A duality principle for the entanglement entropy of free fermion systems

1.9 MB·

by  Jose A. Carrasco

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
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