2.7 TheDoplicher-Robertsreconstructiontheorem . . . . . . . . . . . 37 3 Localquantumphysics 39 3.1 Finitequantumsystems . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Algebraicquantumfieldtheory . . . . . . . . . . . . . . . . . . . . . 42 3.3 Fieldnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4 Quantumlatticesystems . . . . . . . . . . . . . . . . . . . . . . . . . 54 iii Contents 4 Topologicalquantumcomputers 63 4.1 Quantumcomputing . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 Topologicalquantumcomputing . . . . . . . . . . . . . . . . . . . . 67 4.3 Fibonaccianyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5 Thequantumdoubleofafinitegroup 77 5.1 Quantumdoublesoffinitegroups . . . . . . . . . . . . . . . . . . . 78 5.2 Representationtheory . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3 Dualsandribbonstructure. . . . . . . . . . . . . . . . . . . . . . . . 85 5.4 Rep D(G)ismodular. . . . . . . . . . . . . . . . . . . . . . . . . . . 86 f II Stringlikelocalisedsectorsind ˘2¯1 89 6 Stringlikelocalisedsectors 91 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.2 Stringlikelocalisedsectors . . . . . . . . . . . . . . . . . . . . . . . . 94 7 Extensionandrestriction 105 7.1 Thefieldnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.2 Extensiontothefieldnet . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.3 Non-abeliancohomologyandrestrictiontotheobservablealgebra 115 8 Categoricalaspects 119 8.1 Categoricalcrossedproducts . . . . . . . . . . . . . . . . . . . . . . 119 8.2 EssentialsurjectivityofH . . . . . . . . . . . . . . . . . . . . . . . . 123 8.3 Conclusionsandopenproblems . . . . . . . . . . . . . . . . . . . . 127 III Kitaev’smodel 129 9 Kitaev’squantumdoublemodel 131 9.1 Basicdefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 9.3 Ribbonoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 10 Thetoriccode 139 10.1 Themodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 10.2 Stringoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 10.3 Localizedendomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 146 10.4 Fusion,statisticsandbraiding . . . . . . . . . . . . . . . . . . . . . . 153 10.5 EquivalencewithRep D(Z ) . . . . . . . . . . . . . . . . . . . . . . 160 f 2 iv Contents 11 Toriccode:analyticaspects 163 11.1 Conealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 11.2 Haagdualityforcones . . . . . . . . . . . . . . . . . . . . . . . . . . 165 11.3 Distalsplitproperty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 12 Thenon-abeliancase 175 12.1 Thegroundstate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 12.2 Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 12.3 Openproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 A Computingfusionrules 191 SamenvattinginhetNederlands 199 Waaromanyonen? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Kwantumveldentheorieind˘2¯1 . . . . . . . . . . . . . . . . . . . . . . 203 Kitaev’storiccode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Bibliography 207 Index 219 Curriculumvitae 222 v Preface This thesis is the result of the author’s PhD research, under supervision of Klaas Landsman and Michael Müger, the day-to-day supervisor. It was their expert- isethatallowedmetoquicklylearnthebasicsofoperatoralgebraandalgebraic quantumfieldtheory,bothnewtome. Thisthesiswouldnothavebeenpossible without the countless discussions, suggestions, corrections to manuscripts, ad- vice,etcetera,bymyadvisersMichaelMügerandKlaasLandsman.Ithankbothof themfortheirtremendoussupportduringvariousstagesoftheresearch,andfor givingmetheopportunitytodoresearchinmathematicalphysics. Theresearch wassupportedfinanciallybytheNetherlandsOrganisationforScientificResearch (NWO)underprojectno.613.000.608. I feel lucky to have been in the position that I could combine several of my interestsinmyresearch: quantumphysics, mathematics, andquantumcompu- tation. Ihadagreattimethelastfour(-plus)years, duringwhichIlearntagreat dealaboutphysicsandmathematics. Someofthosesubjects,namelythosethat serveasabackgroundtothemainresultsinthisthesis, areincludedinPartIof thisthesis. Ihopethatsomeofmyenthusiasmforthisfieldradiatestothereader ofthisthesis. Mathematics does not have to be a solitary exercise. Therefore, besides my advisers,Iwouldalsoliketothankmyothercolleagues.Inparticular,Iwouldlike tothanktheotherPhDstudents(androom-matesoverthepastyears): Ben,Bert, Dion,Joost,Jord,Maarten,Martijn,Matija,Michiel,Noud,Roel,Roberta,Ruben, Rutger, Sam, Sander (W.) and Sander (R.). Your presence livened up the atmo- sphereintheoffice,andyourcontributionswereessentialinmakingthePhDcol- loquiumandPhDseminarsasuccess.AndofcourseIshouldnotforgettomention theniceget-togethersoutsidework...IwouldalsoliketothankBerndSouvignier forhisinvolvementintheSprint-UpprojectandKarl-HenningRehrenforallhis advice,suggestionsandhelp. Althoughthisisperhapsnotalwaysapparent,lifeisnotallaboutresearch. I wouldliketothankmyfamily,andinparticularmyparents,foralwayssupporting meandstimulatingmetoexploretheworld.InadditionIthankthemanyfriends withwhomIhavespentmyfreetime. InparticularIwouldliketomentionMark & Jessica, Guido & Anneke and Twan for the great evenings, parties and dinners vii Preface spenttogether. IalsothankCees,JosandMarkforintroducingmeto(andbrew- ing!)manygreatbeers. Andlastly, andmostimportantly, IwanttothankKamilla. Fortoleratingme inbusyandstressfulperiods,foralwayssupportingmeinachievingmygoals,for justbeingthere,andforsomuchmore...Камилла,спасибобольшое! Utrecht January2012 PieterNaaijkens viii Introduction In the beginning of the 1980s the mathematician Manin [Man80, Introduction] andthephysicistFeynman[Fey82]suggested,amongothers,thatquantummech- anicalsystemscouldbeusedtodocomputations. Thissuggestionwasbasedon the observation that the (classical) simulation of a quantum mechanical system requiresanextraordinaryamountofcomputations. Perhaps,then,onecoulduse suchsystemsthemselvestodocomputations.1 Awell-knownexampleisShor’salgorithm[Sho94], whichprovidesapolyno- mialtimemethodforfactoringintegers. Thishasapotentiallybigimpactonthe security of most encryption schemes in use today, which rely on factoring be- inghard. Anarguablymoreimportantapplicationissuggestedbytheremarksof ManinandFeynmanmentionedabove: quantumcomputerscanbeusedtosim- ulatequantummechanicalsystems,forinstancethequantummechanicalbeha- viourofamolecule,whichisverydifficult(ifnotdownrightimpossibleforreason- ablycomplexmolecules)withtoday’stechnology. Understandingthisbehaviour isessentialinthedevelopmentofnewdrugs. Afull-fledgedquantumcomputer wouldthereforelikelytogreatlybenefitmedicineresearch. Despite this (potential) power of quantum computation, at the moment no suchquantumcomputerisavailable. Oneofthemainreasonsforthisisthatthe quantumsystemsnecessarytobuildaquantumcomputerareverysensitivetoin- teractionswiththeenvironment. Suchinteractionsleadtodecoherenceofquan- tum superpositions and hence to potential errors in the calculations. Even with theadventofquantumerrorcorrectionprotocols,therequiredaccuraciesareout ofreachofcurrenttechnology. Inrecentyears,however,anewapproachtoquantumcomputinghasemerged. Thisapproachisthoughttobeabletoaddressthisstabilityproblem. Independ- ently, Freedman[Fre98]andKitaev[Kit03]suggestedthattopological featuresof quantumsystemscanbeusedtoovercomethisdifficulty. Becauseoftheirtopo- 1Itshouldbenotedthatquantummechanicsdoesplayaroleinmoderncomputers:itplayedan importantpartinthedevelopmentoftransistors,thefundamentalbuildingblocksofacomputer. Thecomputationsthemselves, however, areclassical: theyworkonfinitebitstrings, withoutthe possibilityofsuperposition. Inaddition,insometypesofmodernflashstoragedevices,quantum mechanicaleffectsareemployedaswell. ix Introduction logical nature, these systems are inherently protected from influences from the environment. This can be seen as a kind of hardware error protection. Kitaev’s proposalisbasedonquantumspinsystems,whereasFreedmanusestopological quantumfieldtheory. Nevertheless,bothapproachesareintimatelyrelated: they bothrevolvearoundthepossibilityofnon-abeliananyons.2 Non-abeliananyonsareageneralisationofbothfermionsandbosons. Recall thatafermionisaparticleobeyingFermi-Diracstatistics(thisimpliesforexample that two identical fermions cannot be in the same state). Bosons satisfy Bose- Einsteinstatistics. The(quantummechanical)stateofasystemissymmetricun- der interchange of two identical bosons, and anti-symmetric under interchange ofidenticalfermions. Forlongitwasthoughtthatthese3 areinfacttheonlypos- sibilities,butintheseventiesitwasrealisedthatinlowdimensionsofspace-time moregeneralbehaviourispossible[LD71,LM77]. Anintroductiontoanyonsand reprintsofclassicpapersonthissubjectcanbefoundin[Wil90]. Inessence,anyonscanbeseenasexcitations(orquasiparticles4)thatbehave non-triviallyunderinterchange.Thisbehaviouriscalledthestatisticsofaparticle. Intuitivelyspeaking,itmeansthatinterchangingtwoidenticalanyonstwiceisnot the same as leaving them in place. This is quite unlike the usual Fermi or Bose statistics, where interchanging two particles twice has the same effect as doing nothingatall.Itturnsoutthatthispropertycanbeexploitedtoperformquantum computations. Itisperhapsinstructivetooutlinehowasystemwithanyonscouldbeusedto doquantumcomputations.Amorein-depthtreatmentcanbefoundinChapter4. Thebasicingredientsofquantumcomputationareasfollows: oneuses(asubset of)thestatesofaquantummechanicalsystemtorepresentthedifferent“inputs” toacomputation,analogouslytothebitsinaclassicalcomputer. Acomputation isperformedbyactingontheinputstatebymeansofunitarytransformations(ef- fected,e.g.,byturningonamagneticfield). Finally,ameasurementisperformed togetananswer.Itshouldbenotedthat,accordingtothelawsofquantummech- anics,theoutcomeofthismeasurementisprobabilistic.Henceonemighthaveto repeatthesamestepsanumberoftimes. So how does this work in a system with anyons? First of all, one once again initialisesthesysteminaknownstate. Thisisdonebycreatingpairsofananyon anditsantiparticlefromthevacuum.Wesupposethatwehavesomemechanism to move the anyons around each other, i.e. “braid” them. Moving them around willchangethestateofthesystem, justlikewhenweinterchangetwofermions. Mathematicallythisisdescribedbyactingwithaunitaryonthestateofthesystem. 2Alternativenamesincludenonabelionsandplektons. 3Andparastatistics,whicharealsorelatedtorepresentationsofthesymmetricgroup. 4Aquasiparticleisanemergentphenomenon,wherecomplexmicroscopiccaneffectivelybe describedbyfictitiousparticles.Awell-knownexampleisaCooperpairinasuperconductor.Inthis casetwoelectronspairupinsuchawaythatthepairessentiallybehaveslikeaboson. x
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