A duality principle for the entanglement entropy of free fermion systems Jose´ A. Carrasco1, F. Finkel1, A. Gonza´lez-Lo´pez1,*, and P. Tempesta1,2 1DepartamentodeF´ısicaTeo´ricaII,UniversidadComplutensedeMadrid,28040Madrid,Spain 2InstitutodeCienciasMatema´ticas(CSIC–UAM–UC3M–UCM),c/Nicola´sCabrera13–15,28049Madrid,Spain *Correspondingauthor. Email: [email protected] ABSTRACT 7 1 0 The analysis of the entanglement entropy of a subsystem of a one-dimensional quantum system is a powerful tool for 2 unravellingitscriticalnature. Forinstance,thescalingbehaviouroftheentanglemententropydeterminesthecentralcharge n oftheassociatedVirasoroalgebra. Forafreefermionsystem,theentanglemententropydependsessentiallyontwosets, a namelythesetAofsitesofthesubsystemconsideredandthesetKofexcitedmomentummodes. Inthisworkwededucea J generaldualityprincipleestablishingtheinvarianceoftheentanglemententropyunderexchangeofthesetsAandK. This 9 principlemakesitpossibletotacklecomplexproblemsbystudyingtheirdualcounterparts. Thedualityprincipleisalsoakey 1 ingredientintheformulationofanovelconjecturefortheasymptoticbehavioroftheentanglemententropyofafreefermion systeminthegeneralcaseinwhichbothsetsAandK consistofanarbitrarynumberofblocks. Wehaveverifiedthatthis ] conjecturereproducesthenumericalresultswithexcellentprecisionforalltheconfigurationsanalyzed. Wehavealsoapplied h p theconjecturetodeduceseveralasymptoticformulasforthemutualandr-partiteinformationgeneralizingtheknownonesfor - thesingleblockcase. t n a u Introduction q [ Oneofthedistinguishingfeaturesofthequantumrealmistheexistenceofentangledstatesincompositesystems,whichhave 1 noclassicalanalogueandplayafundamentalroleinquantuminformationtheoryandcondensedmatterphysics(see,e.g., v Refs.1,2). AwidelyusedquantitativemeasureofthedegreeofentanglementbetweentwosubsystemsA,Bofaquantumsystem 5 A∪Binapurestateρ =|ψ(cid:105)(cid:104)ψ|istheRényientanglemententropy3 S (A)=(1−α)−1logtr(ρα),whereρ isthereduced 5 α A A densitymatrixofthesubsystemAandα >0istheRényiparameter(thevonNeumannentropyisobtainedinthelimitα →1). 3 5 ItiseasytoshowthatSα(A)=Sα(B),andthattheentanglemententropyvanisheswhenthewholesystemisinanon-entangled 0 (product)state. Overthelastdecade,ithasbecomeclearthatthestudyoftheentanglementbetweentwoextendedsubsystems 1. ofamany-bodysysteminonedimensionisapowerfultoolforuncoveringitscriticalityproperties4–7. Thereasonforthisis 0 thatone-dimensionalcriticalquantumsystemsaregovernedbyaneffectiveconformalfieldtheory(CFT)in(1+1)dimensions, 7 whoseentanglemententropycanbeevaluatedinclosedforminthethermodynamiclimit8–10. Inthesimplestcase,whenthe 1 subsystemAconsistsofasingleintervaloflengthLandthewholesystemisinitsgroundstate,thescalingofS (A)forL→∞ α : v isdeterminedsolelybythecentralchargec. InordertoprobethefulloperatorcontentoftheCFT,oneneedstoanalyzemore i complicatedsituationsinwhichthesetAistheunionofafinitenumberofintervals. Infact,inthelastfewyearstherehas X beenaconsiderableinterestinthisproblem,bothforCFTsandone-dimensionallatticemodels(integrablespinchainsorfree r a fermionsystems),aswitnessedbythenumberofpaperspublishedonthissubject(see,e.g.,Refs.11–18). Inthisworkweshallextendthisanalysistothemoregeneralcaseinwhichthesystem’sstateisalsomadeupofseveral blocksofconsecutiveexcitedmomentummodes,whichhasreceivedcomparativelymuchlessattention19–21. Animportant motivationfordealingwiththistypeofstatesisthatitmakesitpossibletotreatpositionandmomentumspaceonamore equalfooting,thusrevealingcertainsymmetriesthathaveremainedunnoticedsofar. Thisapproachnaturallyleadstoanovel dualityprincipleforthebehavioroftheentanglemententropyundertheexchangeofthepositionandmomentumspaceblock configurations,whichinfactcanbeexploitedtosolveproblemsthatupuntilnowhaddefiedananalytictreatment22 with standardtechniquesliketheFisher–Hartwigconjecture23. Wehaveappliedthisdualityprincipletoproposeanewconjecture onthecomposabilityoftheentanglemententropyinthemulti-blockcase,whichyieldsaclosedasymptoticformulaforthe Rényi entanglement entropy of a free fermion system in the most general multi-block configuration, both in position and momentumspace. Thisformula,whichwehavenumericallyverifiedforawiderangeofconfigurationsbothfor0<α <1 andα (cid:62)1,reducestotheknownoneswhentheconfigurationinmomentumspaceconsistsofasingleblock. Italsoleadsto closedasymptoticformulasforthemutualandthetripartite12(orr-partite18)information,whichagainagreewiththegeneral CFTpredictions. Results and methods Preliminariesandnotation ThemodelconsideredisasystemofN free(spinless)hoppingfermionswithcreationoperatorsa†(wherethesubindex j= j 0,...,N−1denotesthesite)andHamiltonianH=∑Ni,j−=10gN(i−j)a†iaj preservingthetotalfermionnumber. Weshallfurther assume that the hopping amplitude g satisfies g (k)=g (−k)∗ =g (k+N), so that H is Hermitian and translationally N N N N invariant. Forthisreason,itisconvenienttointroducetheFourier-transformedcreationoperators 1 N−1 a†= √ ∑ e2πijl/Na†, 0(cid:54)l(cid:54)N−1. (1) (cid:98)j N l l=0 It is straightforward to check that the operators a , a† satisfy the canonical anticommutation relations (CAR), and that (cid:98)j (cid:98)j theydiagonalizeH. Infact,wehaveH =∑Nl=−01εN(l)a(cid:98)l†a(cid:98)l,withεN(l)=∑Nj=−01gN(j)e2πijl/N.Itcanbeshownthatthetotal momentumoperatorPisalsodiagonalinthisrepresentation,namelyP=∑lN=−01pla(cid:98)l†a(cid:98)l,with pl =2πl/N mod2π. Thusthe operatora†createsa(non-localized)fermionwithwell-definedenergyε (l)andmomentum p. Notethatε (l)isobviously (cid:98)l N l N realforallmodesl,andthatthemodeliscritical(gapless)ifε (l)vanishesforsomel. Weshallsupposeinwhatfollowsthat N thesystemisinapureenergyeigenstate |K(cid:105)≡a† ···a† |0(cid:105), K={k ,...,k }⊂{0,...,N−1}, (2) (cid:98)k1 (cid:98)kM 1 M where|0(cid:105)isthevacuum,consistingofMfermionswithmomenta2πk /N. Weshallbeinterestedinstudyingtheentanglement j properties of a subset of sites A≡{x ,...,x }⊂{0,...,N−1} with respect to the whole system when the latter is in the 1 L purestate|K(cid:105). Asiswellknown,thesepropertiesareencodedinthereduceddensitymatrixρ =tr ρ,whereρ ≡|K(cid:105)(cid:104)K| A B andB={0,...,N−1}−A. AsmentionedintheIntroduction,thedegreeofentanglementisusuallymeasuredusingtheRényi entanglemententropyS (A)≡(1−α)−1logtr(ρα)(withα >0). Oneofthemostefficientwaysofcomputingthisentropyis α A toexploittheconnectionbetweenthereduceddensitymatrixρ andthecorrelationmatrixC ,definedby A A (C ) =(cid:104)K|a† a |K(cid:105), 1(cid:54) j,k(cid:54)L. (3) A jk xj xk ThismatrixisobviouslyHermitian,witheigenvaluesν ,...,ν lyingintheinterval[0,1]. Moreover,sincethestate|K(cid:105)is 1 L determinedbytheconditionsa†|K(cid:105)=0fork∈K anda |K(cid:105)=0fork(cid:54)∈K,theexpectationvalue(cid:104)K|a†a |K(cid:105)vanishesfor (cid:98)k (cid:98)k (cid:98)j(cid:98)k k∈/K andequalsδ fork∈K. FromthisfactandEq.(1)weimmediatelyobtainthefollowingexplicitexpressionforthe jk matrixelementsofthecorrelationmatrixC : A 1 (CA)jk= ∑e−2πi(xj−xk)l/N, 1(cid:54) j,k(cid:54)L. (4) N l∈K AsfirstshowninRefs.4,24,thereduceddensitymatrixρ factorsasthetensorproductρ =(cid:78)L ρ(l),whereeachρ(l)isa A A l=1 A A 2×2matrixwitheigenvaluesν and1−ν. Inparticular,thespectrumofρ isthesetofnumbers l l A L ρA(ε1,...,εL)=∏(cid:2)νlεl(1−νl)1−εl(cid:3), εl ∈{0,1}. (5) l=1 SincetheRényientropyS isadditive,itfollowsthat α L L S (A)= ∑S (cid:0)ρ(l)(cid:1)=(1−α)−1∑log(cid:0)να+(1−ν)α(cid:1). (6) α α A l l l=1 l=1 NotethatthelattermethodforcomputingS (A)iscomputationallyveryadvantageous,sinceitisbasedonthediagonalization α oftheL×LmatrixC asopposedtodirectdiagonalizationofthe2L×2L matrixρ . A A Asexplainedabove,itisofgreatinteresttodeterminethe(leading)asymptoticbehaviouroftheentanglemententropyS (A) α asthesizeLofthesubsystemAtendstoinfinity. Tothisend,notefirstofallthatthematrixC isToeplitz(i.e.,(C ) depends A A jk onlyonthedifference j−k)providedthatthesubsystemAunderconsiderationisasingleblock,i.e.,asetofconsecutivesites. LetusfurtherassumethatEq.(4)hasawell-definedlimitasN→∞withLfixed,inthesensethatthereexistsapiecewise smoothdensityfunctionc(p)suchthat(C ) →(2π)−1(cid:82)2πc(p)e−i(j−k)pdpinthislimit. AsfirstshownbyJinandKorepin5, A jk 0 itisthenpossibletoapplyaparticularcaseoftheFisher–Hartwigconjecture23 provedbyBasor25 toderiveanasymptotic 2/12 formulaforthecharacteristicpolynomialofthecorrelationmatrixC ,andhencefortheentanglemententropyS (A)(seealso A α Refs.20,21,26). However,whenthesubsystemAisnotasingleblockitisclearfromEq.(4)thatC isnotaToeplitzmatrix,and A thereforethemethodjustoutlinedcannotbeusedtoderivetheasymptoticbehaviourofS (A)forlargeL. Itshouldalsobe α stressedthattheasymptoticresultinRef.5 isonlyvalidforN(cid:29)L(cid:29)1(i.e.,foraninfinitechain),sincetheN→∞limitwith LfixedistakenbeforelettingL→∞. Inparticular,theasymptoticbehaviourofS (A)whenN→∞withL/N→γ ∈(0,1) α x cannotbedirectlyinferredfromthelatterresult. Asweshallexplainshortly,thesedrawbackscanbeovercomethroughtheuse ofadualityprinciplethatweshallintroducebelow. Thedualcorrelationmatrix Westartbydefiningtheprojectionoftheoperatora†ontothesetL(H )oflinearoperatorsfromtheHilbertspaceH ofthe (cid:98)j A A subsystemAintoitselfintheobviousway,namely(cf.Eq.(1)) 1 a† = √ ∑e2πijl/Na†, (7) (cid:98)A,j N l l∈A andsimilarlyfora . Weshallalsodenotebya† ,a thecorrespondingprojectionsontoL(H ),sothata =a +a , (cid:98)A,j (cid:98)B,j (cid:98)B,j B (cid:98)j (cid:98)A,j (cid:98)B,j a(cid:98)†j =a(cid:98)A†,j+a(cid:98)B†,j.WethendefinethedualcorrelationmatrixC(cid:98)AastheM×Mmatrixwithelements (C(cid:98)A)lm=(cid:10)0(cid:12)(cid:12)a(cid:98)A,kla(cid:98)A†,km(cid:12)(cid:12)0(cid:11), 1(cid:54)l,m(cid:54)M. (8) ThedualcorrelationmatrixC(cid:98)BofthecomplementarysetBisdefinedsimilarly. TheanalogueofthematrixC(cid:98)Aforcontinuous systems,usuallycalledtheoverlapmatrix,wasoriginallyintroducedbyKlich27andhasbeenextensivelyusedintheliterature (see,e.g.,Ref.28). Fromthedefinition(7)oftheprojectedoperatorsa† weimmediatelyobtaintheexplicitformula (cid:98)A,j 1 (C(cid:98)A)lm= ∑e−2πi(kl−km)j/N, 1(cid:54)l,m(cid:54)M. (9) N j∈A ComparisonwithEq.(4)showsthatC(cid:98)AisobtainedfromCAbyexchangingtherolesplayedbythesitesxj∈Aandtheexcited modes k ∈K, which justifies the term “dual correlation matrix”. We shall show in what follows that this duality can be l successfullyexploitedtoobtaintheasymptoticbehaviourofS (A)insituationsinwhichtheusualapproachbasedonthe α correlationmatrixC isnotfeasible. A ThematrixC(cid:98)A isclearlyHermitianandpositivesemidefinite,sinceforallz1,...,zM ∈Cwehave∑Ml,m=1(C(cid:98)A)lmz∗lzm= (cid:13)(cid:13)(cid:0)∑Ml=1zma(cid:98)A†,km(cid:1)|0(cid:105)(cid:13)(cid:13)2.Thustheeigenvaluesν(cid:98)1,...,ν(cid:98)MofC(cid:98)Aarenonnegative.Usingtheidentities(cid:104)0|OAOB(cid:48)|0(cid:105)=(cid:104)0|OB(cid:48)OA|0(cid:105)= 0,whereOAandOB(cid:48) arelinearoperatorsrespectivelysupportedonAandB,itisstraightforwardtocheckthatC(cid:98)B=1lM−C(cid:98)A. SinceC(cid:98)B is also positive semidefinite, from the previous relation it follows that ν(cid:98)i ∈[0,1] for all i=1,...,M. Moreover, the Hermitian character of C(cid:98)A implies that there exists a unitary M×M matrix U ≡(ulm)1(cid:54)l,m(cid:54)M such that UC(cid:98)AU† = diag(ν(cid:98)1,...,ν(cid:98)M), and henceUC(cid:98)BU† =1l−UC(cid:98)AU† =diag(1−ν(cid:98)1,...,1−ν(cid:98)M). We then define the corresponding rotated operatorsc(cid:98)l =∑Mm=1ulma(cid:98)km (1(cid:54)l(cid:54)M),whichtogetherwiththeiradjointssatisfytheCARbytheunitarityofU. Weshall alsoneedtheprojectionsofthelatteroperatorsontothespacesL(H )andL(H ),namely A B M M c = ∑ u a , c = ∑ u a =c −c , (10) (cid:98)A,l lm(cid:98)A,km (cid:98)B,l lm(cid:98)B,km (cid:98)l (cid:98)A,l m=1 m=1 andsimilarlyfortheiradjoints. Fromtheabovedefinitionsitfollowsthatthevacuumcorrelatorsoftheoperators{c ,c† } (cid:98)A,l (cid:98)A,l and{c ,c† }aregivenby (cid:98)B,l (cid:98)B,l (cid:10)0(cid:12)(cid:12)c(cid:98)A,lc(cid:98)A†,m(cid:12)(cid:12)0(cid:11)=ν(cid:98)lδlm, (cid:10)0(cid:12)(cid:12)c(cid:98)B,lc(cid:98)B†,m(cid:12)(cid:12)0(cid:11)=(1−ν(cid:98)l)δlm, (11) andhence(cid:13)(cid:13)c(cid:98)A†,l|0(cid:105)(cid:13)(cid:13)2=ν(cid:98)l, (cid:13)(cid:13)c(cid:98)B†,l|0(cid:105)(cid:13)(cid:13)2=1−ν(cid:98)l. FollowingRef.27, wenotethatthestate|φ(cid:105)=c(cid:98)1†···c(cid:98)M†|0(cid:105)actuallydiffers from|K(cid:105)byanirrelevantphase,sincebydefinitionoftheoperatorsc wehave (cid:98)l M (cid:18) (cid:19) |φ(cid:105)= ∑ u∗ ···u∗ a† ···a† |0(cid:105)= ∑ (−1)σu∗ ··· u∗ a† ···a† |0(cid:105)=detU∗|K(cid:105), m1,...,mM=1 1m1 M,mM(cid:98)km1 (cid:98)kmM σ∈SM 1σ1 M,σM (cid:98)k1 (cid:98)kM 3/12 where (−1)σ denotes the sign of the permutation σ. The latter relation implies that |K(cid:105)(cid:104)K|=|φ(cid:105)(cid:104)φ|, a fact that can be exploited in order to derive an expression for the entanglement entropy Sα(A). To this end, for ν(cid:98)l (cid:54)=0,1 we define the operatorsd(cid:98)A†,l =c(cid:98)A†,l/(cid:112)ν(cid:98)l, d(cid:98)B†,l =c(cid:98)B†,l/(cid:112)1−ν(cid:98)l, sothatbyEq.(11)thestates|1(cid:105)A,l ≡d(cid:98)A†,l|0(cid:105), |1(cid:105)B,l ≡d(cid:98)B†,l|0(cid:105)areproperly normalized. Ontheotherhand,whenν(cid:98)l=0thestatec(cid:98)l†|0(cid:105)=c(cid:98)B†,l|0(cid:105)issupportedonBbyEq.(11),andisnormalized,sincethe operatorsc(cid:98)l,c(cid:98)l†obeytheCAR.Henceinthiscasewesimplysetd(cid:98)B†,l =c(cid:98)B†,l =c(cid:98)l†,|1(cid:105)B,l =d(cid:98)B†,l|0(cid:105). Similarly,whenν(cid:98)l =1we defined(cid:98)A†,l =c(cid:98)A†,l =c(cid:98)l†,|1(cid:105)A,l =d(cid:98)A†,l|0(cid:105),andthepreviousdefinitionswethushavec(cid:98)l†=(cid:112)ν(cid:98)l d(cid:98)A†,l+(cid:112)1−ν(cid:98)l d(cid:98)B†,l (1(cid:54)l(cid:54)M), and therefore |φ(cid:105)=(cid:78)Ml=1(cid:16)(cid:112)ν(cid:98)l |1(cid:105)A,l|0(cid:105)B,l+(cid:112)1−ν(cid:98)l |0(cid:105)A,l|1(cid:105)B,l(cid:17), where |0(cid:105)A,l, |0(cid:105)B,l denote the vacuum state in the l-th mode(withrespecttothec† operators)supportedrespectivelyonAorB. Usingtheidentity|K(cid:105)(cid:104)K|=|φ(cid:105)(cid:104)φ|andtracingover (cid:98)m thedegreesoffreedomofthesubsystemBweeasilyarriveatthefundamentalformula (cid:79)M (cid:16) (cid:17) ρA= ν(cid:98)l|1(cid:105)A,l(cid:104)1|A,l+(1−ν(cid:98)l)|0(cid:105)A,l(cid:104)0|A,l . (12) l=1 Inparticular,thespectrumofthematrixρ isthesetofnumbers A M ρA(ε1,...,εM)=∏(cid:2)ν(cid:98)lεl(1−ν(cid:98)l)1−εl(cid:3), εl ∈{0,1}, (13) l=1 up to zero eigenvalues. From the additivity of the Rényi entropy and Eq. (12) or (13) it follows that the entanglement entropyS (A)isgivenby α M Sα(A)=(1−α)−1∑log(cid:0)ν(cid:98)lα+(1−ν(cid:98)l)α(cid:1), (14) l=1 whichcanbeinterpretedasthedualofEq.(6). Thedualityprinciple Aswehaveseenintheprevioussubsection,theRényientanglemententropyS (A)canbecomputedintwoequivalentways, α usingthe“coordinate”correlationmatrixCAandits“dual”C(cid:98)A(cf.Eqs.(6)-(14)). Thisfactstronglysuggeststheexistenceofa deeperdualityprinciplethatappliestothereduceddensitymatrixρ itself,asevidencedbyEqs.(5)-(13). Toformulatethis A principle,weshallintroducethemoreprecisenotationρ (K)todenotethereduceddensitymatrixofthesubsystemAwhen A thewholesystemisinthepureenergyeigenstate|K(cid:105)givenbyEq.(2). LetspecT standforthespectrumofthematrixT,i.e., thesetofitseigenvalues,eachcountedwithitsrespectivemultiplicity. Likewise,weshalldenotebyspec ρ thespectrumofa 0 (cid:0) (cid:1) densitymatrixρ excludingitszeroeigenvalues,i.e.,spec ρ =spec ρ| .Weshallthensaythattwodensitymatricesρ 0 (kerρ)⊥ i (i=1,2)aresimilaruptozeroeigenvaluesifspec ρ =spec ρ ,i.e.,ρ andρ havethesamenonzeroeigenvalueswiththe 0 1 0 2 1 2 samemultiplicities. Wearenowreadytostatethefollowingfundamentalresult: Theorem1. Thereduceddensitymatricesρ (K)andρ (A)aresimilaruptozeroeigenvalues. A K Proof. Indeed,byEqs.(5)-(13)thespectrumofρ (K)excludingthezeroeigenvaluescanbewritteninthetwoequivalent A ways (cid:26) L (cid:27) (cid:26) M (cid:27) spec0(cid:0)ρA(K)(cid:1)= ∏νlεl(1−νl)1−εl |εl∈{0,1}, νl∈/{0,1} = ∏ν(cid:98)mεm(1−ν(cid:98)m)1−εm|εm∈{0,1}, ν(cid:98)m∈/{0,1} . (15) l=1 m=1 Let us denote by CA(K) and C(cid:98)A(K) the correlation matrix (4) and its dual version (9). We then have C(cid:98)A(K)=CK(A), CA(K)=C(cid:98)K(A), and consequently the sets {νl}Ll=1 and {ν(cid:98)m}Mm=1 are interch(cid:0)anged (cid:1)by the du(cid:0)ality tra(cid:1)nsformation A↔K. ApplyingEq.(15)tothereduceddensitymatrixρ (A)weconcludethatspec ρ (K) =spec ρ (A) ,asclaimed. K 0 A 0 K (cid:0) (cid:1) IfSisanyentropyfunctional,fromnowonweshallusethemoreprecisenotationS(A;K)=S ρ (K) . Obviously,from A theShannon–Khinchinaxiomsitfollowsthattwodensitymatriceswhicharesimilaruptozeroeigenvaluesnecessarilyhave thesameentropy. Fromthisfactandtheprevioustheoremwecanimmediatelydeducetheimportantdualityprinciple S(A;K)=S(K;A), (16) validforanyentropyfunctionalS. 4/12 Asafirstapplicationofthisgeneralprinciple,weshallrigorouslyderiveanasymptoticexpressionfortheRényientanglement entropyofasubsystemAconsistingofr>1disjointblocksofconsecutivespinswhenthesetK ofexcitedmomentaisasingle setofMconsecutiveintegers,validinthelimitN(cid:29)M(cid:29)1. Moreprecisely,letA=(cid:83)r [U,V),K=[P,Q),where[U,V) i=1 i i i i denotesthesetofallintegerslsuchthatU (cid:54)l<V (sothatthecardinalof[U,V)isV −U),andsimilarlyfor[P,Q). Wefirst i i i i i i letN→∞withMfixedandassumethatthefollowinglimitsexist: 2πU 2πV i i lim ≡u , lim ≡v , i i N→∞ N N→∞ N withu,v ∈[0,2π],u −v >0,v −u <2π. WeshallbeinterestedintheasymptoticbehavioroftheRényientropyS i i i+1 i r 1 α asM→∞. ThustheproblemathandispreciselythedualoftheonesolvedinRef.21 withthehelpoftheFisher–Hartwig conjecture. Themainresultofthelatterreferencecanberecastinthepresentcontextastheasymptoticformula (cid:18) s (cid:19) S (cid:0)[U,V);(cid:83)s [P,Q )(cid:1)∼b slogL+∑log(cid:0)2sin(cid:0)qj−pj(cid:1)(cid:1)+logf(p,q) +sc , (17) α j=1 j j α 2 α j=1 where p ≡ lim(2πP/N),q ≡ lim(2πQ /N), j j j j N→∞ N→∞ 1(cid:18) 1(cid:19) 1 (cid:90) ∞(cid:18) 1−α2 (cid:19)dt sin(cid:0)qj−pi(cid:1)sin(cid:0)pj−qi(cid:1) b = 1+ , c = αcsch2t−cschtcsch(t/α)− e−2t , f(p,q)= ∏ 2 2 α 6 α α 1−α 0 6α t 1(cid:54)i<j(cid:54)ssin(cid:0)pj−2pi(cid:1)sin(cid:0)qj−2qi(cid:1) (18) andthe∼notationmeansthatthedifferencebetweentheLHSandtheRHStendsto0asL→∞. Fromthedualityrelation(16) andEqs.(17)-(18)itthenfollowsthatwhenM→∞wehave (cid:20) r (cid:21) S (cid:0)(cid:83)r [U,V);[P,Q)(cid:1)=S (cid:0)[P,Q);(cid:83)r [U,V)(cid:1)∼b rlogM+∑log(cid:0)2sin(cid:0)vi−ui(cid:1)(cid:1)+logf(u,v) +rc . (19) α i=1 i i α i=1 i i α 2 α i=1 Takingintoaccountthat f(u,v)=1whenr=1,fromthepreviousformulawededucethat r S (cid:0)(cid:83)r [U,V);[P,Q)(cid:1)∼∑S (cid:0)[U,V);[P,Q)(cid:1)−I (u,v), with I (u,v)≡−b logf(u,v), (20) α i=1 i i α i i α α α i=1 wherethelasttermcanbenaturallyinterpretedasanasymptoticapproximationtothemutualinformationsharedbytheblocks [U ,V ),...,[U ,V ). 1 1 r r ItisimportanttokeepinmindthelimitingprocessleadingtoEq.(19)inordertocorrectlyassessitslimitofvalidity. For instance,usingtheconnectionbetweenone-dimensionalcriticalsystemsand1+1dimensionalCFTsitwasconjecturedin Ref.22 thattheasymptoticbehaviorofS isgiven(inournotation)by α S (cid:0)(cid:83)r [U,V);[P,Q)(cid:1)∼b (cid:20)rlog(cid:18)Nsin(cid:18)π M(cid:19)(cid:19)+∑r log(v −u)+logf(∞)(u,v)(cid:21)+rc , (21) α i=1 i i α π N i i α i=1 where f(∞)(u,v)istheproductofcrossratios (v −u)(u −v) f(∞)(u,v)= ∏ j i j i . (22) (u −u)(v −v) 1(cid:54)i<j(cid:54)r j i j i TheapparentdiscrepancybetweenthelatterformulasandEqs.(18)-(19)iseasilyexplainedtakingintoaccountthatthelimiting processinthelatterreferenceisthedualofthepresentone,namelyN→∞withfixedU,V and2πP/N→p,2πQ/N→q. In i i otherwords,Eqs.(18)-(19)applywhenN(cid:29)M(cid:29)1andarbitraryL<N,whileEqs.(21)-(22)arevalidforN(cid:29)L(cid:29)1and arbitraryM<N. Itisalsoobviousthatbothapproachescoincideinthe(ratheruninteresting)caseinwhichbothM/N and L/N tendtozero. Ontheotherhand,itshouldbeapparentthatneitherEqs.(18)-(19)nor(21)-(22)arevalidinthegeneral situationinwhichbothL/N andM/N tendtoanonzerolimitasN→∞. Infact,itisclearapriorithatnoneoftheseformulas canholdinthelatterrange,sincetheyarenotconsistentwiththeinvarianceundercomplementsidentityS(A;K)=S(Ac;K) anditsdualconsequenceS(A;K)=S(A;Kc),whereAcandKcrespectivelydenotethecomplementsofAandK withrespectto theset{0,...,N−1}. 5/12 Our next objective is to find an extension of Eqs. (19) and (21) valid in the general case in which both γ and γ ≡ x p lim M/N tend to a nonzero limit as N →∞. To this end, consider first the simplest case in which r =s=1. By N→∞ translationinvarianceandcriticality,asN→∞wemusthaveS ([U,V);[P,Q))∼b logN+σ (γ ,γ ),whereγ =(V−U)/N, α α α x p x γ =(Q−P)/N and σ satisfies: i) σ (γ ,γ )=σ (γ ,γ ) (on account of the duality principle (16)), ii) σ (γ ,γ )= p α α x p α p x α x p (cid:0) (cid:1) σ (1−γ ,γ ) (by the invariance of the entropy under complements), iii) σ (γ ,γ )=b log 2γ sin(πγ ) +c +o(1), α x p α x p α x p α with lim o(1)=0 (by Eq. (21) with r=1). (In fact, combining conditions i) with ii) and iii) it immediately follows γx→0 (cid:0) (cid:1) thatσ (γ ,γ )=σ (γ ,1−γ )andσ (γ ,γ )=b log 2γ sin(πγ ) +c +o(1),whereo(1)→0asγ →0.)Obviously,the α x p α x p α x p α p x α p simplestfunctionsatisfyingthepreviousrequirementsisσ (γ ,γ )=b log(cid:0)2sin(πγ )sin(πγ )(cid:1)+c ,obtainedfromEq.(21) α x p α π x p α withr=1bythereplacementπγ (cid:55)→sin(πγ ). Numericalcalculationsshowthatforallα >0thecorrectasymptoticformula x x forS ([U;V);[P,Q))isindeedthesimplestone,namely α (cid:18) (cid:19) 2N S ([U,V);[P,Q))∼b log sin(πγ )sin(πγ ) +c (23) α α x p α π (see,e.g.,Fig.1(a)forthemost“unfavourable”caseγ =γ =1/2). Thisconclusionisalsoinagreementwiththeanalogous x p resultinRef.29 fortheXX model. Infact,wefoundtheleadingcorrectiontotheapproximation(23)tobemonotonicinN andO(N−2)forα=1,andO(cid:0)cos(2πγ γ N(cid:1)N−2/α)forα>1(cf.Fig.1). Thisbehaviourqualitativelyagreeswiththeresults x p ofRef.30 fortheerroroftheJin–KorepinasymptoticformulafortheRényientanglemententropyofthegroundstateofthe infiniteXX chain(Eq.(23)withsin(πγ )replacedbyπγ ). Ontheotherhand,inthecase0<α <1(whichwasnotaddressed x x inthelatterreference),ournumericalcalculationssuggestthatthecorrectiontoEq.(23)ismonotonicandO(N−2). 0.05 1.5×10–6 0.04 0.002 10–6 0 0.03 ε –0.002 ε 0.02 0.5×10–6 100 110 120 130 140 150 0.01 0.00 0 600 800 1000 1200 1400 1600 1800 2000 600 700 800 900 (a) N (b) N Figure1. Differenceε betweentheexactvalueoftheRényientropy,computedviaEq.(6)bynumericaldiagonalizationof thecorrelationmatrix(4),anditsasymptoticapproximation(23)for(a)γ =γ =1/2and(b)γ =1/8,γ =1/4. Inpanel(a) x p x p wehaveshownthecases(bottomtotop)α =3/5,2/3,3/4,1(vonNeumannentropy)andα =2(inset),whilepanel(b) depictsthecasesα =2,5/2,3(bottomtotop,withthehorizontalaxisdisplacedrespectivelyby0.013and0.035inthelast twocasestoavoidoverlap). ThesolidredlinesrepresentthecurvesprovidingthebestfitsofthedatatothelawsaN−2(main panel(a))andaN−2/αcos(2πγ γ N)(insetofpanel(a)andpanel(b)). x p Atthispoint,itisverynaturaltoassumethatEq.(20)anditsdualarevalidforallvaluesoftheparametersγ ,γ ∈(0,1), x p andnotjustforγ (cid:28)1orγ (cid:28)1,respectively. ThelatterassumptionandEq.(23)thusleadtotheasymptoticformulas p x S (cid:0)(cid:83)r [U,V);[P,Q)(cid:1)∼b (cid:20)rlog(cid:18)2N sin(πγ )(cid:19)+∑r logsin(cid:0)vi−ui(cid:1)(cid:21)−I (u,v)+rc , (24) α i=1 i i α π p 2 α α i=1 S (cid:0)[U,V);(cid:83)s [P,Q)(cid:1)∼b (cid:20)slog(cid:18)2N sin(πγ )(cid:19)+∑s logsin(cid:0)qi−pi(cid:1)(cid:21)−I (p,q)+sc . (25) α i=1 i i α π x 2 α α i=1 Infact, thevalidityofthelatterequationscanbeestablishedbynotingthatonecangofromEq.(17), whichholdsforan infinite chain, to its analogue for a finite chain by the usual procedure18,29 of replacing the “arc distance” L by the chord length(N/π)sin(πL/N)=(N/π)sin(πγ ).InthiswayEq.(17)immediatelyyieldsEq.(25),whichimpliesitscounterpart(24) x bythedualityprinciple(16). Again,ournumericalcalculationsforseveralblockconfigurationsandawiderangeofvaluesoftheRényiparameterα fullycorroboratethevalidityofEqs.(24)-(25)(see, e.g., Fig.2). Moreprecisely, ournumericalanalysissuggeststhatfor 6/12 sufficientlylargeN theerrorterminthelatterequationsbehavesas f(N)O(N−min(2,2/α)),where f(N)isaperiodicfunction ofN. Inparticular,theerrortermmaynotbemonotonicinN evenforα (cid:54)1,incontrastwithwhathappensinther=s=1 case. TheaboveresultsareinagreementwiththosereportedinRef.16 forthe(infinite)XY chainanditscorrespondingfree fermionmodelwithα >1,r=2ands=1. 9 0.04 8 0.02 7 α S ε 0 6 5 –0.02 4 –0.04 2500 3000 3500 2500 3000 3500 (a) N (b) N Figure2. (a)ExactRényientropyS (bluedots)vs.itsasymptoticapproximation(24)(continuousredline)forasubsystem α consistingofthreeequispacedblocksofequallengthN/12whenthewholesystem’sstate(2)ismadeupofasequenceof consecutiveexcitedmodesoflengthN/12(r=3,s=1,γ =1/4,γ =1/12). ThevaluesoftheRényiparameterα x p consideredare(fromtoptobottom)1/2,3/5,3/4,1,3/2,2and3. (b)Differenceε betweentheexactentropyS andits 3 approximation(24)inthepreviousconfigurationasafunctionofthenumberoffermionsN. Thecontinuousredlineisthe graphofthefunction f(N)N−2/3,with f(N)=−5.54238cos(ν N)−0.742586cos(3ν N)−0.39794cos(5ν N)and 0 0 0 ν =2πγ γ /r=π/72. 0 x p Multi-blockentanglemententropy: conjectureforthegeneralcase Weshalladdressinthissectionthegeneralproblem,inwhichbothsetsAandK consistofseveralblocksofconsecutivesites ormodes,respectively. Tothebestofourknowledge,anasymptoticformulafortheentanglemententropyinthiscasehas notpreviouslyappearedintheliterature. Asexplainedabove,themaindifficultyisnowthatneitherthecorrelationmatrixC A noritsdualC(cid:98)A areToeplitz,sothatthestandardprocedurebasedontheuseoftheFisher–Hartwigconjecturetoobtainan asymptoticformulaforthecharacteristicpolynomialofthecorrelationmatrixCA (orofitsdualC(cid:98)A)isnotapplicable. Our approachforderivingaplausibleconjecturefortheasymptoticbehaviorofS inthegeneralcaseconsideredinthissubsection α reliesinsteadonthegeneraldualityprincipleestablishedintheprevioussection(cf.Theorem1andEq.(16)). Inaddition,we shallmakethenaturalassumptionthatwhenthedistancebetweentheblocksA ismuchlargerthantheirlengths(i.e.,when i min1(cid:54)i(cid:54)r(ui+1−vi)(cid:29)max1(cid:54)i(cid:54)r(vi−ui),whereur+1≡u1+2π)theentanglemententropyisasymptotictothesumofthe singleblockentropiesS (A;K). Themotivationbehindthisassumptionisthatwhentheblocksarefaraparttheirmutual α i influenceshouldbenegligible,andtheRényientropyisofcourseadditiveoverindependentevents. ThesimplestasymptoticformulasatisfyingtheaboveassumptionisthetrivialoneSα(A;K)∼∑ri=1Sα(Ai;K). However, thelatterformulacannotbecorrect,sinceitviolatesthedualityprinciple. Theobviouswayoffixingthisshortcomingwouldbe toaddthedualterm∑sj=1Sα(A;Kj)totheRHS,buttheresultingformulaviolatestheaboveassumption. Ontheotherhand, sincebyEq.(20)∑sj=1Sα(A;Kj)∼∑ri=1∑sj=1Sα(Ai;Kj)−sIα(u,v),andIα(u,v)∼0whentheblocksincoordinatespaceare farapart,theasymptoticformula r s r s S (A;K)∼∑S (A;K)+∑S (A;K )−∑∑S (A;K ) (26) α α i α j α i j i=1 j=1 i=1j=1 satisfiestheabovefundamentalassumption. Thisrelationisalsoclearlyconsistentwiththedualityprinciple(16),sincethe RHSofEq.(26)isinvariantundertheexchangeofthesetsAandK onaccountofTheorem1. Wearethusledtoconjecture thatwhenN→∞theRényientropyofaconfigurationwithrblocksA incoordinateandsblocksK inmomentumspace i j satisfiesthepreviousrelation. UsingEqs.(20),itsdualandEq.(23)weimmediatelyarriveattheclosedasymptoticformula S (A;K)∼rs(cid:18)b log(cid:18)2N(cid:19)+c (cid:19)+s(cid:18)b ∑r logsin(cid:0)vi−ui(cid:1)−I (u,v)(cid:19)+r(cid:18)b ∑s logsin(cid:0)qi−pi(cid:1)−I (p,q)(cid:19). (27) α α π α α 2 α α 2 α i=1 i=1 7/12 (a) (b) Figure3. AsymmetricblockconfigurationdiscussedinFig.4(b)in(a)coordinatespace,(b)momentumspace(thethick greenlinesrepresenttheblocks,andthereddotsarethetwoidentifiedendpointsofthechain). The latter equation is manifestly consistent with the duality principle stated in Theorem 1, as expected from the previous remark. ItisalsoapparentthatEq.(27)reducestoEq.(24)or(25)respectivelyfors=1orr=1,astheasymptoticmutual informationI vanishesforasingleblock. Moreover,itisstraightforwardtoexplicitlycheckthatwhentheblocksincoordinate α spacearefaraparttheRHSreducestothesumoftheasymptoticapproximations(25)tothesingle-blockentropiesS (A;K), α i sinceI (u,v)∼0inthislimit. (Byduality,asimilarremarkappliestothecaseinwhichtheblocks[P,Q )inmomentum α j j spacearefarapartfromeachother.) Finally,itisimmediatetocheckthatEq.(27)satisfiestheinvarianceundercomplements identity. Wehaveverifiedthroughextensivenumericalcalculationswithawiderangeofconfigurationsincoordinateandmomentum space that when N (cid:29)1 Eq. (27) is correct. In fact, for symmetric configurations (consisting of equally spaced blocks of thesamelength,bothincoordinateandmomentumspace)theerrorterminthelatterequationbehavesas f(N)N−min(2,2/α), where f isagainaperiodicfunction. Moreprecisely(forrationalγ andγ ), f(N)iswellapproximatedbyatrigonometric x p polynomial∑kkm=a0xakcos(kνN)withsmallkmax(independentofN),wherethemainfrequencyνistheproductofν0≡2πγxγp/rs withasimplefractionthatcanbecomputedfromtheconfigurationparametersr,s,γ ,γ . Thebehavioroftheerrorisvery x p similarinnon-symmetricconfigurations,exceptthatinsomecasesitappearstodecayfasterthanN−2for0<α <1. Asan example,inFig.4wepresentourresultsforthreedifferentconfigurationswith(r,s)=(3,2),(7,4),(10,5). Moreprecisely,the firstandlastoftheseconfigurationsaresymmetric,whilethemiddleoneis(slightly)asymmetric,asdetailedinFig.3. Ascan beseenfromFigs.4(d)-(f),theerrorinEq.(27)behavesinthesethreecasesasdescribedabove,wherethecoefficientsa of k thetrigonometricpolynomial f(N)anditsfundamentalfrequencyν arelistedinTable1. Case k (a ,...,a ) ν ν max 0 kmax 0 (d) 2 (−438.485,105.29,66.716) π/18 ν 0 (e) 14 (−21790.1,76.0009,1602.85,154.097,5143.99,397.121,416.007,1950.55, π/112 2ν /7 0 4556.52,156.444,756.382,168.572,2164.74,232.817,2661.63) (f) 9 (0,−852.969,0,−202.359,0,−99.4396,0,−57.2755,0,−55.2294) π/200 ν 0 Table1. Coefficientsak andfundamentalfrequencyν ofthetrigonometricpolynomial f(N)=∑kkm=a0xakcos(kνN)intheerror ofEq.(27)forcases(d)-(f)inFig.4. Itshouldbenotedthattheasymptoticformula(27),whichwehavenumericallycheckedforafinitechain,easilyyieldsasa (cid:0) (cid:1) limitingcaseananalogousformulaforaninfinitechain. Indeed,ifinEq.(27)weletγ tendto0wehavesin (v −u)/2 (cid:39) x i i π(V −U)/N, andsimilarlyfortheotherargumentsofthesinefunctionsappearingintheasymptoticmutualinformation i i termI (u,v). InthiswayweeasilyarriveattheanalogueofEq.(27)foraninfinitechain,namely α S(∞)∼sb log(cid:20)∏r (V −U)· ∏ (Vj−Ui)(Uj−Vi)(cid:21)+r(cid:18)b ∑s logsin(cid:0)qi−pi(cid:1)−I (p,q)(cid:19)+rs(b log2+c ). (28) α α i i (U −U)(V −V) α 2 α α α i=1 1(cid:54)i<j(cid:54)r j i j i i=1 Tothebestofourknowledge,thisgeneralasymptoticformulahasnotpreviouslyappearedintheliterature. Notealsothat fors=1(i.e.,whenthereisasingleblockofexcitedmomenta)Eq.(28)impliestheasymptoticexpressionforthemutual informationofrblocksincoordinatespaceconjecturedinRef.22. Fromtheasymptoticapproximation(27)(oritsequivalentversionEq.(26))onecanalsodeducearemarkableexpression forthe(asymptotic)mutualinformationofrblocksA ≡[U,V)(1(cid:54)i(cid:54)r)inpositionspacewhenthechainisinanenergy i i i eigenstate|K(cid:105)madeupofsblocksK ≡[P,Q )(1(cid:54) j(cid:54)s)ofexcitedmomentummodes,definedasI (cid:0)A ,...,A ;K(cid:1)≡ j j j α 1 r ∑ri=1Sα(cid:0)Ai;K(cid:1)−Sα(cid:0)(cid:83)ri=1Ai;K(cid:1).Indeed,usingEqs.(20)and(26)weimmediatelyobtaintheasymptoticformula s (cid:20) r (cid:21) s I (cid:0)A ,...,A ;K(cid:1)∼ ∑ ∑S (cid:0)A;K (cid:1)−S (cid:0)(cid:83)r A;K (cid:1) ∼ ∑I (u,v)=sI (u,v). (29) α 1 r α i j α i=1 i j α α j=1 i=1 j=1 Thus(inthelargeN limit)themulti-blockmutualinformationI issimplystimesthemutualinformationwhenthechain’s α state|K(cid:105)consistsofasingleblockofconsecutivemomenta. Inparticular,weseethatI dependsonlyonthetopologyof α 8/12 22 120 20 70 110 18 100 60 α16 α α 90 S 14 S 50 S 80 12 70 10 40 60 8 50 1000 1500 2000 2500 3000 3500 4000 2800 3000 3200 3400 3600 3800 4000 2000 2500 3000 3500 4000 (a) N (b) N (c) N –0.00003 0.0000 0.3 –0.0005 –0.00004 –0.0010 0.2 0.1 –0.0015 ε–0.00005 ε ε 0.0 –0.0020 –0.1 –0.00006 –0.0025 –0.2 –0.0030 –0.00007 –0.0035 –0.3 2860 2880 2900 2920 2940 2960 2980 3000 2800 3000 3200 3400 3600 3800 4000 3600 3800 4000 4200 4400 (d) N (e) N (f) N Figure4. (a)-(c): exactRényientropyS (bluedots)anditsasymptoticapproximation(27)(continuousredline)forα=1/2, α 3/5,3/4,1,3/2,2,3(toptobottom)in(a)asymmetricconfigurationwithr=3,s=2,γ =1/2,γ =1/3,(b)anasymmetric x p configurationwithr=7,s=4,γ =1/2,γ =1/4,and(c)asymmetricconfigurationwithr=10,s=5,γ =1/2,γ =1/4. x p x p (d)-(f)Differenceε betweentheexactentropyS anditsapproximation(27)fortheaboveconfigurationsand(d)α =1/2, α (e)α =1(vonNeumannentropy),and(f)α =2. Theredlinesrepresentthecorrespondingcurves f(N)N−min(2,2/α), with f(N)=∑kkm=a0xakcos(kνN)giveninTable1. thestate|K(cid:105)(i.e.,thenumberofblocksofexcitedmomenta),notonitsgeometry(i.e.,theparticulararrangementandthe lengthsoftheseblocks). OnecouldalsodefinethemutualinformationofsblocksofexcitedmomentaK ≡[P,Q )(1(cid:54) j(cid:54)s) j j j for a fixed configuration A≡(cid:83)r A in position space. It easily follows from Eq. (29) and the duality principle that this i=1 i mutualinformationisasymptotictorI (p,q). Ofcourse,ananalogousformulashouldholdfortheinfinitechainreplacing α thefunctionI byitsN→∞limitI(∞)(U,V)=b logf(∞)(U,V).Inparticular,fors=1thelatterexpressionimpliesthatthe α α α model-dependentoverallfactorappearinginthegeneralformulaforthemutualinformationofa1+1dimensionalCFT(see, e.g.,Refs.11,13,18) isequalto1forthemodelsunderconsideration. An alternative measure of the information shared by the blocks A (1 (cid:54) i (cid:54) r) discussed in Ref.18 is the quantity i I(cid:101)α(A1,...,Ar)≡∑rl=1(−1)l+1∑1(cid:54)i1<···<il(cid:54)rSα((cid:83)lk=1Aik)(weomitthedependenceonthechain’sstate|K(cid:105)forconciseness’s sake). Inparticular,forr=3weobtainthetripartiteinformationintroducedinRef.12,whosevanishingcharacterizestheexten- sivityofthemutualinformationIα. Itcanbereadilycheckedthattheasymptoticrelation(27)impliesthatI(cid:101)α(A1,...,Ar)van- ishesasymptoticallyforthemodelsunderconsideration.ThisfollowsimmediatelyfromEq.(29)—whichisitselfaconsequence of (27)— and the identities ∑1(cid:54)i1<···<il(cid:54)r∑lk=1Sα(Aik) = (cid:0)rl−−11(cid:1)∑ri=1Sα(Ai), ∑1(cid:54)i1<···<il(cid:54)rIα(cid:0)(ui1,...,uil),(vi1,...,vil)(cid:1) = (cid:0)r−2(cid:1)I (u,v). Inparticular,thisshowsthattheconjecture(27)impliestheasymptoticextensivityofthemutualinformationI l−2 α α forthemodelsunderconsideration. (Fortheinfinitechainwiths=1,thishadalreadybeennotedinRef.22.) Anothernoteworthyconsequenceoftheasymptoticformula(27)isthefactthatforlargeN theentanglemententropycan beapproximatelywrittenas(omitting,forsimplicity,itsarguments) S ∼rs(cid:18)b log(cid:18)2N(cid:19)+c (cid:19)+b g, with g≡s(cid:18)∑r logsin(cid:0)vi−ui(cid:1)+logf(u,v)(cid:19)+r(cid:18)∑s logsin(cid:0)qi−pi(cid:1)+logf(p,q)(cid:19). α α π α α 2 2 i=1 i=1 (30) Theterminparenthesisinthelatterformula,whichcontainstheleadingcontributionrsb logN toS asN →∞,depends α α onlyonthetopologyoftheconfigurationconsidered. Inparticular,fromthecoefficientofthelogN termwededucethatthe modelsunderconsiderationarecritical,behavingasa1+1dimensionalCFTwithcentralchargers. Notealsothatthefactthat theleadingasymptoticbehavioroftheRényientanglemententropyS dependsonlyonthetopologyoftheconfigurationin α bothpositionandmomentumspaceisageneralizationofthewidespreadhypothesis(forthecaser=1)thattheentanglement propertiesofcriticalfermionmodelsaredeterminedbythetopologyoftheirFermi“surface”(see,e.g.,Ref.31). 9/12 Ontheotherhand,thenumericalconstantginthepreviousequationisindependentofNandα,andissolelydeterminedby thegeometryoftheconfigurationinbothpositionandmomentumspace. Forinstance,forthetwosymmetricconfigurations discussedinFig.4(a),(c)thisconstantisrespectivelyequalto−3log12and−25log1250. Theasymptoticformula(30)makesitpossibletotackleseveralrelevantproblemsthatwouldotherwisebeintractablein practice. Forinstance,itisnaturaltoconjecturethatfixingr,s,γ andγ theblockconfigurationwhichmaximizestheentropy x p isthesymmetricone(i.e.,requallyspacedblocksofequallengthinpositionspace,andsimilarlyinmomentumspace). Our numericcalculationsforseveralconfigurationssuggestthatthisisindeedthecase(see,e.g.,Fig.5(a)forthecaseα =2). As weseefromEq.(30),thisproblemreducestoastandard(constrained)maximizationproblemforthegeometricfactorg,which inturnssplitsintotwoseparateproblemsforthefunctiong1(u,v)≡∑ri=1logsin(cid:0)vi−2ui(cid:1)+logf(u,v)anditsmomentumspace counterpart. Forinstance,whenr=2wecanexpressg (u,v)intermsofthelengthL ≡V −U ofthefirstblockandthe 1 1 1 1 interblockdistanced≡U −V as 2 1 g (u,v)=σ(θ)+σ(2πγ −θ)+σ(2πγ +δ)+σ(δ)−σ(θ+δ)−σ(2πγ −θ+δ)≡h(θ,δ), (31) 1 x x x where σ(x)≡logsin(x/2), θ =2πL /N ∈(0,2πγ ), δ =2πd/N ∈(0,2π(1−γ )). Moreover, from the symmetry of h 1 x x under θ (cid:55)→2πγ −θ and δ (cid:55)→2π(1−γ )−δ, it suffices to find the maximum of this function in the rectangle (0,πγ ]× x x x (0,π(1−γ )]. Anelementarycalculationshowsthathhasalocalmaximumatθ =πγ ,δ =π(1−γ ),i.e.,atthesymmetric x x x configuration,andthat∇hhasnootherzeroson(0,πγ ]×(0,π(1−γ )]. Thisprovestheconjectureinthecaser=2(cf.Fig.5 x x (b)). Forinstance,forr=s=2themaximumvalueoftheentropyiseasilyfoundfromthelatterargumentandEq.(30)tobe 4[b log(Nsin(πγ )sin(πγ )/2π)+c ]. α x p α 20 18 2 16 S 14 12 1500 2000 2500 3000 3500 4000 N (a) (b) Figure5. (a): RényientropyS vs.itsasymptoticapproximation(24)(redline)insymmetric(bluepoints)andsome 2 non-symmetric(bluetriangles)configurationswithγ =1/3,γ =1/2and(bottomtotop)4+2,5+2and4+3blocks. (b): x p 3Dplotofthefunctionh(θ,δ)inEq.(31)forγ =1/2(theredpointcorrespondstothesymmetricconfiguration x (θ,δ)=(π/2,π/2)). Discussion InthisworkwehaverigorouslyestablishedageneraldualityprinciplewhichpositstheinvarianceoftheRényientanglement entropyS(A;K)ofachainoffreefermionsunderexchangeofthesetsofexcitedmomentummodesK andchainsitesAof thesubsystemunderstudy,wherebothAandK aretheunionofanarbitrary(finite)numberofblocksofconsecutivesitesor modes. Bymeansofthisprinciple,wehavederivedanasymptoticformulafortheRényientanglemententropywhentheset K consistsofasingleblock. Fromthisformulaandanaturalassumptionconcerningtheadditivityoftheentropywhenthe blocksarefarapartfromeachotherineitherpositionormomentumspacewehaveconjecturedanasymptoticapproximation fortheentanglemententropyinthegeneralcasewhenbothsetsAandK consistofanarbitrarynumberofblocks. Wehave presentedamplenumericalevidenceofthevalidityofthisformulafordifferentmulti-blockconfigurations,andhaveanalyzed itserrorcomparingitwithitscounterpartfortheXX modeldiscussedbyCalabreseandEssler30. Ourconjecturealsoyields anasymptoticformulaforthemutualinformationofacertainnumberofblocksinposition(ormomentum)spacevalidfor arbitrarymulti-blockconfigurations,whichfors=1andinthecaseofaninfinitechainisconsistentwiththegeneralonefor 1+1dimensionalCFTs. Thepreviousresultsopenupseveralnaturalresearchavenues. Inthefirstplace,itwouldbedesirabletofindarigorous proofofthefundamentalasymptoticrelation(26),whichleadstotheexplicitasymptoticformula(27). Inparticular,itwould 10/12