Entanglement negativity after a local quantum quench in conformal field theories

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by  Xueda Wen

Entanglementnegativityafteralocalquantumquenchinconformalfieldtheories
Xueda Wen,1 Po-Yao Chang,1 and Shinsei Ryu1
1DepartmentofPhysics, UniversityofIllinoisatUrbana-Champaign, Urbana, IL61801, USA
(Dated:August11,2015)
Westudythetimeevolutionoftheentanglementnegativityafteralocalquantumquenchin(1+1)-dimensional
conformalfieldtheories(CFTs),whichweintroducebysuddenlyjoiningtwoinitiallydecoupledCFTsattheir
endpoints.Wecalculatethenegativityevolutionforbothadjacentintervalsanddisjointintervalsexplicitly.For
twoadjacentintervals,theentanglementnegativitygrowslogarithmicallyintimerightafterthequench. After
developing a plateau-like feature, the entanglement negativity drops to the ground-state value. For the case
5
oftwospatiallyseparatedintervals,alight-conebehaviorisobservedinthenegativityevolution;inaddition,a
1
long-rangeentanglement,whichisindependentofthedistancebetweentwointervals,canbecreated.Ourresults
0
agreewiththeheuristicpicturethatquasiparticles,whichcarryentanglement,areemittedfromthejoiningpoint
2
andpropagatefreelythroughthesystem. Ouranalyticalresultsareconfirmedbynumericalcalculationsbased

g onacriticalharmonicchain.
u
A
I. INTRODUCTION respecttoA2’sdegreesoffreedomisdefinedas
8
] A. Introduction (cid:104)e(i1)e(j2)|ρTA21∪A2|e(k1)e(l2)(cid:105)=(cid:104)e(i1)e(l2)|ρA1∪A2|e(k1)e(j2)(cid:105),
h (3)
c where |e(1)(cid:105) and |e(2)(cid:105) are arbitrary bases in H and H ,
e Recently, it has been recognized that quantum entangle- i j A1 A2
respectively. Thenthelogarithmicnegativityisdefinedas
m ment provides us a powerful tool to study quantum proper-
- ties of many-body systems in condensed matter physics [1– E :=ln||ρT2 ||=lnTr|ρT2 |, (4)
at 4]. When the system is prepared in a pure state |Ψ(cid:105), a good A1,A2 A1∪A2 A1∪A2
t quantity that describes the bipartite entanglement is the von whereT indicatesthepartialtranspositionwithrespecttoA ,
s 2 2
. Neumannentropy,whichisdefinedas andthetracenorm||ρT2 ||isdefinedasthesumofallthe
at absolute values of theAe1i∪gAen2values of ρT2 . Recently, the
m S =−Trρ lnρ (1) A1∪A2
A A A logarithmicnegativityhasbeenextensivelyusedtostudyvari-
- ousmany-bodysystems,includingone-dimensionalharmonic
d
n where ρA = TrBρ is the reduced density matrix of subsys- chains[9,10],quantumspinchains[11–17],freefermionsys-
o temA,withρ = |Ψ(cid:105)(cid:104)Ψ|. Analternativemeasureofbipartite tems[18],andtopologicallyorderedsystems[19,20]. Inpar-
c entanglementinpurestatesistheRenyientropy ticular, the universal features of the entanglement negativity
[ inone-dimensionalcriticalsystemshavebeenunderstoodby

2 S(n) = 1 lnTrρn. (2) developingaconformalfieldtheory(CFT)approach[21,22].
v A 1−n A Lateron,thecomparisonofCFTresultsandnumericalcalcu-
8 lations of one-dimensional critical systems were studied in a
6 Theseentanglementmeasureshaveprovedtobeofgreatuse seriesofworks[23–25].
5 incharacterizingquantumentanglementofmany-bodystates. Althoughmanyworkshavebeendoneontheentanglement
0
However, for a mixed state, neither the von Neumann en- negativity,thereislessunderstandingonthenon-equilibrium
0
tropy nor the Renyi entropy is a good measure of entangle- propertiesoftheentanglementnegativity. Mostrecently,time
.
1 mentsincequantumandclassicalcorrelationsarenotclearly evolution of the logarithmic negativity after a global quench
0
separatedinthesemeasures.Nowsupposeweareinterestedin wasstudiedwithCFTapproach[26]. InRef.[27], thenega-
5
theentanglementbetweentwosubsystemsA andA ,which tivityevolutionfortwoadjacentintervalsafteralocalquench
1 1 2
: arenotnecessarilycomplementarytoeachotherand,areem- was numerically studied in a harmonic chain. However, a
v
bedded in a larger system, the union A ∪A cannot be de- thoroughstudyofthenegativityevolutionafteralocalquan-
1 2
i
X scribedbyapurestateafterintegratingoutdegreesoffreedom tum quench is still lacking, and it is appealing to unveil the
inthecomplementofA ∪A . Inthiscase,weneedtosearch universalfeaturesofthedynamicalbehavioroftheentangle-
r 1 2
a for other quantities that may characterize quantum entangle- mentnegativitypropagation.
ment for a general mixed state. Among different proposals Inthispaper, ourmotivationistostudythetimeevolution
[5,6],acomputablemeasurementofentanglement,theloga- of the entanglement negativity after a local quantum quench
rithmic negativity [7], turns out to be very useful and practi- analytically. Forsimplicity,weconsidera(1+1)-dimensional
cal. In particular, it is proved that the logarithmic negativity criticalsystem,whichisphysicallycutintotwopartsthatare
is a proper entanglement monotone in Ref. [8]. Following preparedintheirowngroundstates. Thenattimet = 0, we
Ref. [7], the negativity can be obtained by first taking a par- join the two parts together at their endpoints, and study the
tialtransposition. Tobemoreprecise,givenadensitymatrix time-evolutionoftheentanglementnegativityafterwards. As
ρ which describes a bipartite mixed state in a Hilbert shown in Fig. 1, once the two CFTs are joined at the end-
A1∪A2
spaceH = H ⊗H ,thepartialtranspositionwith points,theinteractionbetweenthemisintroducedsimultane-
A1∪A2 A1 A2
2
intervalτ ∈[0,β]:
1 (cid:90) (cid:89)
ρ= [dφ(x,τ)] δ(φ(x,0)−φ(cid:48)(x(cid:48)))
Z
x (6)
(cid:89)
× δ(φ(x,β)−φ(cid:48)(cid:48)(x(cid:48)(cid:48)))e−SE,
x
wheretherowsandcolumnsofthedensitymatrixarelabeled
by the fields {φ(x,τ)} at τ = 0 and β respectively, with β
beingtheinversetemperature,S istheEuclideanactionand
E
Z =Tre−βH isthepartitionfunction. Nowweconsidersub-
FIG.1: Setupforalocalquantumquench. TwoseparateCFTsde- systems A1 and A2 located in intervals [u1,v1] and [u2,v2],
finedontwosemi-infinitelinesarejoinedtogetherattheirendpoints. respectively. ThenthereduceddensitymatrixρA1∪A2 maybe
Then quasiparticles, which may be viewed as entangled pairs, are obtainedbysewingtogetherallthepointsalongedgesτ = 0
generatedatthejointingpoint,andpropagatefreelythroughthesys- andτ =βexceptthepointsinA ∪A . Thatis,weleavetwo
1 2
tem. Theentanglementnegativitybetweentwointervalswhichare opencutsat[u ,v ]and[u ,v ]alongτ =0.
1 1 2 2
farfromeachothermaybebuiltwiththehelpofthesepropagating Next,beforewecomputeTr(ρT2 )ne,itisbeneficialto
entangledpairs. seehowtocalculateTr(ρ )nA1fi∪rAst2. Inordertocalculate
Tr(ρ )n,weconsiderAn1∪cAo2piesofthecutplane,andthen
A1∪A2
ously,whichgeneratesquasiparticles(excitations)atthejoint- sewtogetherthecut[ui,vi]jτ=0− withthecut[ui,vi]jτ+=10+ for
ing point. These quasiparticles may be viewed as entangled i=1,2andallthecopiesj =1,··· ,n. Notethatforj =n,
pairs [28–32] which carry entanglement information. When wesewtogetherthecut[ui,vi]jτ==n0− withthecut[ui,vi]jτ==10+.
the entangled pairs arrive at two intervals separately, the en- Inthisway,wedefinean-sheetedRiemannsurfaceRn. The
tanglement negativity can be built immediately. Because the traceof(ρA1∪A2)nisthengivenby
(1+1) dimensional critical system is Lorentz invariant at the
Z
lowenergylimit,wecanutilizethepowerofconformalfield Tr(ρ )n = Rn, (7)
theoryandunderstandtheuniversalfeatureofthisdynamical A1∪A2 Zn
phenomenon.
where Z is the partition function for the orbifold CFT on
The rest of the paper is organized as follows. In part B of Rn
R . Ratherthandealingwiththefieldsonanontrivialmani-
n
this section, we give a brief review of the path integral rep-
fold,itisfoundmoreconvenienttoworkonasinglecomplex
resentationoftheentanglementnegativity,andthenintroduce
plane. It turns out Eq. (7) can be expressed in terms of lo-
theCFTsetupforalocalquantumquenchinpartC.InSection
caltwistedfieldsdefinedat(u ,0)and(v ,0)onthecomplex
i i
II,byusingCFTapproach,wecomputethetimeevolutionof
planeasfollows
theentanglementnegativityfortwoadjacentintervalsinpart
A,andtwodisjointintervalsinpartB.Weconsiderbothcases Tr(ρ )n =(cid:10)T (u )T¯ (v )T (u )T¯ (v )(cid:11). (8)
wherethetwointervalsaresymmetricallyandasymmetrically A1∪A2 n 1 n 1 n 2 n 2
located. In section III, we describe the numerical method of Intuitively, the effect of twist fields T and T¯ is shown in
n n
calculatingtheentanglementnegativityforaharmonicchain, Fig. 2. Winding anticlockwise (clockwise) around the twist
based on which we study the local quench of the entangle- fieldT (T¯ ),oncethebranchcutiscrossed,onewillgofrom
n n
mentnegativity. Thenwecomparethenumericalresultswith layerj tolayerj+1.
theCFTresults. InsectionIV,weconcludeourworkandlist With the introduction of twist fields, the expression of
someinterestingfutureproblemstobestudied. Tr(ρT2 )n is very straightforward. As discussed in Refs.
A1∪A2
[21,22],theeffectofpartialtranspositionwithrespecttoA
2
isequivalenttochangingthetwotwistoperatorsT (u )and
n 2
B. Entanglementnegativityinquantumfieldtheory T¯ (v ). Thenonehas
n 2
Adetaileddescriptionofpathintegralrepresentationofthe Tr(cid:16)ρT2 (cid:17)n =(cid:10)T (u )T¯ (v )T¯ (u )T (v )(cid:11). (9)
entanglement negativity can be found in Ref. [22]. For the A1∪A2 n 1 n 1 n 2 n 2
completenessofthispaper,wegiveabriefreviewhere.
If the two intervals [u ,v ] and [u ,v ] are adjacent to each
First,asdiscussedinRef.[22],byusingareplicatrick,one 1 1 2 2
other,wesimplysetu →v ,andthenEq.(9)canbewritten
canrelatetheentanglementnegativitywiththeintegerpowers 2 1
as
ofρT2 as
A1∪A2 E = lim lnTr(cid:16)ρT2 (cid:17)ne, (5) Tr(cid:16)ρTA21∪A2(cid:17)n =(cid:10)Tn(u1)T¯n2(u2)Tn(v2)(cid:11). (10)
A1,A2 ne→1 A1∪A2
Therefore,fromEqs.(5),(9)and(10),itisfoundthatthecom-
wheren isaneveninteger,andthedensitymatrixρmaybe putationoftheentanglementnegativityreducestothecompu-
e
expressedasa(Euclidean)pathintegralintheimaginarytime tationofexpectationvaluesoftwistfieldsinacomplexplane.
3
FIG.2:Pathintegralrepresentationof(a)Tr(ρ )nand(b)Tr(ρT2)nfortwodisjointintervals.
A A
C. CFTapproachtoalocalquench ofthepathintegralonamodifiedword-sheet,wherethephys-
ical cut corresponds to two ‘walls’ with one extending from
Before we study the CFT approach to a local quantum τ = −∞to−(cid:15)andtheotherextendingfromτ = +(cid:15)to+∞
quench,itisbeneficialtocommentonthedifferencebetween in a complex z-plane. No energy nor momentum can flow
localquenchesandglobalquenches.Localquenchesaremore through the two ‘walls’, and therefore conformal boundary
complicated than global quenches because they are inhomo- conditions are imposed on the wall (As will be shown later,
geneous. Forglobalquenches,wechangetheparametersofa theconcreteboundaryconditiondoesnotaffecttheuniversal
translationalinvariantHamiltonianglobally,andthereforethe resultweconsider.). Forconvenienceofcalculation,wemap
system before and after global quenches are always transla- the z-plane to a right half plane (RHP) with Rew > 0 by
tionalinvariant. Inthiscase,asdiscussedinRefs.[31,32],the usingtheconformalmapping
initial state can flow to a conformal invariant boundary state
underrenormalizationgroup(RG).Forlocalquenches,how-
(cid:114)
ever,beforewejointhetwodecoupledCFTstogether,theto- z (cid:16)z(cid:17)2
talsystemisapparentlynottranslationalinvariant. Therefore, w = + +1. (11)
(cid:15) (cid:15)
theinitialstatecannotflowtoaconformalinvariantboundary
state. In addition, for global quenches, quasiparticle excita-
tionsareemittedfromeverywhereinthebulkofCFT;forlo-
Then the local quench problem is reduced to the calculation
calquenches, quasiparticleexcitationsareemittedonlyfrom
ofcorrelationfunctionsoftwistfieldsintheRHP[33],which
thepointwheretwoCFTarejoinedtogether.
wewillstudyindetailinthenextsection.
Thetimedependentdensitymatrixcanbewrittenasρ(t)=
|φ(x,t)(cid:105)(cid:104)φ(x,t)|, where |φ(x,t)(cid:105) = e−iHt|φ (x)(cid:105). In path
0
integralrepresentation,onehas
(cid:104)φ(cid:48)(cid:48)(x(cid:48)(cid:48))|ρ(t)|φ(cid:48)(x(cid:48))(cid:105)
1
= (cid:104)φ(cid:48)(cid:48)(x(cid:48)(cid:48))|e−iHt−(cid:15)H|φ (x)(cid:105)(cid:104)φ (x)|e+iHt−(cid:15)H|φ(cid:48)(x(cid:48))(cid:105),
Z 0 0
wherethefactore−(cid:15)H isintroducedtodampouthigh-energy
modes and make the path integral absolutely convergent. If
theCFTarisesasthelowenergylimitofalatticemodel,then
(cid:15)maybeviewedasthelatticespacing. Inthestudyofglobal
quenches[31–33],|φ (x)(cid:105)maybeconsideredasaconformal
0
invariant boundary state under the RG. For local quenches,
as we discussed above, the initial state cannot flow to a con-
formal invariant boundary state under the RG. In this case, FIG.3: IllustrationoftheconformalmappinginEq.(11),basedon
onemayintroduceboundaryconditionchangingoperators,as whichthez-planeismappedtoarighthalfplane(RHP)withRew>
utilizedinRefs.[34,35]. Inthiswork,however,wewillfol- 0.Forlateruse,wealsolabelw(cid:101)i =−w¯i,whichistheimageofwi.
low the method proposed by Calabrese and Cardy [33]. As
showninFig.3,thedensitymatrixcanbeexpressedinterms
4
Withananalyticalcontinuationτ →it,onecanobtain
(cid:10)T2(z )(cid:11)=c˜ (cid:18) a(cid:15) (cid:19)∆(n2). (16)
n 1 n 2((cid:15)2+t2)
As discussed in Refs. [21, 22], the scaling dimension ∆(2)
n
dependsontheparityofnas
(cid:26)∆ oddn,
∆(2) = n , (17)
n 2∆ evenn,
n/2
where
FIG.4:Configurationsoftwointervalsconsideredinthiswork:two
adjacentintervals(up)andtwodisjointintervals(bottom). c (cid:18) 1(cid:19)
∆ = n− . (18)
n 12 n
II. ENTANGLEMENTNEGATIVITYAFTERALOCAL ThenbyusingtheexpressionsinEqs. (5)and(9),onecanget
QUENCH:CONFORMALFIELDTHEORYAPPROACH
c a(cid:15)
E =− ln +c˜(cid:48), (19)
In this section, we calculate the time evolution of the en- 4 2((cid:15)2+t2) 1
tanglement negativity after a local quench in conformal field
wherec˜(cid:48) =lnc˜ . AsinRefs.[3,33],theshorttimebehavior
theories. We will consider adjacent intervals in part A and 1 1
of E(t) allows us to fix the regulator (cid:15) in terms of the non-
disjointintervalsinpartB,respectively.
universalconstantc˜ byrequiring
1
c a
E(t=0)=− ln +c˜(cid:48) =0, (20)
A. Twoadjacentintervals 4 2(cid:15) 1
basedonwhichonegets
1. Semi-infiniteintervals
a
(cid:15)= e−4c˜(cid:48)1/c. (21)
Asawarmup,weconsiderthesimplestcase,i.e.,thetotal 2
systemisbipartitionedintotwosemi-infinitepartsA andA .
1 2 Nowitispossibletoeliminateaandc˜(cid:48) inEq.(19)intermsof
Inthiscase,ρ ispure,andthelogarithmicnegativityis 1
A1∪A2 (cid:15),andthenonecanget
the same as the Renyi entropy with n = 1/2 [21, 22]. This
casewasalsostudiedinRef.[27]. c (cid:15)2
For two adjacent semi-infinite intervals, we only need to E =− ln . (22)
4 (cid:15)2+t2
consider a single twist field T2(z ) in z-plane, which is in-
n 1
sertedat Inthelimitt(cid:29)(cid:15),oneendswith
c t
z =l+iτ. (12)
1 E = ln , (23)
2 (cid:15)
By choosing the insertion position at the origin l = 0, i.e.,
whichwasobservedinthenumericalcalculationsbasedona
A ∈ (−∞,0]andA ∈ (0,+∞), onesimplyhasz = iτ.
1 2 1 criticalharmonicchain[27].
TheexpectationvalueofT2(z )canbeexpressedas[33]
n 1
(cid:10)Tn2(z1)(cid:11)=c˜n(cid:32)(cid:12)(cid:12)(cid:12)(cid:12)ddwz(cid:12)(cid:12)(cid:12)(cid:12) 2Rea(w )(cid:33)∆(n2), (13) 2. Symmetricfiniteintervals
z1 1
Inthispart,weconsiderthecaseofsymmetricfiniteinter-
wherec˜ isanonuniversalconstantwhichdependsonthepar- vals with l = l = l, i.e., A ∈ [−l,0] and A ∈ (0,l], as
n 1 2 1 2
ticularboundaryCFT,aisanUVcutoff(e.g.,thelatticespac- showninFig.4.Inthiscase,ρ representsamixedstate,
A1∪A2
inginaharmonicchain),and∆(2)isthescalingdimensionof andthereisnocorrespondencebetweenthelogarithmicneg-
n
T2(z). ByusingtheconformalmapinEq.(11),onehas ativityandtheRenyientropies. ByusingEq.(10)anddoing
n
aconformalmappingontotheRHP,onehas
τ 1(cid:112)
w1 =i(cid:15) + (cid:15) (cid:15)2−τ2, (14) Tr(cid:16)ρT2 (cid:17)n =(cid:10)T (z )T¯2(z )T (z )(cid:11)
A1∪A2 n 1 n 2 n 3
and (cid:12)(cid:12)(cid:12)dw(cid:12)(cid:12)(cid:12) = √ 1 . (15) =i(cid:89)=31(cid:12)(cid:12)(cid:12)(cid:12)ddwz(cid:12)(cid:12)(cid:12)(cid:12)∆zi(i)(cid:10)Tn(w1)T¯n2(w2)Tn(w3)(cid:11)RHP,
(cid:12)dz(cid:12) (cid:15)2−τ2 (24)
z1
5

3.0 3.0
(a) l=25 (b) l=25
l=50 l=50
2.5 l=75 2.5 l=75
l=100 l=100
y 2.0 2.0
vit
gati 1.5 1.5
e
N
1.0 1.0
0.5 0.5
0.0 0.0
0 20 40 60 80 100 0 20 40 60 80 100
Time Time
FIG. 5: Entanglement negativity E for two symmetric adjacent intervals as a function of time. Here we choose the central charge c = 1,
(cid:15)=0.1,l=25,50,75and100,respectively.Shownin(a)istheCFTresult,and(b)isthenumericalcalculationbasedonacriticalharmonic
chain.
wherethescalingdimensions∆ =∆ =∆ and∆ = whichmaybefurtherexpressedas
(1) (3) n (2)
∆(n2). Thethree-pointcorrelationfunctionontheRHPcanbe c (cid:15)2 c w w w
expressedas E =− ln − ln 13 1˜2 2˜3, (28)
4 (cid:15)2+t2 4 w w w
1˜3 12 23
(cid:10)T (w )T¯2(w )T (w )(cid:11) wherewehavedefinedw =|w −w |,w =|w −w |and
n 1 n 2 n 3 RHP ij i j i˜j i (cid:101)j
c˜ η∆(n2)−2∆n1/2 w˜i˜jW=ith|w(cid:101)thie−exw(cid:101)pjr|e,srseisopnesctoifvewly. that are calculated in the Ap-
=(cid:81)3i=1|(wi−nw(cid:101)i)/a|∆(i) η11∆,,23(n2)η2∆,3(n2) F({ηj,k}), pfuenncdtiixo,noonfetimcaenaosbfotalilnowthse iejntanglement negativity E as a
(25)
 c (cid:15)2 c l+t
whereη arecrossratioswhichcanbeconstructedfromthe  − 4ln(cid:15)2+t2 − 4lnl−t, t<l
i,j E = (29)
eansdfoplolionwtsswi(andtheirimagesw(cid:101)i)oftheintervalsintheRHP  − 4c ln(cid:15)2(cid:15)+2t2 − 4c ln4(cid:15)tl2, t>l.
(w −w )(w −w ) Inthelimitl,t(cid:29)(cid:15),E canbesimplifiedas
η = i j (cid:101)i (cid:101)j (26)
i,j (wi−w(cid:101)j)(w(cid:101)i−wj)  c t2+(cid:15)2(l−t)
4ln (cid:15)2 (l+t), t<l
with w = −w¯ being the image of w (see Fig.3). The E = (30)
nonun(cid:101)iviersalfuncitionF({ηj,k})dependsionthefulloperator  c ln l +const, t>l.
4 2(cid:15)
contentoftheCFT.F({η })isusuallydifficulttocalculate
j,k
and only known for several specific CFTs and BCFTs. But ShowninFig.5(a)istheplotofE withdifferentl. Atthevery
it is found that in the limits ηi,j → 0, 1, or ∞, the function beginning of the local quench t (cid:28) l, based on Eq. (30), one
F({ηj,k})isjustaconstant,whichfollowsfromthelong-and has
short-distanceexpansionsofthecorrelationfunctionsoftwist
c t
operators[3,32,36–38]. Forsymmetricintervalsinthispart, E = ln , (31)
2 (cid:15)
wecalculatethecrossratiosη explicitlyintheappendix. It
i,j
isfoundthatonealwayshasη =1or0forthecasest(cid:28)l, which agrees with the result of semi-infinite intervals as
ij
t=l+0−andt>l. Inotherwords,ourresultsareuniversal shown in Eq. (23). This is reasonable because in the limit
fortheabovethreecases. t (cid:28) l,thequasiparticlesessentiallypropagatewithoutnotic-
ByusingEqs. (5)and(10),andneglectingvariousnonuni- ingthefinitesizeeffect. Fort<l,theentanglementsaturates
versalterms,wehave for a certain time. Then for t > l, we get the ground-state
valueofE,i.e.,
E =− c ln (cid:15)2 − c ln η1,3 , (27) E = c ln l , (32)
4 (cid:15)2+t2 8 η1,2η2,3 G 4 2(cid:15)
6
which is also observed in the numerical calculations in Ref. Ontheotherhand,fromthescalingbehaviorinRef.[27],one
[27]. Note that in the numerical calculations, E tends to the canfind
ground-state value gradually. In our CFT results, E drops
to the ground-state value immediately after t = l. This is l 1 l
E (t= )= ln +0.13ln3−0.28ln2. (35)
because all quasiparticles propagate at the same velocity in numerical 2 2 2
CFTs. In lattice models, however, the dispersion relation is
not linear for all momentum vectors, and therefore not all
Inthelargellimit,i.e.,l(cid:29)(cid:15),onealwayshas
quasiparticlespropagateatthesamevelocity,asdiscussedin
detailinsectionIV.
l l 1 l
Inaddition,thescalingbehaviorofE foraharmonicchain
E (t= )(cid:39)E (t= )(cid:39) ln . (36)
wasnumericallystudiedinRef.[27].Fort<l,theyproposed CFT 2 numerical 2 2 2
theansatz
Beforeweendthispart,wementionthatitisinterestingto
tα (l−t)β
E ∼ln . (33) checkhowE(t)behavesinotherlatticemodels. Considering
l−ρ(l+t)−γ
E(t < l) depends on the non-universal functions F({η }),
j,k
whichvariesfordifferentCFTs,weexpectthatforothercrit-
By fitting the numerical results, they found α = 1/2, β (cid:39)
ical lattice models such as the critical Ising model one may
0.15, γ (cid:39) 0.13 and ρ = −(β +γ) (cid:39) −0.28. For our CFT
observedifferentscalingbehaviorsinE(t<l).
resultsinEq.(30),bysettingc=1andtakingthelimitt(cid:29)(cid:15),
one has α = 1/2, β = 0.25, γ = −0.25 and ρ = −(β +
γ) = 0. On the other hand, in the limit t (cid:28) l, both Eq.
(30) and Eq. (33) collapse to E = 1lnt. We attribute the
2
abovedisagreement/agreementtothefollowingfact. Fort ≤ 3. Asymmetricfiniteintervals
l, because we neglect the non-universal functions F({η })
j,k
whichmaynotbeconstants, ourresultsarenotaccurateand
Inthispart,weconsiderthecaseofasymmetricfiniteinter-
thereforemaynotobtainthecorrectscalingbehavior.Fort(cid:28)
valswithA ∈ [−l ,0]andA ∈ (0,l ],asshowninFig.4.
l, however, ourCFTresultsareuniversalandindependentof 1 1 2 2
Withoutlossofgenerality,wesupposel < l . Thecalcula-
thespecificCFT.Toreproducethenumericalresultsfort≤l 1 2
tionsaresimilartothesymmetriccase, andweneedtoeval-
in Ref. [27], we have to consider the nonuniversal functions
uatethethreepointcorrelationfunctionsinEq.(24). First,as
F({η }), which is a difficult task, and out of the scope of
j,k shownintheappendix,wecalculatethecrossratioη explic-
ourwork. ij
itly. Itisfoundthatonealwayshasη = 1or0forthecases
Nevertheless, by comparing the values of the plateau be- ij
t (cid:28) l , t = l +0− and t > l . That is to say, our results
tweenCFTresultsandnumericalresultsinFig.5,itisfound 1 2 2
areuniversalintheseregions. Second,byneglectingvarious
they are very close to each other. To be concrete, let’s take
nonuniversalterms,wearriveatthesameresultasinEq.(28).
E(t= l)forexample. BasedonEq.(30),onecanget
2 The difference is that for asymmetric intervals, we have dif-
ferentexpressionsofw ,asexplicitlygivenintheappendix.
l 1 l ij
E (t= )= ln +const. (34) Bypluggingw intoEq.(28),oneobtains
CFT 2 2 2(cid:15) ij
 c (cid:15)2 c (l +t)(l +t)
 − 4c ln(cid:15)2(cid:15)+2t2 − 8c ln(cid:32)(l112−(cid:115)t()l(l22+−lt))(,t+l )(t−l )t2(cid:33) t<l1
E = − ln − ln 1 2 1 1 , l <t<l
 − 44c ln(cid:15)(cid:15)22(cid:15)++2tt22 − 44c ln2((cid:15)l1(cid:15)l+ll2)t2, (l2−l1)l12 1 t>l22.
1 2
Inthelimitl,t(cid:29)(cid:15),E canbesimplifiedas Notethatinthelimitl =l ,wecanreproducethesymmetric
1 2
adjacentintervalsresultinEq.(30).TheplotofE fordifferent
8cc ln((ll11+−(ttl))((ll−22+−l )ttl))2(cid:15)tt442, t<l1 (cclaa1ns,elfi.2n)Adicstthusehaltoliwym,nebeianvseoFdliugot.ino6nE(aoq)f..E(F3(i7tr))s,ti,sofnsoiemrctialan(cid:28)rctohmetchinke[lst1hy,amlt2mi]n,eottrhniece
E = ln 2 1 1 , l <t<l (37)
84c ln((cid:15)l(1ll+1+l2l2l)()t++clo1n)(stt,−l1) 1 t>l22.
1 2
7

3.0 3.0
(a) l1=25, l2=75 (b) l1=25, l2=75
2.5 l1=50, l2=75 2.5 l1=50, l2=75
l1=75, l2=75 l1=75, l2=75
y 2.0 2.0
vit
gati 1.5 1.5
e
N
1.0 1.0
0.5 0.5
0.0 0.0
0 20 40 60 80 100 0 20 40 60 80 100
Time Time
FIG.6:EntanglementnegativityEfortwoasymmetricadjacentintervalsasafunctionoftime.Herewechoosecentralchargec=1,(cid:15)=0.1,
(l ,l )=(25,75),(50,75),and(75,75),respectively.Shownin(a)istheCFTresult,and(b)isthenumericalcalculationbasedonacritical
1 2
harmonicchain.
limitt(cid:28)min[l ,l ] 4(b). In this case, we need to consider the correlation func-
1 2
tionoffourtwistfieldsasshowninEq.(9). Byapplyingthe
c t
E = ln , (38) conformalmapinEq.(11),onehas
2 (cid:15)
which shows the lnt behavior again, as expected. For t > (cid:10)T (z )T¯ (z )T¯ (z )T (z )(cid:11)
n 1 n 2 n 3 n 4
max[l ,l ],weobtainthegroundstatevalueoftheentangle-
mentn1eg2ativity[21,22],i.e., =(cid:89)4 (cid:12)(cid:12)(cid:12)(cid:12)ddwz(cid:12)(cid:12)(cid:12)(cid:12)∆n(cid:10)Tn(w1)T¯n(w2)T¯n(w3)Tn(w4)(cid:11)RHP,
E = c ln l1l2 . (39) i=1 zi
G 4 (cid:15)(l1+l2) wherethefour-pointcorrelationfunctionontheRHPhasthe
Interestingly,itisfoundthatthesuddendropofE happensat form
t=min[l1,l2], (40) (cid:10)Tn(w1)T¯n(w2)T¯n(w3)Tn(w4)(cid:11)RHP
which is again straightforward to understand based on the c˜2 1 (cid:18)η η (cid:19)∆(n2)/2−∆n
= n 1,4 2,3
hveieuwriesdticaspheynstiacnagllepmicetnutrepathirast otfhetwqouaqsiupaanrttaic.lesFomrayt b<e (cid:81)4i=1|(wi−w(cid:101)i)/a|∆n η1∆,2nη3∆,4n η1,3η2,4
min[l1,l2],twoentangledquasiparticlesareinA1andA2sep- ×F({ηj,k}).
arately, andcreatetheentanglementbetweenA andA . At (41)
1 2
t = min[l ,l ] = l (herewesupposel < l ),althoughone
1 2 1 1 2 For the nonuniversal functions F({η }), as explicitly cal-
quasiparticleisstillinA , theotherquasiparticlepropagates j,k
2 culated in Ref. [36], they are simply a constant in the limit
outofA ,andthereforetheentanglementbetweenA andA
1 1 2 l/d(cid:28)1. Inotherwords,whenthetwointervalsarefarapart,
decreasessuddenlyatt=min[l ,l ].
1 2 we do not need the knowledge of F({η }). By using the
Beforeweendthispart, weemphasizethatourresultsare j,k
definitioninEq.(5),anddroppingvariousmultiplicativecon-
universalfortheregimest(cid:28)l ,t=l +0− andt>l . For
1 2 2 stants,wehave
thecaseoft ≤ l ,however,similarwiththesymmetriccase,
2
because we neglected the nonuniversal functions F({ηj,k}) E =−c ln(cid:18)η1,4η2,3(cid:19), (42)
whichmaynotbeconstants,ourresultsarenotaccurate,and 8 η η
1,3 2,4
onehastocalculateF({η })fordifferentCFTs.
j,k
whichisalternativelywrittenas
c w w w w w w w w
B. Twodisjointintervals E =− ln 14 1˜3 23 2˜4 ˜1˜4 ˜13 ˜2˜3 ˜24. (43)
8 w w w w w w w w
1˜4 13 2˜3 24 ˜14 ˜1˜3 ˜23 ˜2˜4
1. Symmetricfiniteintervals By noting that wij = w˜i˜j and wi˜j = w˜ij, Eq. (43) can be
simplifiedas
In this part, we consider the symmetric disjoint intervals, E =− c lnw14w1˜3w23w2˜4. (44)
i.e., A1 ∈ [−d−l,−d] and A2 ∈ [d,d+l], as shown Fig. 4 w1˜4w13w2˜3w24
8

(a)
(b)
0.4 0.06
d=140, l=10 d=140, l=10
d=160, l=10 d=160, l=10
d=180, l=10
y 0.3 d=180, l=10
gativit 0.04
e 0.2
N
0.02
0.1
0.0 0.00
100 120 140 160 180 200 220 100 120 140 160 180 200 220
Time Time
FIG.7:EntanglementnegativityEfortwosymmetricdisjointintervalsasafunctionoftime.Herewechoosethecentralchargec=1,(cid:15)=1,
(d,l) = (140,10), (160,10) and (180,10), respectively. Shown in (a) is the CFT result, and (b) is the numerical calculation based on a
criticalharmonicchain.
Withtheexplicitformsofw givenintheAppendix,wecan simultaneously. Note that at t = d + l/2, taking the limit
ij
obtaintheentanglementnegativityE asafunctionoftimeas d(cid:29)l,onehas
follows
c l
0c ln(2d+l)(d+l−t)(t2−d2) d<t<td<+dl which is independenEtto=fd+th2le(cid:39)dis4talnnce2(cid:15)d,, as also can be(4o6b)-
E = 4 (cid:15)dl(d+l+t)
served in Fig. 7. That is to say, with the help of entangled
 c ln (2d+l)2 t>d+l pairs, we can create a long-range entanglement between two
4 4d(d+l) intervals which are far from each other. Note that this long-
(45) range entanglement was also observed in the time evolution
Notethatinthestudyofthenegativityevolutionafteraglobal ofmutualinformationI(t)inRef.[36],whereitisfoundthat
quench, it was found that E(t) shows the same behavior as I (cid:39) c ln l .
t=d+l 3 2(cid:15)
the Renyi mutual information apart from the prefactor [26]. 2
Forthelocalquenchstudiedhere,bycomparingourresultin
Eq.(45)withtheresultofmutualinformationinRef.[36],it
2. Asymmetricfiniteintervals
isfoundthattheexpressionsarealsothesameexceptforthe
prefactor. Inotherwords,ourresultsparallelwiththestoryin
thenegativityevolutionafteraglobalquench.Therelationbe- Inthispart, weconsidertheasymmetricdisjointintervals.
tweentheentanglementnegativityandthemutualinformation Wehavemultiplechoicesasfollows: (i)d1 (cid:54)=d2,l1 =l2,(ii)
afteralocalquantumquenchwillbesystematicallydiscussed d1 = d2, l1 (cid:54)= l2 and (iii) d1 (cid:54)= d2, l1 (cid:54)= l2. For simplicity,
insectionIV. weconsiderthecase(i),i.e.,A1 ∈ [−d1+l,−d1]andA2 ∈
As shown in Fig. 7(a), we plot the evolution of the entan- [d2,d2+l]. Withoutlossofgenerality,wechoosed1 <d2 ≤
glementnegativitywithdifferent(d,l)accordingtoEq.(45). d1+l.
A ‘light-cone’ effect can be observed: For t < d, there is The calculation of the negativity evolution is similar with
no entanglement negativity between A and A . At t = d, thesymmetriccase,andweobtainthesameresultinEq.(44).
1 2
theentanglementnegativitybeginstodevelop,andreachesthe The difference is that we should express w in terms of d ,
ij 1
maximumapproximatelyatt=d+l/2. Att=d+l,theen- d and l, as explicitly shown in the appendix. By plugging
2
tanglementnegativitydecreasessuddenly,whichcorresponds the expressions of w into Eq. (44), one arrives at the time
ij
totheentangledpairspropagatingoutofintervalsA andA evolutionoftheentanglementnegativity
1 2
9

0.6

(a) (b)

d1=150, d2=150, l=15 d1=150, d2=150, l=15
0.10
0.4 d1=150, d2=155, l=15 d1=150, d2=155, l=15
y
vit d1=150, d2=160, l=15 d1=150, d2=160, l=15
gati d1=150, d2=165, l=15 d1=150, d2=165, l=15
e
N
0.05
0.2
0.0 0.00
100 120 140 160 180 100 120 140 160 180
Time Time
FIG.8:EntanglementnegativityEfortwoasymmetricdisjointintervalsasafunctionoftime.Herewechoosethecentralchargec=1,(cid:15)=1,
l=15,(d ,d )=(150,150),(150,155),(150,160),and(150,165),respectively. Shownin(a)istheCFTresult,and(b)isthenumerical
1 2
calculationbasedonacriticalharmonicchain.
0 t<d
 −− 8cc llnn((dd(cid:15)11(++d1dd+22)+d(2dl)2)(−d(cid:115)2t)+((ddl12−++tll)−(+d2dt)2+()d(t2d)2(−d+2d1−l−+d1dl))1)(d1+l+t)(d2+l+t) d <d1t<<td<+d21l
E = 4 2(d +d +l) (d +l−t)(d +l−t)(t2−d2)(t2−d2) 2 1 (47)
1 2 1 2 1 2
 −− 84cc llnn[(td21−+((ddd21+++(dd2l2)l)2+−(](ddld1)112+)+(ddd212)+)2(dd22++l)l3−(td21−)(dd211)+d2+2l) d1+l<tt<>dd22++ll
1 2
Onecancheckthatwhend = d = d,theresultinEq.(45) entanglement negativity for a harmonic chain has been nu-
1 2
isreproduced. merically studied in several works [9, 22, 26, 27, 39]. Here
AccordingtoEq.(47),weplotE(t)withdifferent(d ,d ) wefollowthemethoddevelopedintheseworks,andapplyit
1 2
inFig.8(a).Comparedtothesymmetriccase,the‘light-cone’ to the local quench problem. We will first introduce the lat-
effect is still observed. The difference is that the time when ticemodelandthecovariancematrixinpartA.InpartB,we
E(t)increasesquicklynowhappensat introducetheevolutionmatrixandshowhowtocalculatethe
entanglement negativity. In part C, we apply the method to
t=max[d ,d ], (48)
1 2 thecasesstudiedwithCFTapproach,andcomparetheresults
andthetimewhenE(t)decreasesquicklyhappensat accordingly.
t=min[d +l,d +l], (49)
1 2
whichisalsoinagreementwiththequasiparticlepicture. A. Harmonicchainandthecovariancematrix
TheHamiltonianoftheharmonicchainis
III. NUMERICALEVALUATIONOFTHENEGATIVITY
FORAHARMONICCHAINAFTERALOCALQUENCH (cid:88)N p2 Mω2 K
H = [ n + 0q2 + (q −q )2], (50)
2M 2 n 2 n+1 n
In this section, to confirm our CFT results, we study the n=1
timeevolutionofthelogarithmicnegativityafteralocalquan- where N is the number of sites of the chain, M is the
tumquenchinalatticemodel,acriticalharmonicchain. The mass scale, ω is the characteristic frequency, and K is the
0
10
nearest-neighborcoupling. p andq denotethemomentum ForDBC,thecorrelatorsare
n n
and position operators with canonical commutation relations
L−1 (cid:18) (cid:19) (cid:18) (cid:19)
[p ,p ]=[q ,q ]=0and[q ,p ]=iδ . 1 (cid:88) 1 πkn πkm
n m n m n m n,m (cid:104)0|q q |0(cid:105)= sin sin ,
For periodic boundary condition (PBC), the Fourier trans- n m L Mω L L
k
k=1
formofthecanonicalvariablesare
L−1 (cid:18) (cid:19) (cid:18) (cid:19)
 1 (cid:88) πkn πkm
 qn =(cid:88)L q(cid:101)k√1Le2πikn/L, (cid:104)0|pnpm|0(cid:105)= L k=1Mωksin L sin L ,
k=1 (cid:104)0|q p |0(cid:105)=iδ /2. (58)
(51) n m n,m
 q(cid:101)k =(cid:88)L qn√1Le−2πikn/L,
n=1 B. Evolutionmatrixandthelogarithmicnegativity
wheren = 1,··· ,L. Forp ,theFouriertransformisidenti-
n FromtheHeisenbergequationofmotion,q˙ (t) = 1 p (t)
caltoqn. TheHamiltonianisdiagonalizedinthemomentum (cid:101)k M(cid:101)k
space andp(cid:101)˙k(t)=−Mωk2q(cid:101)k(t),wehave
 1
H =(cid:88)L (cid:18) 1 p2 + Mωk2q2(cid:19), (52) q(cid:101)k(t)= √M(cosωktq(cid:101)k(0)+ωk−1sinωktp(cid:101)k(0)),
2M(cid:101)k 2 (cid:101)k √
k=1 p (t)= M(−ω sinω tq (0)+cosω tp (0)).
(cid:101)k k k (cid:101)k k (cid:101)k
where
Thetime-dependentcanonicalvariablesintherealspaceare
(cid:114)
4K πk  1 (cid:88)
ωk = ω02+ M sin( L )2, k =1,··· ,L, (PBC()5.3) qn(t)=√×M(cid:0)coks,mωφt∗kq(n()0φ)k+(mω)−1sinω tp (0)(cid:1)
k m k k m
For the Dirichlet boundary condition (DBC), the Fourier √
(cid:88)
tmraentrsyf.orHmowisenvoetr,vtahliedFdouueriteortshienberteraaknisnfogromftcraannsblaetdioenfianlesdymas- pn(t)= M k,mφ∗k(n)φk(m)
×(−ω sinω tq (0)+cosω tp (0)),
 L−1 (cid:114) (cid:18) (cid:19) k k m k m
qn =(cid:88)q(cid:101)k L2 sin πLkn , where
k=1 (54) 1
q(cid:101)k =L(cid:88)−1qn(cid:114)L2 sin(cid:18)πLkn(cid:19), φn(k)= √2Le−2πikn/L, (PBC),
n=1 φ (k)= √ sin(πkn/L), (DBC). (59)
n
L
wheren = 1,··· ,L. Forp ,theFouriersinetransformation
n
isdefinedsimilarly. TheHamiltonianinthemomentumspace Therefore,thetimeevolutionofthecovariancematrixis
isidenticaltoEq. (52). Butthefrequencyω hasadifferent
k γ(t)=S(t)γ(0)S(t)T, (60)
form,
wheretheevolutionmatrixis
(cid:114)
4K πk
ωk = ω02+ M sin(2L)2, k =1,··· ,L−1, (DBC). Sn,m(t)=(cid:88)φ∗k(n)φk(m)
(55) k
(cid:32) (cid:33)
The covariance matrix is constructed from the two-point × √√1M cosωkt √1√Mωk−1sinωkt .
− Mω sinω t Mcosω t
correlators k k k
(61)
(cid:18) (cid:19)
(cid:104)0|q q |0(cid:105) (cid:104)0|q p |0(cid:105)
γ =Re n m n m . (56)
n,m (cid:104)0|p q |0(cid:105) (cid:104)0|p p |0(cid:105) The entanglement properties are encoded in the reduced
n m n m
densitymatrix,whichcanbeextractedfromthethecovariance
ForPBC,thecorrelatorsare matrixγ associatedwiththethesubsystemA. Thelogarith-
A
mic negativity is defined by the partial transposition of the
L (cid:20) (cid:21)
1 (cid:88) 1 2πk(n−m) reduceddensitymatrixρ withthesubsystemA = A ∪A
(cid:104)0|q q |0(cid:105)= cos , A 1 2
n m 2L Mω L as E = lnTr|ρT2|. We first consider the partial transposition
k=1 k A
ofγ ,whichcanbeconstructedbyinvertingthesignsofthe
A
L (cid:20) (cid:21)
1 (cid:88) 2πk(n−m) momentacorrespondingtoA [9].
(cid:104)0|p p |0(cid:105)= Mω cos , 2
n m 2Lk=1 k L γT2 =(cid:18) IlA 0lA (cid:19)·γ ·(cid:18) IlA 0lA (cid:19), (62)
(cid:104)0|q p |0(cid:105)=iδ /2. (57) A 0 R A 0 R
n m n,m lA A2 lA A2