Evolution of entanglement entropy in the D1-D5 brane system

0.86 MB·English

by  Curtis T. Asplund

NSF-KITP-11-179
IMSc/2011/8/7
Evolution of entanglement entropy in the D1–D5 brane system
Curtis T. Asplund1,∗ and Steven G. Avery2,3,†
1Dept. of Physics, University of California, Santa Barbara, California 93106, USA
2The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai, India 600113
3KavliInstituteforTheoreticalPhysics,UniversityofCalifornia,SantaBarbara,California93106,USA
We calculate the evolution of the geometric entanglement entropy following a
local quench in the D1D5 conformal field theory, a two-dimensional theory that
describesaparticularboundstateofD1andD5branes. Thequenchcorrespondsto
a localized insertion of the exactly marginal operator that deforms the field theory
2 off of the orbifold (free) point in its moduli space. This deformation ultimately
1 leads to thermalization of the system. We find an exact analytic expression for the
0
entanglemententropyofanyspatialintervalasafunctionoftimeafterthequench
2
and analyze its properties. This process is holographically dual to one stage in the
n
a formationofastringyblackhole.
J

6

]
h
I. INTRODUCTION
t
-
p
e Consider an initial, smooth configuration of matter that collapses into a black hole.
h
There are longstanding questions about how the information in the initial configuration,
[

such as the entanglement between various subsystems, becomes encoded in the result-
3
v ing black hole. As a quantum theory of gravity, string theory addresses many of these
0 questions. While the AdS/CFT correspondence leads immediately to the proposal that
1
certainblackholesaredualtothermalmixedstatesofadualconformalfieldtheory(CFT)
5
2 [1],thissayslittleabouttheformationprocess. Togofurthertowardansweringsuchdy-
.
8 namicalquestions,oneneedstostudytheunitaryevolutionofaCFTwithagravitational
0 dual, undergoing thermalization. In this paper we begin such an investigation in the
1
D1D5 CFT, which describes a bound state of D1 and D5 branes, and is well-known as a
1
: usefulsystemforstudyingblackholesinstringtheory.
v
i Westudythisprocessusingtheentanglemententropy,definedbelow,whichmeasures
X
entanglementbetweensubsystemsinaquantumsystem. Asopposedtothemanystudies
r
a ofblackholeentropyasentanglemententropyofdifferentpartsofthebulkspacetimeor
between different boundary CFTs ([2] reviews many of these), we are considering the
evolution of the entanglement entropy of subsystems of a single CFT, in order to study
itsthermalization. Theconnectionbetweenquantumentanglementandthermodynamics
has a long history. See [3] for a short review, [4, 5] for relevant early investigations and
[6–10]forrecentgeneralresults.
Asabriefreview,beginwithaquantumsystemwithHilbertspaceHandHamiltonian
H. Then factorize, or coarse-grain, the Hilbert space of the full system as H = H ⊗H ,
A E
where the first factor contains all states describing degrees of freedom in a subsystem
A and the second factor all states for the exterior of A—the “environment” E. We call
a quantum system, in a state specified by a density matrix ρ, thermalized if for general
∗[e-mail]
†[e-mail]
2
small subsystems A the reduced density matrices ρ , obtained by tracing over H , are
A E
approximately(Gibbs/canonicalensemble)thermalmixedstates,e.g.,
e−βHA
ρ ≈ (1.1)
A
Z
A
for some β (the inverse temperature), where H = Tr H and Z = Tr e−βHA. This is an
A E A A
admittedly imprecise definition of “thermalized,” but is sufficient for our purposes. The
expression in (1.1) is appropriate for those cases where the Hamiltonian is the only con-
served quantity. For systems with more symmetries, including integrable systems with
infinitely many conserved quantities, one can still define generalized Gibbs ensembles
and appropriate thermal reduced density matrices, characterized by generalized chemi-
cal potentials in addition to β [6, 8]. This is relevant for CFTs like the one we consider,
which undergo a quench but which subsequently evolve as free theories and are charac-
terizedbyasetofmomentum-dependenttemperatures[10,11].
We are concerned here with the case that the full system AE is in a pure state, i.e., a
closed quantum system. Then the fact that ρ is mixed comes entirely from the entan-
A
glement of A with E. This entanglement can be measured in a variety of ways, but in
this paper we consider the Re´nyi and von Neumann entropies, which are given by (3.3)
below. We choose these quantities for a number of reasons, including nice analytic prop-
erties and calculability, to be discussed in the paper. Here we just emphasize that they
let one track the thermalization of various subsystems as well as deviations from strictly
thermal behavior (see [12] §8.2 for an introductory discussion of this topic). The entan-
glemententropies,asafunctionofsubsystem,time,andotherparameters,canalsoyield
much other information about the system. This is explored in voluminous recent work
incondensedmatterphysics(see[13]forseveralrecentreviews).
One can use these quantities, in principle, to investigate the recently conjectured ther-
malization time for black holes that saturates a causality bound [14], although we don’t
get that far in this paper. As we discuss below, it is technically difficult to quantitatively
compare the nonequilibrium dynamics we study here to the system in equilibrium and
the full thermalization process. However, we can still learn a lot from the results we
present.
Motivated by the rapid thermalization observed in heavy ion collisions as well as
the theoretical questions already mentioned, there are many investigations that use
AdS/CFT to study strongly coupled CFTs far from equilibrium or undergoing ther-
malization, e.g., [15–21]. Some of these [22–25] also use entanglement entropy via the
holographic entanglement entropy proposal [26, 27] (see also [28] for a recent study of
the holographic Re´nyi entropies). These latter investigations use the dual classical ge-
ometryandsocannotaddresssomeofthemostpuzzlingquestionsaboutinformationin
blackholes,whichinvolvequantummechanicsofthebulktheoryinanessentialway.
We study the D1D5 CFT at weak coupling, which has long been used in string theory
to study black holes [29, 30]. Early studies focused on the extremal, zero-temperature
configurations, whereas we consider exciting to a state that is far from extremal. Recent
workhasstudiedHawkingradiationinthissystemindetail[31–35]anddeformationsof
the CFT away from the orbifold (free) point in the moduli space [36–38] (see also [39, 40]
for leading order calculations away from the orbifold point). Because we work at weak
coupling, i.e. near the orbifold point, we are not in the regime where supergravity is
a good approximation. Nonetheless the above work indicates that this regime contains
3
much information about black holes. Additional support for this includes several pre-
cise matchings between gravitational calculations and calculations from the free theory
data [41–43]. More evidence for the surprising efficacy of the orbifold CFT in describing
black holes comes from the very recent paper [44]. There are also general arguments for
thermalization of CFTs with gravity duals in the large N limit at any finite value of the
coupling [45] corroborated by exact calculations in simplified models [46, 47]. The re-
sulting weakly coupled thermal state is dual to a “stringy black hole,” in that the string
length is large compared to the size of the black hole (see [45] for further discussion of
stringy black holes). The above work indicates that such a state should also tell us about
traditionalblackholes.
In particular, the results illustrated in §VI have a natural explanation in terms of free
excitations traveling at the speed of light around the S1. This picture is similar to the
CFT description in [48, 49] of near-extremal supergravity excitations, which, when the
decoupling limit is relaxed, can periodically escape the AdS throat to the asymptotic flat
space. The period and rate of emission were reproduced from the same kind of CFT
dynamics we observe. Thus our results correctly capture some qualitative aspects of
the supergravity description. On the other hand, we expect that large energy (far from
extremal) supergravity excitations can back-react to form black holes. In the CFT, this
corresponds to thermalization. Since the entanglement entropy that we find does not
persist, but rather has short-time periodic dynamics, we conclude that, as expected, the
orbifoldCFTdoesnotcapturethisimportantprocess. Wehopetoaddressthisissuemore
quantitativelyinfutureworks.
A closely related precursor to our work is [50], which also calculates the evolution
of entanglement entropy in a weakly coupled CFT. The authors proposed this as a way
to study quantum black hole formation and emphasized that the entanglement entropy
can be thought of as a coarse-grained thermodynamic entropy. However, their CFT (a
single fermion) has no clear dual black hole interpretation, although it does illustrate
somegeneralfeaturesofthekindofproblemweareconsidering.
The general process of thermalization in weakly coupled theories is well studied, but
the D1D5 system exhibits some novel features. In particular, we consider dynamics aris-
ing from a localized insertion of a particular marginal deformation of the orbifold CFT
thatactsasalocalquench,tobedescribedbelow. Thecalculationoftheentropyproduced
by this quench is the main result of this paper. This is the basic process by which en-
tropy is generated. [Note added in proof: After this article was written we learned of
[51], which contains calculations of similar quantities and appears consistent with our
results.]
As emphasized in [45], the familiar semiclassical dynamics of black holes, including
thepuzzlingapparentlossofinformation,shouldappearinthelimitN → ∞ofthedual
holographictheory. HerethatwouldcorrespondtothelimitofinfinitelymanyD1andD5
branes. We do not consider that limit here, rather we consider a finite number of branes
unitarily evolving toward a thermalized pure state as described above. In this paper we
justanalyzethebasicprocessinthatevolution.
In §II we review the D1D5 CFT and the marginal deformations that we study. In §III
we set up the calculation of the entanglement entropies in the CFT, including a review
of the replica trick. Next, in §IV we compute the four-point function that we need to
calculate the entropies, which we do in §V. We illustrate some of their properties in §VI.
Weconcludewithadiscussionofourresultsandfuturedirections.
4
II. D1D5REVIEW
The D1D5 system is realized in IIB string theory compactified on1 T4 × S1 with the
bound state of N D1-branes wrapping the S1 and N D5-branes wrapping T4 ×S1. We
1 5
take the S1 to be large compared to the T4. The near-horizon limit of the geometry is
AdS ×S3×T4,whichisdualtoatwo-dimensionalCFTlivingontheboundaryofAdS .
3 3
Thetwo-dimensionalD1D5CFThasN = (4,4)supersymmetrywithSU(2) ×SU(2)
L R
R-symmetrycorrespondingtotheisometryoftheS3. Thetwo-dimensionalbasespaceof
the CFT is given by the cylindrical boundary of AdS parametrized by time and the S1.
3
In addition, we can organize the field content using the SO(4) (cid:39) SU(2) ×SU(2) sym-
I 1 2
metry broken by the compactification on T4. One can also fix the total central charge
c = 6N N from the algebra of diffeomorphisms that preserve the asymptotic AdS .
1 5 3
The CFT has a twenty-dimensional moduli space that corresponds to the near-horizon
twenty-dimensionalmodulispaceoftheIIBsupergravitycompactification.
There is a point in moduli space called the “orbifold point,” analogous to free SYM
in AdS /CFT , where the D1D5 CFT is a sigma model with orbifolded target space,
5 4
(T4)N1N5/S . JustasthedualoffreeSYMdoesnothaveagoodgeometricdescription,
N1N5
theorbifoldCFTisfarfrompointsinmodulispacethatarewelldescribedbysupergrav-
ity. We wish to study the effect of certain (4,4) exactly marginal deformations that move
theorbifoldCFTtowardpointsinmodulispacethatshouldhavegeometricdescriptions
and, in particular, should include black hole physics. Even though we work far from the
supergravity regime, as discussed in the introduction, we can still capture some black
holephysics.
WecanthinkoftheorbifoldmodelasN N copiesofa(4,4)c = 6CFT.Eachcopyhas
1 5
four real bosons that are vectors of SO(4) , Xi, and their fermionic superpartners. See,
I
e.g., [52] for details. For computational purposes, we map the real cylinder coordinates,
t ∈ Randy ∈ [0,2πR),todimensionlesscomplexcoordinatesonthecylinder
t y
w = τ +iσ (cid:55)→ −iτ = θ. (2.1)
R R
Note that we have also incorporated a Wick rotation in this step. We prefer to perform
mostofthecomputationinthecomplexplanebyfurthermappingtocoordinates
z = ew z¯= ew¯. (2.2)
Inadditiontothelocalbosonicandfermionicexcitationsofeachcopy,theorbifoldthe-
oryalsohastwistedsectors: stateswhichcomebacktothemselvesonlyuptoanelement
of the orbifold group S upon circling the S1. The twist operators σ (z) are labeled
N1N5 n
by n-cycles and change the twist sector of the theory. More concretely, consider opera-
tors O(i)(z) in the ith copy. In the presence of σ (z ), the operators have boundary
(12...n) 0
conditions

O(i+1)(z +z) i = 1,...,n−1
 0
O(i)(z +ze2πi) = O(1)(z +z) i = n . (2.3)
0 0

O(i)(z +z) i = n+1,...,N N
0 1 5
1OnemayalsoconsiderK3insteadofT4.
5
Letusemphasizethatthetwistoperatorsconsideredherearephysicalcomponentsofthe
orbifold CFT, and should not be confused with twist operators introduced as part of the
replicatrick.
Following [36–38], we focus on four of the marginal deformations that involve twist
operators. These operators are believed to be responsible for thermalization in the D1D5
CFT. The (4,4) supersymmetric deformations are singlets under SU(2) × SU(2) . To
L R
obtain such a singlet we apply modes of the supercharges G∓ to σ±, where we use
A˙ 2
plus/minusindicestolabelelementsofSU(2) doubletsanddottedcapitalLatinindices
L
fordoubletsofSU(2) . In[36]itwasshownthatwecanwritethedeformationoperator(s)
2
as
(cid:90) (cid:90)
(cid:104) dw (cid:105)(cid:104) dw¯ (cid:105)
O(cid:98)A˙B˙(w0) = 2 2πiG−A˙(w) 2πiG¯−B˙(w¯) σ2++(w0), (2.4)
w0 w¯0
wherethefactorof2normalizestheoperator. Theoperatorσ+ isnormalizedtohaveunit
2
OPEwithitsconjugate
1
σ (z(cid:48))σ+(z) ∼ . (2.5)
2,+ 2 z(cid:48) −z
ThisimpliesthatactingontheRamondvacuum[36]
σ+(z)|0−(cid:105)(1)|0−(cid:105)(2) = |0−(cid:105)+O(z). (2.6)
2 R R R
Here |0−(cid:105) is the spin down Ramond vacuum of the CFT on the doubly wound circle
R
produced after the twist. The normalization (2.6) has given us the coefficient unity for
the first term on the RHS and the O(z) represent excited states of the CFT on the doubly
woundcircle.
III. SETUP
Let us now outline the precise calculation we perform. Since we are interested in
the dynamics of thermalization or scrambling in the CFT, we quench the system and
then look at the entanglement entropy of spatial subsystems as a function of time. The
entanglemententropy ofsubsystems, asdiscussed, isavery naturalquantity toexamine
whendiscussingthermalization. Happily,thereisalreadysomeconsiderabletechnology
for computing the entanglement entropy after both global and local quenches in two-
dimensionalCFTs[53,54].
Thespecificquenchweconsiderisalocalinsertionofthedeformationoperatorintro-
ducedabove. Sincethisoperatorisbelievedtoberesponsibleforthermalization,itseems
natural to consider the dynamics after its application. Moreover, these results should tie
instronglywithpreviousinvestigations[36–38],whichshowedthatthedeformationop-
erator, in essence, effects a Bogolyubov transformation. For instance, the deformation
operator,whenactingonthevacuum,producesasqueezedstateoftheform[36]
(cid:34) (cid:35)
σ+(z )|0−(cid:105)(1)|0−(cid:105)(2) = exp −1 (cid:88)γB α αAA˙ +(cid:88)γF ψ+Aψ− |0−(cid:105). (3.1)
2 0 R R 2 mn AA˙,−m −n mn −m A,−n R
m,n m,n
In this equation, we only show the left (holomorphic) sector and consider just the twist
part of the deformation. On the left-hand side we have the two-twist operator acting
6
on the untwisted Ramond vacua, which produces many pairs of bosonic and fermionic
excitations on the two-twisted Ramond vacuum. Note that the Ramond vacua have an
SU(2) × SU(2) spin structure. The coefficients γB and γF are functions of z given
L R 0
explicitly in [36]. The calculation we propose, then, computes the time dependence of
theentanglemententropyofthissqueezedstate;although,wedonotusetheaboveform
explicitly.
The physical setup is as follows. We apply the deformation operator (2.4) to the Ra-
mond vacuum at time t = 0 and y = 0. We then look at the time dependence of the
entanglement entropy of an arbitrary spatial interval. Since we are mostly interested in
how the entanglement entropy changes due to the quench, we subtract off the entangle-
mententropyofthevacuum. WeactintheRamondsectorsincethatisthesectorrelevant
for black holes. We will sketch the gravitational picture this corresponds to in the final
section.
III.1. Reviewofthereplicatrickforcomputingentanglemententropy
Intheremainderofthissection,wesetupthecalculationoftheentanglemententropy
after the quench. Consider a system S with some subsystem A, and its complement B.
Recall that the (von Neumann) entanglement entropy of A in S is defined as the von
Neumannentropyofthereduceddensitymatrix,
S(A) = −Tr ρˆ logρˆ ρˆ = Tr ρˆ . (3.2)
A A A A B S
Thedensitymatrixρˆ isthedensitymatrixforthefullsystemS. If,asistruethroughout
S
this paper, the total system S is in a pure state |ψ(cid:105), then ρˆ = |ψ(cid:105)(cid:104)ψ|. For our calculation,
S
the subsystem A corresponds to degrees of freedom living on some interval of S1. This
definition has a number of nice properties that make it the natural measure of entangle-
ment including positive definiteness, strong subadditivity, and S(A) = S(B) for a pure
state. In fact, this is essentially the unique measure of entanglement satisfying the above
properties[55].
Computing the von Neumann entanglement entropy is computationally difficult be-
cause of the log, so instead we follow [56–58] and use the replica trick: we first compute
theRe´nyientropyofordernandthenanalyticallycontinuetothevonNeumannentropy.
RecallthattheRe´nyientropyofordernisdefinedas
1
S (A) = log(Tr ρˆn) and S (A) = limS (A). (3.3)
n 1−n A A vN n→1 n
The Re´nyi entropies are an interesting measure of entanglement even before taking the
limit to the von Neumann entanglement entropy. In particular, they serve as a lower
boundonS andvanishonanunentangledstate.
vN
Before showing how to compute the Re´nyi entropy, we first review how to write the
density matrix ρˆ as a path integral. From there we can easily compute Trρn as a path
S A
integral with twisted boundary conditions. Let us work in some basis with states that
we will write as |ϕ(cid:105); it is perhaps most natural to think of these as shape states (field
eigenstates), but any basis works. Then, the ϕ –ϕ element of the density matrix at time
1 2
T canbewrittenas
(cid:104)ϕ |ρˆ(T)|ϕ (cid:105) = (cid:104)ϕ |ψ(T)(cid:105)(cid:104)ψ(T)|ϕ (cid:105)
2 1 2 1
7
= (cid:104)ϕ |e−iHˆT |ψ (cid:105)(cid:104)ψ |eiHˆT |ϕ (cid:105)
2 0 0 1
= (cid:104)ψ |eiHˆT |ϕ (cid:105)(cid:104)ϕ |e−iHˆT |ψ (cid:105), (3.4)
0 1 2 0
where we have suggestively switched the order of the two amplitudes for reasons that
should become clear. The state |ψ (cid:105) is the state at t = 0, which for us is the state imme-
0
diately after the quench. We have the product of two amplitudes, each of which can be
writtenasaseparatepathintegral;however,itismorefruitfultothinkofthisasonepath
integralwithdiscontinuousintermediateboundaryconditions. Morespecifically,
(cid:104)ϕ2|ρˆ(T)|ϕ1(cid:105) = (cid:90) Dφ(t)(cid:12)(cid:12)(cid:12)φ(0)=ϕ2 ei(cid:82)−0TdtL(φ(t))(cid:90) Dφ(t)(cid:12)(cid:12)(cid:12)φ(−T)=ψ0ei(cid:82)0−TdtL(φ(t))
(cid:12) (cid:12)
φ(−T)=ψ0 φ(0)=ϕ1
(cid:90) (cid:12)
(cid:12) (cid:82)
= Dφ(t)(cid:12) ei CdtL(φ(t)), (3.5)
(cid:12)
BCs
where “BCs” in the last line indicates the boundary conditions from the previous line,
and the contour C starts at t = −T goes to t = 0 and then backwards to t = −T. We
see, then, that we can think of the density matrix as a path integral which accepts two
boundary conditions at t = 0. We have translated the ψ boundary condition down to
0
t = −T tomatchwithpreviouscalculations[53,54].
While this is formally correct, there are a couple of subtleties to address. First of all,
we should clarify what we mean by the above path integral, since as written we need
double-valuedfields. ItismoreprecisetoparametrizeC as
(cid:40)
s s ∈ [−T,0]
t = s ∈ [−T,T], (3.6)
−s s ∈ (0,T]
inwhichcasetheactioninthepathintegralbecomes
(cid:90) (cid:90) 0 (cid:90) T
dtL(φ(t)) = dsL(φ(s))− dsLT(φ(s)), (3.7)
C −T 0
and φ is single-valued on s. We put the superscipt T on L in the second term to indicate
that it is the time-reversed Lagrangian. The second issue we need to address is Wick-
rotating the path integral. We usually Wick-rotate the path integral to imaginary time to
maketheoscillatorytermiS intoaconvergent−S ;however,wenowhaveaminussign
E
between the two terms in (3.7), which means that we should Wick-rotate the two terms
oppositely, see Fig. 1. When we Wick-rotate the second part in the opposite direction,
we get rid of the minus sign and the time-reversal: we get a smoothly defined Euclidean
pathintegral
(cid:90) (cid:12) (cid:18) (cid:90) τ (cid:19)
(cid:12) f
Z(τ ,τ ;ψ ;ϕ ,ϕ ) = Dφ(τ)(cid:12) exp − dτ L (φ(τ))
0 f 0 1 2 E
(cid:12) (3.8)
BCs τ0
BCs: φ(τ ) = φ(τ ) = ψ , φ(0−) = ϕ , φ(0+) = ϕ .
0 f 0 2 1
We can compute this path integral for τ < 0 < τ with real τ and τ , and finally analyt-
0 f 0 f
icallycontinuetothedesiredmatrixelementvia
(cid:104)ϕ |ρˆ|ϕ (cid:105) = lim N Z(−iT −(cid:15),−iT +(cid:15);ψ ;ϕ ,ϕ ), (3.9)
2 1 (cid:15) 0 1 2
(cid:15)→0+
8
τ
ψ φφ ψ
0 2 1 0
τ 0-0+ τ
0 f
-iT ± ε
FIG. 1. The contour along the real axis of the complex τ-plane for the Euclidean path integral.
TheanalyticcontinuationbacktoLorentziantimeisshowninlightgray,whichshowshowthe(cid:15)
regularizationarises.
where we put in (cid:15) to “remember” which direction we Wick-rotated the two terms. The
factorofN isanormalizationconstantthatensuresTrρˆ= 1,
(cid:15)
(cid:90)
1
= (cid:104)ψ |e−(2(cid:15))Hˆ |ψ (cid:105) = DφZ(−iT −(cid:15),−iT +(cid:15);ψ ;ϕ,ϕ). (3.10)
0 0 0
N
(cid:15)
The limit as (cid:15) → 0 is both delicate and crucial to getting the right physics, since φ(τ) has
abranchcutalongthenegativeimaginaryaxis. Later,itshouldbecomeclearthat(cid:15)plays
theroleofaUVcutoff.
Beforecontinuing,letusremarkthattheaboveshouldbereminiscentoftheSchwinger–
Keldysh, or closed time path, formalism with temperature T = 1/(2(cid:15)) (see, e.g., [59] for
a review of this formalism). Indeed, if one integrates over ψ , then it is exactly the
0
Schwinger–Keldysh formalism, with some insertions at t = 0. Also note that if one iden-
tifies ϕ = ϕ = ϕ and integrates over ϕ, then one computes Trρˆ, which is unity for a
1 2
purestate.
We now have all of the tools to understand how to compute the Re´nyi entanglement
entropyasafunctionoftimeafterthequench. Firstnotethatitshouldnowbeclearhow
tocomputethereduceddensitymatrix(3.2):
(cid:104)a |ρˆ |a (cid:105) = (cid:104)a |Tr ρˆ|a (cid:105)
2 A 1 2 B 1
(cid:90)
(cid:0) (cid:1)
= DbN Z −iT −(cid:15),−iT +(cid:15);ψ ;ϕ = {a ,b},ϕ = {a ,b}
(cid:15) 0 1 1 2 2
B
(cid:0) (cid:1)
≡ N Z −iT −(cid:15),−iT +(cid:15);ψ ;a ,a . (3.11)
(cid:15) A 0 1 2
Here we indicate a field taking values a on A and b on B by {a,b}. We can compute
this quantity from the same path integral in (3.8) with altered boundary conditions at
t = 0. In the region B, we now demand that φ be continuous at t = 0. We started with a
full cut, which we sew together in region B. The boundary conditions on the remainder
determinethematrixelementcomputed. ThismanifoldispicturedinFigure2.
(cid:82)
Now,tocomputeTr ρˆn westartbyinserting Da|a(cid:105)(cid:104)a|inbetweeneachρˆ andthen
A A A
performthetraceinthe|a(cid:105)basis. Thisbecomesndistinctcopiesoftheabovepathintegral
9
a
B A 1
a
2
FIG. 2. The reduced density matrix as a path integral. Note that the flat piece on top is there
for illustrative purposes only. The boundary conditions on the bottom two edges are both |ψ (cid:105),
0
whereastheboundariesinregionAare“inputs”thatdeterminethematrixelementofthereduced
densitymatrix.
withappropriateintegralsoverthea :
i
(cid:90) (cid:90) (cid:90)
Trρˆn = Da Da ··· Da (cid:104)a |ρˆ |a (cid:105)···(cid:104)a |ρˆ |a (cid:105)(cid:104)a |ρˆ |a (cid:105)
A 0 1 n−1 0 A n−1 2 A 1 1 A 0
(cid:90) (cid:90)
= Da ··· Da (N )n Z (τ ,τ ;ψ ;a ,a )···Z (τ ,τ ;ψ ;a ,a ),(3.12)
0 n−1 (cid:15) A 0 f 0 n−1 0 A 0 f 0 0 1
whereτ andτ getanalyticallycontinuedasdescribed. Onecanthenputallofthepieces
0 f
into a single path integral over n replicas, with n-twisted boundary conditions in region
Aconnectingthereplicasandsingly-twistedboundaryconditionsoutsideofA.
III.2. EntanglemententropyintheD1D5CFT
Let us apply the above general discussion to the matter at hand. We need to compute
thetwistedpathintegraldescribedabove. Wecanrewritethepathintegralasacorrelator
of local twist operators that induce the appropriate monodromy, and then compute the
correlator using techniques in [60]. Let us note that there is an extra layer of obfuscation
beyond computations in other CFTs since our quench involves a distinct twist operator
thatispartofthephysicalspectrumoftheCFT.
Wepreparethestate|ψ (cid:105)bystartingwiththevacuumatτ = −∞,evolvingforwardto
0
τ where we insert our quench O(cid:98)(w ) from (2.4). To compute the Re´nyi entropy, we need
0 0
nreplicasof|ψ (cid:105). Thetracein(3.12)isthenproportionaltothefour-pointfunction
0
(cid:10) (cid:11)
W (τ ,τ ;θ ,θ ) = [O†(w )]nσ (w )σ (w )[O(w )]n , (3.13)
n 0 f 1 2 f n 2 n 1 0
where
w = τ w = τ w = iθ w = iθ . (3.14)
0 0 f f 1 1 2 2
SinceeachoftheO’shasa2-twistitbecomesnecessarytoclarifythebranchingstructure
ofthecorrelator. Letusspecifytheindicesofthetwistfieldsinvolvedinthecorrelator:
[σ ]n = σ σ ···σ σ = σ . (3.15)
2 (12) (34) (2n−1,2n) n (135...2n−1)
10
The indices are labels for the 2n sheets involved in the correlator, and we use the par-
enthetical notation for single cycles of S . Note that the bare twist σ introduced by the
n n
replicatricktwistsonecopyfromeachpairoftwistedreplicas. Thisfixesthetopologyof
thecorrelator.
WecannowwritetheRe´nyientanglemententropyas
(cid:18) (cid:19)
1 W (−iT −(cid:15),−iT +(cid:15);θ ,θ )
n 1 2
S (T,θ ,θ ) = − log . (3.16)
n 1 2 1−n [W (−iT −(cid:15),−iT +(cid:15))]n
1
Notethatσ istheidentityoperatorandsothereisnoneedtospecifytheθ andθ forW .
1 1 2 1
Also note that any normalization issues from defining |ψ (cid:105) in terms of the local operator
0
O(w )canceloutbetweenthenumeratoranddenominator.
0
IV. THEFOUR-POINTFUNCTION
The four-pointfunction inEquation (3.13)factorizes into afour-point functionof bare
twist operators that we compute by mapping to a covering space and a correlator of
insertionsinthecoveringspace.
We first compute the correlator of the bare twists, and then treat the non-twist su-
percharge insertions that appear in the covering space. We map the correlator in Equa-
tion (3.13) to the plane via the exponential map (2.2). We will then treat the associated
Jacobianfactorsin§IV.4.
IV.1. Thetwistcorrelator
Letusbegin,then,withjustthetwistpartofthecorrelator
(cid:104)[σ (z )]nσ (z )σ (z )[σ (z )]n(cid:105). (4.1)
2 f n 3 n 2 2 0
NotethatthispartofthecorrelatorappliestoanyCFTwithtwocopiesthataresuddenly
joinedbyσ . Thuswhendiscussingthebaretwistresults,wekeepcthecentralchargeof
2
a single copy. The SL(2,C) symmetry determines the form of the 4-pt function up to an
arbitraryfunctionofthecross-ratio. Therefore,wecancomputethe4-ptfunction
F (u) = (cid:104)[σ (∞)]nσ (u)σ (1)[σ (0)]n(cid:105), (4.2)
n 2 n n 2
andthenfindthe4-ptfunctionofinterestinEquation(4.1).
To compute the four-point function we need to find a map to the covering space, and
thencomputetheLiouvilleactionassociatedwiththemap[60]. Fortunately,AppendixD
of [52] gives an explicit formula for spherical genus correlation functions of S -twist
N
operators as a function of the coefficients of the map. Once we find the map, we can
makeuseoftheformulatoavoidcomputingtheLiouvilleactiondirectly.
Let us now list the properties the map z = z(t) from the z-plane to the t-plane must
have,asdeterminedbytheindexstructureshowninEquation(3.15). First,onecanshow
fromtheRiemann–Hurwitzformulathatthecoveringspacemusthavesphericalgenus:
1 (cid:88) 1
g = r −s+1 = [1·n+(n−1)+(n−1)+1·n]−(2n)+1 = 0, (4.3)
i
2 2
i