Boundary-bulk relation in topological orders

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Boundary-bulk relation in topological orders

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Kong, Liang, et al. “Boundary-Bulk Relation in Topological Orders.”

Nuclear Physics B 922 (September 2017): 62–76 © 2017 The

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http://dx.doi.org/10.1016/j.nuclphysb.2017.06.023

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ScienceDirect

Nuclear Physics B 922 (2017) 62–76

www.elsevier.com/locate/nuclphysb

Boundary-bulk

relation

in

topological

orders

Liang Kong

a,∗

_,

_{Xiao-Gang Wen}

b

_,

_{Hao Zheng}

c

a_Department_of_Mathematics_and_Statistics,_University_of_New_Hampshire,_Durham,_NH_03824,_USA b_Department_of_Physics,_{Massachusetts}_Institute_of_Technology,_Cambridge,_{MA 02139,}_USA

c_Department_of_Mathematics,_Peking_University,_Beijing,_100871,_China Received 25February2017;receivedinrevisedform 26June2017;accepted 28June2017

Availableonline 5July2017 Editor: HubertSaleur

Abstract

Inthispaper,westudytherelationbetweenananomaly-freen+1Dtopologicalorder,whichareoften calledn+1Dtopologicalorder inphysicsliterature,anditsnDgappedboundaryphases.Wearguethat the n+1D bulkanomaly-freetopological orderforagiven nDgappedboundaryphaseisunique.This uniquenessdefinesthenotionofthe“bulk”foragivengappedboundaryphase.Inthispaper,weshowthat then+1D“bulk”phaseisgivenbythe“center”ofthenDboundaryphase.Inotherwords,thegeometric notionofthe“bulk”correspondspreciselytothealgebraicnotionofthe“center”.Weachievethisbyfirst introducingthenotionofamorphismbetweentwo(potentiallyanomalous)topologicalordersofthesame dimension,thenprovingthatthenotionofthe“bulk”satisfiesthesameuniversalpropertyasthatofthe “center”ofanalgebrainmathematics,i.e.“bulk= center”. Theentireargumentdoesnotrequireusto knowtheprecisemathematicaldescriptionofa(potentiallyanomalous)topologicalorder.Thisresultleads toconcretephysicalpredictions.

1. Introduction

Topologicalordershaveattractedalotofattentioninrecentyearsamongcondensedmatter physicists becauseit isanewkindof orderbeyondLandau’ssymmetrybreaking theory (see

* _{Corresponding}_author.

E-mailaddresses:kong.fan.liang@gmail.com(L. Kong),xgwen@mit.edu(X.-G. Wen),hzheng@math.pku.edu.cn (H. Zheng).

http://dx.doi.org/10.1016/j.nuclphysb.2017.06.023

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reviews[42,38]).Therearealotofattemptsatdefiningthenotionofatopologicalorderatthe physicslevelofrigor[10,43].Amathematicallyrigorousdefinitionisstilloutofreach.

Ann+1D(spacetimedimension)topologicalorderiscalledanomaly-freeifitcanberealized byann+1Dlatticemodel;andiscalledanomalous ifotherwise[28].Inthiswork,weareonly interestedinn+1Danomaly-freetopologicalorders that allowgapped nDboundaries, which areingeneralnotunique. Ifthen+1Dbulkphaseisnottrivial,thenthe boundaryphasesare anomalousnDtopologicalorders.Inthiswork,whenwerefertobothanomaly-freeand anoma-loustopologicalorders,weusetheterm apotentiallyanomaloustopologicalorder, orsimply atopologicalorder.Themaingoalof thispaperistogiveaprecisedescriptionof therelation betweenananomaly-freen+1DtopologicalorderanditsgappednDboundaryphases.

Whenananomaly-free2+1Dtopologicalorderadmitsgappedboundaries,itiscompletely determinedbyitstopologicalexcitations(seeareview[24]).The1+1Dgappedboundaryphases arealsodeterminedbyitstopologicalexcitations.Thetopologicalexcitationsongapped1+1D boundarieswerefirststudiedinthe2+1Dtoriccodemodel[23]byBravyiandKitaevin[8]. ItwaslatergeneralizedtoLevin–Wenmodels[36]withgappedboundariesin[25],wherethe topologicalexcitationsontheboundaryof suchalatticemodel wereshowntoformaunitary fusioncategoryC,andthoseinthebulkformaunitarymodulartensorcategorywhichisgiven bytheDrinfeldcenterZ(C) ofC (seealso[33]).Buttheseworksdidnotaddresstheuniqueness ofthebulkphaseforagivenboundary.In[16],itwasshownmodel-independentlythatamong allpossiblebulk phasesassociatedtothesameboundary phaseC,theDrinfeldcenterZ(C) is

theuniversalone(aterminalobject).Onewaytocompletetheproofof theuniquenessof the bulk istoviewthe gapped boundary asaconsequence of anyon condensation[4] of agiven bulktheoryD tothetrivialphase.Thisidealeadstoaclassificationofgappedboundariesfor abelian2+1Dtopologicaltheories[20,41,35,5].Thisresult,togetherwithresultsin[16],implies theuniquenessofthebulkforabelian2+1Dtopologicaltheories.Theproofforgeneral2+1D topologicalordersappearedinthemathematicaltheoryofanyoncondensationdevelopedin[26], inwhichitwasshownthatsuchacondensationisdeterminedbyaLagrangianalgebraAinD, andC is monoidally equivalentto the category DA of A-modules inD. Moreover,we have

Z(DA) D.Thiscompletestheproofofthebulk-boundaryrelation in2+1D,whichsaysthat

the2+1DbulkphaseD foragiven1+1Dboundary C isunique,andisgivenbytheDrinfeld centerofC,i.e.bulk= centerforsimplicity.

Doesthisbulk-boundaryrelation(i.e.bulk= center)holdinhigherdimensions?Inthiswork, weproposethattheanswerisyes,andprovideaformalproofofthisrelationundersomenatural assumptions.Therearethreekeystepsinthisformalproof:

1. Forany givennDpotentially anomaloustopological orderCn,we assumethat there isa

uniqueanomaly-freen+1Dtopologicalorder,denoted byZn(Cn),suchthatCn canbe

re-alized as agapped boundary of Zn(Cn) (called unique-bulk hypothesis, see Sec. 2). We

willrefer toZn(Cn)asthebulk of Cn.Moreover,byrestrictingthen+1Dtopological

or-der Zn(Cn) to a1-codimensional subspace, we obtain an nD topological order, denoted

byPn(Zn(Cn)).WeassumethatPn(Zn(Cn))containsalltopologicalexcitationsinZn(Cn)

(a consequenceoftheself-detectionhypothesis,seeSec.2).

2. Althoughwedonot knowhowtodefineatopologicalorderrigorously,assuming the ex-istenceofsuchadefinitionandusingthenotionofthebulk,wecandefinethenotionofa morphismbetweentwotopologicalordersofthesamedimension(seeSec.4).Inparticular,

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weshowthatthereisacanonicalmorphismρ: Pn(Zn(Cn))Cn→ Cn,where denotes

thestackingoftwotopologicalordersofthesamedimension.

3. Weshowthat thepair(Pn(Zn(Cn)),ρ),satisfies theuniversalpropertyof thecenterofan

algebrainmathematics(seeTheorem 5.1).Thisimpliesthatbulk= center.

Thisresultisindependentof howwedescribetheboundary/bulk phasemathematically.Itisa non-trivialresultthatleadstoconcretephysicalpredictions(seeRemark 5.4).

Wedenoten+1Dtopologicalorders byAn+1,Bn+1,Cn+1,Dn+1,etc.Ifthe spacetime

di-mensionn+1 isclearfromthecontext,we abbreviateCn+1as C.Wedenotethetrivialn+1D

topologicalorderby1n+1.Inphysics,thetrivialtopologicalorder1n+1correspondstothe

equiv-alenceclassoftheproductstates[10].

Thispaperiswrittenforworkingcondensedmatterphysicists,especiallyforthoseworking inthefieldoftopologicalphasesofmatters.Inordertoconveythesimplyidea,wetrytokeep thecategoricallanguagetotheminimum.Inparticular,wecollectallmathematicallytechnical partsinRemarksandExamples.Themaintextshouldbereadabletothosewhodonothaveany extrabackgroundincategorytheorybeyondthosebasicsthathavealreadybeenwidelyusedin condensedmatterphysics(see[24]).

Acknowledgments: X-G.W is supported by NSF Grant No. DMR-1506475 and NSFC 11274192.HZissupportedbyNSFCunderGrantNo.11131008.

2. Basicsoftopologicalorders

Inthissection,werecallsomebasicfactsabouttopologicalorders,stateourkeyassumptions andsetournotations.

ApotentiallyanomalousnDtopologicalorderCncanalwaysberealizedasagappedboundary

ofananomaly-freen+1DtopologicalorderEn₊₁[28].Inphysics,itisgenerallybelievedthatthe

bulkanomaly-freetopologicalorderEn+1isuniquelydeterminedbyitsgappedboundaryphase

Cn.Onewaytoseethisistonotethatthereisnopreferredlengthscale.Inordertodefinethe

boundaryphaseCn,weneeddefinetheequivalenceclassofquantumstatesbyallowingproper

deformationofthestateinanarbitrarylargeneighborhoodoftheboundary(withoutclosingthe gap).Asaconsequence,En+1shouldbeunique(see[28]fordetails).Thisuniquenessisthekey

assumptionofthispaper.Wehighlightithere.

Unique-bulkhypothesis:foranygivennDpotentiallyanomaloustopologicalorderCn,there

isauniqueanomaly-freen+1Dtopologicalorder,denotedbyZn(Cn),suchthat Cn canbe

realizedasagappedboundaryofZn(Cn).

WedenoteEn+1byZn(Cn)andrefertoitasthebulk ofCn.ItisclearthatZn(1n)= 1n+1.Inthis

work,weoftenusethefollowingpicture:

toillustratethegeometricrelationbetweentheboundaryphaseCnandthebulk phaseZn(Cn).

Moreprecisely,then+1Dbulk phaseZn(Cn)(definedonanopenn-diskasthespacemanifold)

isdepictedbyanopenintervalandthenDboundaryphaseCn(onanopenn−1-disk)isdepicted

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Remark2.1.Wewanttoremarkthatthenotionofn+1Dphaseofmatterisalocalconceptwhich isdefinedonlyonanopenn-disk(asthespacemanifold).Similarly,theboundaryphaseisalso alocalconcept.Strictlyspeaking,theboundaryphaseshouldbedefinedonaneighborhoodof theboundaryinthebulk.Namely,thespacemanifoldfortheboundaryphaseis[0,)× Dn−1, whereDn−1isanopenn−1-disk.Forexample,whenn= 2,thebulk2+1Dphase,definedonan open2-disk,canbedescribedbyaunitarymodulartensorcategorytogetherwithacentralcharge

c∈ R.Its1+1Dboundary(togetherwithanopenneighborhoodinthebulk)cansupportafew differentboundaryphases,eachofwhichlivesonanopen1-diskasthespacemanifold.Different boundaryphasesareseparatedbyhighercodimensionaldefects[25,9].Aboundaryphasecanbe transformedtoanotherboundaryphaseviaapureboundaryphasetransitionwithoutalteringthe bulkphase[39].

Itiswell-knownthattopologicalexcitationsinananomaly-free2+1DtopologicalorderC3

areallparticle-like(of codimension2),andtheyformaunitary modulartensorcategory,still denotedbyC3(seeforexample[24]).Theonly1-codimensionaldefect(ordomainwall)inC3

isthetrivialone.Byrestrictingtothetrivial1-codimensionaldomainwall,alltheparticleshave tofusealongthewall.Nobraidingstructureremainsonthewall.Therefore,restrictingC3tothe

trivial1-codimensionaldomainwall,weobtaina1+1Dtopologicalorder,denotedbyP2(C3),

whichisgivenbythesameunitaryfusioncategoryasC3butforgettingitsbraidingstructure.We

wouldliketogeneralizethisfacttoalldimensionsunderthefollowingassumption:

Self-detectionhypothesis: alltopologicalexcitationsinananomaly-freetopologicalorder

Cn+1shouldbeabletodetectthemselvesviadoublebraidings[28,29].

Asaconsequence,Cn+1cannotcontainanynon-trivialtopologicalexcitations(ordefects)of

codimension 1 because two is the smallestcodimension for an excitation tobe braidedwith anotherexcitation.Therefore,byrestrictingthetopologicalorderCn+1tothetrivialexcitationof

codimension1(i.e.thetrivialdomainwall),weobtainannDtopologicalorderPn(Cn+1).Inthis

restrictingprocess,wedonotloseanynon-trivialtopologicalexcitations,noranyinformationof thefusionamongtheminnspatialdimensions,butonlyforgettheinformationoftheirfusionin then+1th direction,whichfurtherencodesthebraidingsamongexcitationsofcodimension 2. Moreover,bydoublefoldingtheanomaly-freetopologicalorderCn+1,wecreateadoublelayered

systemCn+1Cn+1withagappedboundaryphasePn(Cn+1)(i.e.Eq.(2.5)),where isthe

stackingoperationexplainedbelowandCn+1isthetimereverseofCn+1(becausetheorientation

orthenormal(orthetime)directionofoneofthetwolayersisflipped).

OnecanstackannDpotentiallyanomaloustopologicalorderAnonthetopoftheanotherone

Bnwithoutintroducinganycouplingbetweenthetwolayersasshownin(2.1).Thisoperationis

denotedby .Moreprecisely,thedashedbox,whenviewedfromfaraway,canbeviewedasa singlenDtopologicalorderAnBnwithasingle(buttwo-layer)bulkphaseZn(An)Zn(Bn).

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Thisstackingoperationiscompletelysymmetric.ItdoesnotmatterifweputAnonthetopor

the bottomof Bn becausethereisnocoupling betweenthem.Theresultingnewphaseisthe

same,i.e.

An Bn= Bn An. (2.2)

Clearly,we have1n An= An.Sincethe bulk isunique, we shouldalso havethe following

identity:

Zn(An Bn)= Zn(An)Zn(Bn).

More generally,wecanglueAn withBn byapotentiallyanomalousn+1DphaseCn+1to

obtainanewnDtopologicalorder,denotedbyAn _C_n₊₁Bn,asshownin(2.3).Moreprecisely,

the dashedbox,when viewedfromfaraway, canbe viewedasasinglenDtopologicalorder

An _C

n+1Bn,whichhasasinglebulkphasegivenbyA

n Zn+1(Cn+1)B

n.1Itisclearthat =

₁

n+1.Thisoperation Cn+1 isnotsymmetricingeneral.

2

= (2.3)

Agappeddomainwall(orawall)Mnbetweentwoanomaly-free3n+1Dtopologicalorders

Cn₊₁andDn₊₁isitselfapotentiallyanomalousnDtopologicalorder.Moreover,wehave

Zn(Mn)= Cn+1Dn+1, (2.4)

whereDn+1isthetimereverseofDn+1.Asaspecialcase,wehave

Zn(Pn(Cn+1))= Cn+1 Cn+1. (2.5)

AnnDtopologicalorderAncanbeviewedasawallbetweenZn(An)and1n+1.

AwallMnbetweenCn+1andDn+1canbefusedwithawallNnbetweenDn+1andEn+1to

obtainawallMn _D_n₊₁Nn betweenCn+1andEn+1.Thisfusionoperationofwallsisclearly

associative,i.e.forawallOnbetweenEn+1andFn+1,

(Mn _D

n+1Nn)En+1On= Mn Dn+1(Nn En+1On). (2.6) Forsimplicity,wedenotethetwosidesofEq.(2.6)byM_DN_EO intherestofthispaper. Wehavethefollowingidentities:

1 _In_(2.3)_,_note_that_A

n+1= Zn(An)andBn+1= Zn(Bn).Actually,bytheuniquenessofthebulk,asageneralization

ofEq.(2.5),wehaveZn(An) = Cn+1_Z

n+1(Cn+1)An+1.

2 _But_we_can_rotate_the_left_picture_in_(2.3)_around_a_horizontal_line_pass_through_the_middle_point_of_C

n+1by180

degrees.WeobtainthesameboundaryphaseAn_C

n+1Bn= BnCn+1An,whereCn+1isthemirrorreflectionof

Cn+1alongthesamelineandisnotequivalenttoCn+1ingeneral.ThisalsoexplainsEq.(2.2)because1n+1= 1n+1. 3 _A_more_general_notion_of_a_(potentially_anomalous)_gapped_domain_wall_between_two_potentially_anomalous topo-logicalorderscanbeintroduced(see[29,Sec. 6.1]).Butwedonotneeditinthiswork.

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Fig. 1.Mnisaninvertibledomainwallbetweentwoanomaly-free n+1DtopologicalordersCn+1andDn+1,andNn

isitsinverse.The“×”inthesepicturesrepresentsanon-trivialtopologicalexcitationintheCn+1-phase.Notethat thenon-trivialnessrequires“×”tobeatleast2-codimensional.Thesepicturesdepictaprocessoftheexcitation“×” tunnelingthroughtheMnwall.Inparticular,thesecond“”isobtainedbyannihilatingNnwiththeMnontheright

sideofNn.Similarly,thereisatunnelingprocessfromDn+1toCn+1,whichisinversetoit.Thisgivesawaytoidentify theCn+1-phasewiththeDn+1-phase.

Pn(Dn+1)_D

n+1Nn= Nn= Nn En+1Pn(En+1). (2.7) AgappeddomainwallMnbetweentwoanomaly-freeCn+1andDn+1iscalledinvertible if

thereisagappeddomainwallNnbetweenDn+1andCn+1suchthat

Mn _D_n₊₁Nn= Pn(Cn+1), Nn _C_n₊₁Mn= Pn(Dn+1). (2.8)

SuchaninvertibledomainwallMnprovidesawaytoidentifyCn+1withDn+1asdepictedin

Fig. 1.

Remark2.2.Gappeddomainwallsbetweentopologicalordersinarbitrarydimensionshavenot beenextensivelystudied(seesomediscussionin[29]).Theyare relativelywellunderstoodin 2+1D(see[26,16,15,34,21,1,22])andin1+1D(see[29]).

3. Theuniversalpropertyofthecenterofanalgebra

Inthissection,werecalltheuniversalpropertyofthecenterofanordinaryalgebra.

AnalgebraAoverafieldkisatriple(A,A⊗ A−→ A,m k−→ A),ιA wheremisthe multiplica-tionmapandιAistheunitofA.WealsodenoteιA(1)= 1A.ItscenterZ(A)isdefinedtobethe

subalgebra:

Z(A)= {z ∈ A | az = za, ∀a ∈ A}.

Thisdefinitionis,however,very limitedandnot usefultous atall.Abetterdefinition of the centerofanalgebraisgivenbyitsuniversalproperty (seeforexample[18],[37,Section6.1.4]) whichisapplicabletomanyothertypesofalgebras.

Moreprecisely,letm: Z(A)⊗ A→ A bethemultiplicationmap,i.e.m(z⊗ a)= za forz∈ Z(A)anda∈ A.NotethatmdefinesaunitalactiononA,i.e.m(1Z(A)⊗ a)= a.Equivalently,

wehavethefollowingcommutativediagram:

Z(A)⊗ A m k⊗ A = A idA ιZ(A)⊗idA A . (3.1)

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Moreover,misanalgebrahomomorphism,i.e.m(z⊗ a)m(z⊗ a)= m(zz⊗ aa)forz,z∈ Z(A)anda,a∈ A.

Thepair(Z(A),m)satisfiesthefollowinguniversalproperty:

• Givenanotherpair(X, X⊗ A−→ A) wheref X isanalgebra,f isaunitalactionandan algebrahomomorphism,thereisauniquealgebrahomomorphismf : X → Z(A) suchthat

m◦ (f ⊗ idA)= f ,ordiagrammatically,wehavethefollowingcommutativediagram:

Z(A)⊗ A m X⊗ A f f⊗idA A . (3.2)

Indeed,iff: X ⊗ A→ A isaunitalactionandanalgebrahomomorphism,then

f (x⊗ 1A)a= f (x ⊗ 1A)f (1X⊗ a) = f (x ⊗ a) = f (1X⊗ a)f (x ⊗ 1A)= af (x ⊗ 1A)

foralla∈ A,wherethefirstandthelastequalitiesholdbecausef isaunitalaction,thesecond andthethirdequalitiesholdbecausef isanalgebrahomomorphism.Therefore,theassignment

x→ f (x ⊗ 1A)definesanalgebrahomomorphismf : X → Z(A) rendering(3.2)commutative.

By restricting thediagram tothe subset X⊗ 1A⊂ X ⊗ A,we see that f is theunique map

making the diagram commutative. This shows that the pair (Z(A),m) satisfies the universal property.Notethatinthespecialcase(X,f )= (Z(A),m),wehavef = idZ(A).

On theotherhand,suppose (Y,g)is anothersuchapair satisfyingthisuniversalproperty. Thenginducesanalgebrahomomorphismg: Y → Z(A).Since(Y,g)alsosatisfiesthe univer-salproperty,themapm: Z(A)⊗A→ A alsoinducesanalgebrahomomorphismm: Z(A)→ Y .

Theng◦ m hastobetheidentitymapidZ(A) bytheuniquenessintheuniversalproperty.

Sim-ilarly,m◦ g = idY.Namely, g andmareinverse toeach other,henceidentifyY withZ(A).

Therefore,theuniversalpropertydetermines(Z(A),m)uniquelyuptocanonicalisomorphism, henceprovidesanalternativeapproachtodefinethenotionofcenter.

Remark 3.1.ThecenterZ(A)ofanordinaryalgebraAisalwaysasubalgebra.Thusthedata

minthepair(Z(A),m)isredundant.However,thisisnotthecaseforothertypesofalgebras naturallyarisinginmathematics.Oneshouldkeepinmindthatthecenterofanalgebraisapair rathermerelyanalgebra.

Inmathematicallanguage,thecollectionofalgebrasoverkformasymmetricmonoidal cat-egory Alg. Roughlyspeaking, an algebra A is referred to as an object of Alg. An algebra homomorphism f : A→ B isreferred toas a morphism betweenthe objects Aand B.Two morphismsf: A→ B andg: B → C canbecomposedtogiveanewmorphismg◦ f : A→ C.

OnehasanidentitymorphismidA: A→ A foreveryobjectA.Moreover,thereisabinary

oper-ation⊗ onAlgwhichcarriesapairofobjectsA,B totheirtensorproductA⊗ B.Thisbinary operationissymmetric,i.e.A⊗ B = B ⊗ A,andunital,i.e.thereisadistinguishedobjectksuch thatk⊗ A= A forallA.ThissymmetricmonoidalcategoryAlgsatisfiesanadditionalproperty: thereexistsauniquemorphismιA: k → A foreveryobjectA.Theseareallthedatathatwehave

used tostatethe universalpropertyofthe centerof analgebra.Oncegivensuch asymmetric monoidalcategorynomatterhowcrazytheobjectsare,oneisabletowritedowntheuniversal propertyanddefinethenotionofcenter.

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The collectionof nDtopological orders almost form asymmetric monoidalcategory. For example,there isasymmetric binaryoperation (recall Eq. (2.2)) that carriesapair of nD topologicalordersAn,BntoAn Bn,andthereisadistinguishednDtopologicalorder1nsuch

that1n An= AnforallAn.OnceweknowwhatisamorphismbetweentwonDtopological

orders,wecandefinethecenterofannDtopologicalordersbyapplyingtheuniversalproperty. Thisisthesubjectofthenexttwosections.

4. Amorphismbetweentwotopologicalorders

Althoughwedonothavearigorousdefinitionoftopologicalorder,wemaytreatitasablack boxanduseittogiveaphysicaldefinitionofamorphismbetweentwotopologicalorders.

Amorphism f : Cn→ DnbetweentwonDtopologicalordersCnandDnisagappeddomain

wallfn,viewedasannDtopologicalorder,betweentwon+1Danomaly-freetopologicalorders

Zn(Cn)andZn(Dn)suchthatfn _Z_n₍_C_n₎Cn= Dn.

Thegeometricidealizationofthephysicalconfigurationassociatedtothismorphismcanbe depictedasfollows:

(4.1) TheequalityfnZn(Cn)Cn= Dnmeansthatthisconfigurationisidenticaltothefollowingone:

Thisdefinitionisquiteunconventionalineitherphysicsormathematics.Wewouldliketo ex-plaintheintuitionbehindthisconcept.Inmathematics,amorphismbetweentwomathematical objects,suchasgroups,ringsandalgebras,areoftenrequiredtopreservetheinternalstructures oftheobjects.Inourcase,theinternalstructuresofatopologicalordershouldincludethe fusing-braidingstructuresofthetopologicalexcitations.Inordertopreservetheinternalstructures,we realizethenotionofamorphismf : Cn→ DnbetweentwonDtopologicalordersCnandDnby

aphysicalprocessof“screening”Cnfromoutside.Moreprecisely,weglueanewnDtopological

orderfntoCnbyann+1DglueZn(Cn)suchthatfn _Z

n(Cn)Cn= Dn.Notethatatopological excitationinCnisautomaticallyatopologicalexcitationinfnZn(Cn)Cn= Dn.Inthisway,the morphismf suppliesamapfromthesetofthetopologicalexcitationsinCntothatinDn.

Intu-itively,itisreasonablethatthisprocessof“screening”fromoutsideshouldpreservetheinternal structuresofCn.Whenn= 2,weexplaininExample 4.5thatthisnotionofamorphismexactly

coincideswiththatofaunitarymonoidalfunctorbetweentwounitaryfusioncategories,which describetwo2dtopologicalorders.

Remark4.1.Ananalogofsuch amorphisminmathematicsisanalgebrahomomorphismφ: A→ B betweentwomatrixalgebrasAandB.Actually,φalwaysfactorsasA−−−−−→ C ⊗ A1C⊗idA BwhereCisanothermatrixalgebra.Indeed,ifAisthealgebraofp× p-matricesandB isthe algebraofq× q-matrices,thentheexistenceofanalgebrahomomorphismφ: A→ B implies pdividesq andC isthealgebraof _pq ×_pq-matrices.Amiraclehappenshereisthatanalgebra homomorphismφisencodedbyanalgebraC.Thisphenomenonbecomeshighlynontrivialin categoricalsettings(seeExample 4.5).

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Remark4.2.Strictlyspeaking,wecannotsaythatannDtopologicalorderisequaltoanother unlesswespecifyhowtheyareidentified.Suchanidentificationcanberealizedbythechoiceof aninvertiblen−1Ddomainwallaswedidin[29,Def. 4.3].Inordertoconveythesimpleidea, however,wewouldliketousetheequality“=”forsimplicity.

Two morphismsf : Cn→ Dn andg: Dn→ En canbecomposedtogetanewmorphism

g◦ f : Cn→ En whichisdefinedbythe followinggapped domainwallbetweenZn(Cn)and

Zn(En):

(g◦ f )n:= gn _Z_n₍_D_n₎fn.

Thephysicalconfigurationassociatedtothiscomposedmorphismisdepictedasfollows:

NotethatthiscompositionlawisassociativebyEq.(2.6).Thatis,h◦ (g ◦ f )= (h◦ g)◦ f

foranymorphismsf : Cn→ Dn,g: Dn→ En andh: En→ Fn.Moreover,thetrivialdomain

wall Pn(Zn(Cn)) between Zn(Cn) andZn(Cn)defines the identitymorphism idCn: Cn→ Cn by Eq.(2.7).Thatis,f ◦ id_C_n= f foranymorphismf : Cn→ Dn andidCn◦ g = g forany morphismg: En→ Cn.Amorphismf : Cn→ Dnisanisomorphism(i.e.thereisamorphism

g: Dn→ Cnsuchthatg◦ f = id_Cn andf◦ g = idDn)ifandonlyiffnisaninvertibledomain wall(seeEq.(2.8)).

Remark4.3.Amorphismbetweentwomanybodysystems(notnecessarilytopological)ofthe samedimensioncanbeintroducedinasimilarway(see[29,Sec. A.3]).Thecompositionlaw is,however,notassociativeingeneral.Forthisreason,theresultofthisworkdoesnotapplyto non-topologicaltheories.

Example4.4.Wegiveafewmoreexamplesofmorphismsthatwillbeusedlater.LetCnbean

nDtopologicalorder.

1. Thereisauniquemorphismι_C_n: 1n→ Cn from1n toCn whichisdefinedbytheobvious

gappeddomainwallCnbetween1n+1andZn(Cn)asdepictedbelow.

(4.2) NotethatCn 1n+11n= Cn.

2. There is a morphism ρ : Pn(Zn(Cn)) Cn → Cn defined by the gapped domain wall

Pn(Zn(Cn)) Pn(Zn(Cn))betweenZn(Cn)Zn(Cn)Zn(Cn)andZn(Cn)asdepictedin

thefollowingpicture:

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Notethatwehave Pn(Zn(Cn))Pn(Zn(Cn)) _Z n(Cn)Zn(Cn)Zn(Cn) Pn(Zn(Cn))Cn = Cn.

Tosummarize,thecollectionofnDtopologicalordersformasymmetricmonoidalcategory

TOn:anobjectisannDtopologicalorderCn,amorphismf : Cn→ Dn betweentwoobjects

Cn,DnisagappeddomainwallfnbetweenZn(Cn)andZn(Dn)asdefinedabove,andthereisa

symmetricbinaryoperation onTOn.Moreover,thereexistsauniquemorphismιCn: 1n→ Cn foreveryobjectCnaswehaveseeninExample 4.4.Thesedatasufficetodefinetheuniversal

propertyofthecenterinTOn.Beforeproceedingon,wewouldliketotakealookatthespecial

casen= 2.

Example4.5.A2DtopologicalorderC2isdescribedmathematicallybyaunitaryfusion

cate-gory,anditsbulk Z2(C2)isgivenbytheDrinfeldcenterofC2(see[25,26]).Inparticular,the

trivial2Dtopologicalorder12isdescribedbythecategoryoffinite-dimensionalHilbertspaces

H.Moreover,P2(Z2(C2))istheunderlyingunitaryfusioncategoryofZ2(C2)byforgettingthe

braidingstructure.Thestackingoperation correspondstoDelignetensorproductand,more generally,theoperation _Z

2(C2) istherelativetensorproduct(see[1,Sec. 5.2]).Bydefinition,

theDelignetensorproductC2D2oftwounitaryfusioncategoriesC2,D2isalsoaunitary

fu-sioncategorywiththemonoidalstructure(cd)⊗ (cd):= (c ⊗ c)(d⊗ d)forc,c∈ C2

andd,d∈ D2.

Theresultfrom[30,Sec. 3.2]thenstatesthatamorphismbetweentwo2Dtopologicalorders isnothingbut an(isomorphismsclass of) monoidalfunctor.More explicitly,if f : C2→ D2

isaunitary monoidalfunctor betweentwo unitaryfusion categoriesC2andD2,thenthe

cor-responding unitary fusion category f2 is given by the category of C2-D2-bimodule functors FunC2|D2(D2,D2).Namely,thereisacanonicalmonoidalequivalence:

FunC2|D2(D2,D2)Z2(C2)C2 D2.

Conversely,onerecoversf fromf2asthemonoidalfunctorC2

1_f2_Z2(C2)id_C2

−−−−−−−−−→ f2Z2(C2)C2 D2.NotethattheidentitymorphismidC2 : C2→ C2ispreciselytheidentitymonoidalfunctor.

Moreover,thereexistsauniquemonoidalfunctorι_C₂: H→ C2,theonethatcarriesthetensor

unitC ofH tothetensorunit1_C₂ ofC2.

Insummary, thesymmetricmonoidalcategoryTO2 isdescribedas follows.Anobjectisa

unitaryfusioncategory.Amorphismisan(isomorphismclassof)monoidalfunctor.The sym-metricbinaryoperation isDelignetensorproduct,andthereisadistinguishedobjectH such

thatHC2 C2forallobjectsC2inTO2.

5. bulk= center

Inthissection,weprovethatthebulk satisfiestheuniversalpropertyofthecenter.

Let Cn be an nD topological order andlet Zn(Cn)be its bulk. First, we observe that the

morphismρ: Pn(Zn(Cn))Cn→ CnfromExample 4.4isaunitalaction.Thatis,thefollowing

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Pn(Zn(Cn))Cn ρ 1n Cn= Cn ι_Pn(Zn(Cn))id_Cn id_Cn Cn.

Indeed,thisfollowsfromthefollowingtworealizationsofthesamephysicalconfiguration:

= ,

wherethelefthandsidedepictsthemorphismρ◦ (ιPn(Zn(Cn)) idCn)andtherighthandside depictstheidentitymorphismid_Cn.

Nowwearereadytostateandprovethemainresultofthispaper.

Theorem5.1.Thepair(Pn(Zn(Cn)),ρ) satisfiestheuniversalpropertyofthecenter.More

pre-cisely,if(Xn,f ) isanotherpair,whereXnisannDtopologicalorderandf : Xn Cn→ Cnis

amorphismandaunitalaction,thenthereexistsauniquemorphismf : Xn→ Pn(Zn(Cn)) such

thatthefollowingdiagram

Pn(Zn(Cn)) Cn ρ Xn Cn fid_Cn f Cn (5.1) iscommutative.

Proof. Sincef : Xn Cn→ Cn isaunitalaction,f ◦ (ιXn idCn)= idCn.Thephysical con-figurationassociatedtothisequalityisdepictedasfollows:

= ,

(5.2) whichimpliesthat

fn _Z_n₍_X_n₎Xn= Pn(Zn(Cn)) (5.3)

as gappeddomainwallsbetweenZn(Cn)andZn(Cn).Nowwe regardfn asagapped domain

wallbetweenZn(Xn)andZn(Cn)Zn(Cn).Accordingtothedefinitionofamorphism,Eq.(5.3)

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Suchdefinedf makesthediagram(5.1)commutative.Thisfollowsfromthefollowingtwo realizationsofthesamephysicalconfiguration:

= ,

(5.4) wherethelefthandsidedepictsthemorphismρ◦ (f id_Cn)andtherighthandsidedepictsthe morphismf.

The uniqueness of f also follows from above equality. More precisely, if g : Xn →

Pn(Zn(Cn))isanothermorphismmakingthediagram(5.1)commutative,thentheequality(5.4),

withthefnonthelefthandsidereplacedbygn,holds.Thisnewidentityimmediatelyimplies

thatgn= fn,i.e.g= f . 2

Notethattheuniversalpropertydeterminesthepair(Pn(Zn(Cn)),ρ)uptocanonical

isomor-phism.Thatis,if(Yn,γ )isanotherpairsatisfyingtheuniversalpropertyofthecenter,thenthe

morphismγ: Yn→ Pn(Zn(Cn))inducedbyγ isinversetothemorphismρ: Pn(Zn(Cn))→ Yn

inducedbyρhenceidentifiesYnwithPn(Zn(Cn)).Inanotherword,thepair(Pn(Zn(Cn)),ρ)is

determinedbytheuniversalpropertywithoutambiguity.Undertheterminologyofmathematics, thepair(Pn(Zn(Cn)),ρ)isthecenter ofthenDtopologicalorderCn.

RecallthatthenDtopologicalorderPn(Zn(Cn))canbeobtainedbydoublefoldingthen+1D

topologicalorderZn(Cn).TorecoverZn(Cn)fromPn(Zn(Cn)),itsufficestoreversethedouble

foldingprocess,i.e.tosplitZn(Pn(Zn(Cn))).Thisispossibleif weusethe additionaldataρ,

becauseρn isaninvertibledomainwallbetweenZn(Pn(Zn(Cn)))andZn(Cn)Zn(Cn)hence

identifiesthem(recallthe discussionsassociated toFig. 1).Namely,ρ providesasplittingof

Zn(Pn(Zn(Cn)))thusrecovers Zn(Cn). Thisshowsthat the pair (Pn(Zn(Cn)),ρ)containsthe

sameinformationasZn(Cn).

Wereachtheconclusion“bulk= center”.Thisresultisindependentofhowwedescribethe boundaryphaseandthebulk phasemathematically.

Example5.2.LetusproceedonExample 4.5.Themorphismρ: P2(Z2(C2))C2→ C2isgiven

bythemonoidalfunctor

P2(Z2(C2))C2→(P2(Z2(C2))P2(Z2(C2)))

_Z

2(C2)Z2(C2)Z2(C2)(P2(Z2(C2))C2) C2,

orequivalently,bythecomposedfunctorP2(Z2(C2)) C2→ C2C2−→ C⊗ 2.Itisclearthatρ

isaunitalaction,i.e.ρ(1P2(Z2(C2)) a) a fora∈ C2.

Let f : X2 C2→ C2 be amonoidalfunctor which isalso a unitalaction. In particular, f isequippedwithnaturalisomorphismsf (x a)⊗ f (y b) f ((x ⊗ y) (a⊗ b)) and f (1_X₂ a) a.Thenthereisamonoidalfunctorf : X2→ C2 definedbyx→ f (x 1C2).

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f (x 1_C₂)⊗ a f (x 1_C₂)⊗ f (1_X₂ a) f (x a) f (1_X₂ a)⊗ f (x 1_C₂) a ⊗ f (x 1_C₂).

Inotherwords,f definesamonoidalfunctorfromX2toP2(Z2(C2)).Moreover,thefollowing

diagram P2(Z2(C2))C2 ρ X2C2 f fid_C2 C2

iscommutative.Theuniquenessoff isalsoeasytosee.Thisshowsthatthepair(P2(Z2(C2)),ρ)

satisfiestheuniversalpropertyofthecenter.

Recallthat P2(Z2(C2))isobtainedfromZ2(C2)byforgettingthebraidingstructure.Aswe

haveargued,onecanuseρtorecoverZ2(C2)fromP2(Z2(C2)).Indeed,bytheuniversal

prop-erty of the center,the composedmonoidalfunctorμ: P2(Z2(C2)) P2(Z2(C2)) C2 idρ

−−−→ P2(Z2(C2)) C2

ρ

−→ C2 determines a monoidal functor μ : P2(Z2(C2)) P2(Z2(C2)) → P2(Z2(C2)).The uniqueness of μ forcesit tobe the obviousone, the tensorproductfunctor ⊗,andthemonoidalnessofμsuppliesanaturalisomorphismμ(a b)⊗ μ(cd) μ((a ⊗ c)(b⊗ d)) fora,b,c,d∈ P2(Z2(C2)).ThentheforgottenbraidingstructureonP2(Z2(C2))

can be recovered by the following natural isomorphisms b ⊗ c μ(1 b)⊗ μ(c 1)

μ((1⊗ c) (b⊗ 1)) μ(cb) c ⊗ b.

Example 5.3.The onlyanomaly-free 1+1D topological orderis the trivial one 12, whichis

described bythecategoryof finite-dimensionalHilbertspacesH.Itwasexplainedin[29,

Ex-ample2.25] that the only1D topological orders can be described by apair (H,u), whereu

isanobjectofH.Moreover,thestackingoperationisgivenby(H,u)(H,v)= (H,u⊗ v).

NotethatZ1(H,u)= H,P1(Z1(H,u))= 11= (H,C) andρ: P1(Z1(H,u))(H,u)→ (H,u)

is the identitymorphismof (H,u).That is,bothof Z1(H,u) and(P1(Z1(H,u)),ρ)are

triv-ial. By definition, a morphism (H,u)→ (H,u) is a 1D topological order (H,v) such that

(H,v) (H,u)= (H,u);thisisequivalenttoafunctorH→ H thatcarriesutou (and car-riesC tov).Thepair(P1(Z1(H,u)),ρ)satisfiestheuniversal propertyof thecentertrivially:

if X1 (H,u)→ (H,u)isamorphismthenX1= (H,C) whichisidenticaltoP1(Z1(H,u)).

This“bulk= center”relationstillholdsevenifweincludeunstable1+1Dphases,whichoccur naturallyindimensionalreductionprocesses(see[29,Example3.7, 6.4]andalso[1,Sec. 5.2]).

Remark 5.4.Theorem 5.1 is a non-trivial result, which gives concrete physical predictions. For example, in the 3+1D Walker–Wang model [40] built on a unitary premodular tensor 1-categoryC,whichcanbeviewedasamonoidal2-category,Theorem 5.1impliesthatthe topo-logicalexcitationsinthe3+1Dbulkshallformthebraidedmonoidal2-category[19,12]given bythemonoidalcenterofthemonoidal2-categoryC constructedbyBaezandNeuchl[3].We believethatWalker–Wang’sconstructioncanbegeneralizedto3+1Dlatticemodelsbasedona genericunitaryfusion2-categoryC[29],andthebulkexcitationsinthisconjecturalmodelshould formaunitarybraidedfusion2-categorygivenbythemonoidalcenterofC.Moreover,“bulk= center”providesaseriousconstrainttotheprecisemathematicalformulationoftopological or-dersinalldimensions.Itled ustoproposein[29]acategoricaldescription ofthetopological excitationsinpotentiallyanomaloustopologicalordersinalldimensions.

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Remark5.5.Itwasexplainedin[29]that“bulk =center”discussedinthisworkisonlythefirst layerofthecompleteboundary-bulkrelation,whichcanbesummarizedasthefunctorialityof thecenter.For2+1Dtopologicalorderswithgappedboundaries,thisfunctorialityofDrinfeld centerofunitaryfusioncategorieswasprovedrigorouslyin[30].

Remark5.6.Ourmainresult alsosheds lightson asimilarboundary-bulkduality(i.e. open-closedduality)in2DrationalCFT’s[13,27,11,7].Itwillalsobeinterestingtostudyitsrelation tosomeresultsinfactorizationalgebras(see[37,14,2,17,6,1]).

Beforeweconcludethispaper,wewouldliketogiveanotherimportantremark,whichhas alreadyledtosomenewresults[31,44,32].

Remark5.7.Ifann+1Danomaly-freetopologicalorderhasatopologicallyprotectedgapless

nDboundaryphaseandiftheunique-bulkhypothesisstillholds,thenitiseasytoseethatour proofalsoworksforthegaplessboundarycase.Therefore,thegappedbulkphaseassociatedtoa gaplessboundaryphaseshouldalsobegivenbythecenteroftheboundaryphase.Inparticular,in 2+1DquantumHallsystems,thisresultsuggeststhatthereshouldbeamathematicaldescription ofthe1+1DgaplessedgemodessuchthatitscentergivesthemodulartensorcategoryC that describesthe2+1Dbulkphase.In[31],itwasshownthatthereisanenrichedmonoidalcategory

C_(enriched_by_boundary_CFT’s)_such_that_its_Drinfeld_center_is_exactly_C._In_[32]_,_we_showed

thatsuchenrichedmonoidalcategoriesgiveamathematicaldescriptionofthegaplessboundary phasesof 2+1D topological orders. Sucha mathematicaldescription also provides awayto extendtheReshetikhin–Turaev2+1DTQFTdowntopoints[44].

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