Monodromy methods for torus conformal blocks and entanglement entropy at large central charge
PPrepared for submission to JHEP
Monodromy methods for torus conformal blocks andentanglement entropy at large central charge
Marius Gerbershagen
Institut für Theoretische Physik und Astrophysikand Würzburg-Dresden Cluster of Excellence ct.qmat,Julius-Maximilians-Universität Würzburg, Am Hubland, 97074 Würzburg, Germany
E-mail: [email protected]
Abstract:
We compute the entanglement entropy in a two dimensional conformal fieldtheory at finite size and finite temperature in the large central charge limit via the replicatrick. We point out that the correlation function of twist operators commonly used to calculateentanglement entropy is not applicable to the torus case and it is necessary to compute thefull partition function on the replica surface. We first generalize the known monodromymethod for the calculation of conformal blocks on the plane to the torus. Then, we derivea monodromy method for the zero-point conformal blocks of the replica partition function.We explain the differences between the two monodromy methods before applying them tothe calculation of the entanglement entropy. We find that the contribution of the vacuumexchange dominates the entanglement entropy for a large class of CFTs, leading to universalresults in agreement with holographic predictions from the RT formula. While the twistcorrelator agrees with the replica partition function and the RT formula for small intervals orlow temperatures, it cannot reproduce the entanglement entropy phase transition encounteredfor large intervals and high temperatures. a r X i v : . [ h e p - t h ] F e b ontents – 1 – Introduction
Entanglement entropy is a measure for the amount of entanglement between two parts of aquantum system. It is defined as the von Neumann entropy of the reduced density matrix ρ A for a subsystem A . In general, the entanglement entropy depends on details of the theoryand state in question such as the spectrum and operator content. However, certain universalfeatures are common to all quantum field theories. For example, the leading order divergencein the UV cutoff usually scales with the area of the boundary of the subregion A [1, 2]. Con-formal field theories in two dimensions admit more general universal features. In particular,the entanglement entropy of a single interval A at zero temperature is given by [3] S A = c l/(cid:15) UV ) , (1.1)depending only on the central charge, irrespective of any other details such as the OPEcoefficients or the spectrum of the theory. For subsystems A consisting of multiple intervals,the entanglement entropy is no longer universal for any CFT. However, as shown in [4],in the semiclassical large central charge limit and at zero temperature, the entanglemententropy again becomes universal for a large class of conformal field theories. These CFTs arecharacterized by a sparse spectrum of light operators and at most exponentially growing OPEcoefficients. By using conformal transformations, the universal results of [3, 4] translate tothe case of either finite temperature or finite size. This publication is dedicated to the studyof universal features of the entanglement entropy in a system with both finite size and finitetemperature.The computational approach most commonly used to determine entanglement entropiesis the replica trick. It is based on the calculation of the Rényi entropies S ( n ) A = 11 − n log Tr ρ nA , (1.2)via a partition function Z n on a higher genus Riemann surface R n obtained by gluing n copies of the complex plane cyclically together along the entangling interval A . This parti-tion function is then mapped to a correlation function of twist operators. For a subsystem A consisting of N disjoint intervals, the correlation function contains 2 N twist operator inser-tions at the endpoints of the N intervals. Finally, the entanglement entropy is obtained byanalytically continuing n to the real numbers and taking the limit n →
1. The universalityof the entanglement entropy for a single interval follows immediately from the universalityof the two-point function in any conformal field theory [3]. For multiple intervals, the Rényientropy is mapped to a higher-point correlation function of twist operators which decomposesinto a sum over conformal blocks. The universality in the semiclassical limit observed in [4] isexplained by the fact that only a single conformal block (the vacuum block) contributes to theentanglement entropy. More precisely, in the semiclassical limit of large central charge, thecontribution of other conformal blocks is exponentially suppressed in the central charge, as-suming the aforementioned restrictions on the theory, i.e. a sparse spectrum of light operatorsand at most exponentially growing OPE coefficients.– 2 –n the case of a system with both finite size and finite temperature, the replica trickinstructs us to calculate the partition function on a higher genus surface constructed bygluing n copies of the torus along the entangling interval A . It is commonly assumed thatthe partition function on this higher genus surface is equal to the correlation function of twistoperators on the torus, and a number of explicit calculations of the entanglement entropyhave been performed using this approach (see e.g. [5–8]). However, there is a simple argumentshowing that this cannot be correct. It is well known that for a pure state ρ , the entanglemententropy for A is equal to the entanglement entropy of the complement A c , S A = S A c . However,for mixed states such as the thermal states described by the CFT on the torus, this propertyno longer holds. Since the correlation function of twist operators contains no informationabout whether we compute the entanglement entropy for A or for A c (the location of thebranch cuts between the twist operators is not fixed by the twist correlator), it cannot givethe correct answer on the torus. In the case of a free CFT, a discrepancy between the twistoperator result and the higher genus partition function has already been observed in [9, 10]based on previous work [11–13]. In particular, it was found in [9] that the twist operatorresult is not modular covariant and violates Bose-Fermi equivalence.A convenient way to calculate the entanglement entropy on the plane works by expand-ing the twist correlator in conformal blocks and calculating these using the well-known mon-odromy method first described in [14]. This monodromy method is derived as follows. Byinserting a degenerate field into the correlation function, one obtains a differential equationfor an auxiliary function Ψ( z ). This differential equations contains derivatives of the soughtafter conformal block as accessory parameters. The accessory parameters are fixed by de-manding a certain monodromy of the solution Ψ( z ) around cycles which encircle a numberof operator insertion points. Which insertion points are encircled depends on the channel inwhich the correlation function is expanded. We review this method in detail in sec. 2.1 beforegeneralizing it to the case of finite temperature and finite size.A different perspective on this method was offered in [15], where it was related to auniformization problem on the replica surface R n . In general, a compact Riemann surfaceΣ can be obtained as the quotient of the complex plane by a subgroup of P SL (2 , C ) [16].Thus, there exists a single valued map w → z from the complex plane to R n . This uni-formization map is given as the quotient w = Ψ ( z ) / Ψ ( z ) of two independent solutions of adifferential equation. It turns out that this differential equation is equal to the one from themonodromy method for the conformal blocks of the twist correlator on the plane [15]. Usingthe equivalence between the CFT partition function and the gravitational action in the dualAdS space, this yields a proof of the RT formula at zero temperature [15]. We use similararguments from the uniformization problem on the replica surface of the torus to determinea monodromy method for the zero-point conformal blocks of the replica partition functionat finite temperature. This new monodromy method is closely connected to the monodromymethod for the conformal blocks of the twist correlator on the torus, with the crucial differ-ence being that the new method allows for choosing a larger set of cycles around which toimpose the monodromy. These new cycles are necessary to reproduce the S A = S A c property– 3 –or thermal states.In the context of the AdS/CFT correspondence, universal features of the entanglemententropy for holographic CFTs at large central charge are predicted by the Ryu-Takayanagiformula (RT formula for short) [17]. The RT formula states that the entanglement entropyof a subregion A in the boundary field theory corresponds to the area of a minimal surface γ A in the bulk, anchored at ∂A on the boundary of the AdS space. Apart from reproducingthe universal results obtained in [3, 4], the RT formula also predicts interesting universalfeatures of the entanglement entropy in the case of both finite temperature and finite size.In particular, there are two phase transitions [5]. First, there is a Hawking-Page transitionin the bulk from thermal AdS to the BTZ black hole phase as the temperature increases.This induces a corresponding phase transition in the entanglement entropy. Second, theentanglement entropy in the BTZ phase also shows a phase transition as the size of theentangling interval increases. We explain how these features appear from the CFT side.Related work on the entanglement entropy in conformal field theories at finite size andfinite temperature includes [18–22]. [18–20] is concerned with the entanglement entropy invarious limits of high and low temperature or size of the entangling interval A , in whichcase universal results for arbitrary values of the central charge can be obtained. In [21, 22],the holographic entanglement entropy for a single entangling interval on the boundary of athermal AdS space and the BTZ black hole was calculated using a monodromy method onthe gravity side. The monodromy method derived from the CFT side in this publication willturn out to be equivalent to the monodromy method on the gravity side used in [21, 22].Related work on torus conformal blocks includes [23] which derived the monodromy methodfor the special case of one-point Virasoro conformal blocks on the torus and [24, 25] whichperformed explicit calculations of one- and two-point Virasoro conformal blocks in variouslimits including the semiclassical one which we study in this publication.Our paper is organized as follows. In sec. 2, we derive the monodromy methods used inthis publication. After a review of the standard monodromy method for conformal blocks onthe plane in sec. 2.1, we generalize to torus conformal blocks in sec. 2.2. Sec. 2.3 explains howto obtain a monodromy method for zero-point conformal blocks of the partition function onthe replica surface and the differences between it and the monodromy method for conformalblocks on the torus. Following this, we apply the newly derived monodromy methods tothe calculation of the entanglement entropy in sec. 3. Assuming that the vacuum exchangedominates the partition function on the higher genus Riemann surface, we find universalresults in agreement with the RT formula. For the partition function on the replica surface,we find in particular agreement with the phase transition in the entanglement entropy atlarge interval size and high temperature. This feature cannot be reproduced from the twistcorrelator. We check the assumption on the dominance of the vacuum exchange numericallyin sec. 3.4. Finally, we conclude with a brief discussion and outlook in sec. 4.– 4 – Monodromy methods
This section contains an overview over the monodromy methods used in this publication. Westart with a review of the standard monodromy method for conformal blocks on the plane,then turn to the case of conformal blocks on the torus and finally explain how to derivea monodromy method for zero-point blocks of the partition function on the replica surfacerelevant to the computation of entanglement entropy on the torus.
In this section, we review the monodromy method for the calculation of four-point semiclas-sical conformal blocks on the plane first derived in [26] (see also [27] for a more detailedexplanation). The starting point of the derivation is the correlation function of four primaryfields O i hO ( z , ¯ z ) O ( z , ¯ z ) O ( z , ¯ z ) O ( z , ¯ z ) i . (2.1)Let us parametrize the central charge as c = 1 + 6( b + 1 /b ) and take the semiclassical limit c → ∞ , b → h i of the operators O i as well as the internalconformal weight h p scale proportional to the central charge. In the correlation function (2.1),we insert a degenerate operator Ψ( z, ¯ z ) with conformal weight h Ψ = − / − b / ∼ O ( c )obeying (cid:18) ˆ L − + 1 b ˆ L − (cid:19) Ψ( z, ¯ z ) = 0 (2.2)From the conformal Ward identities, this yields the following differential equation known asthe decoupling equation, " b ∂ z + X i (cid:18) h i ( z − z i ) + ∂ z i z − z i (cid:19) hO O Ψ O O i = 0 . (2.3)To get to the s -channel conformal block, we then insert the operator product expansion O ( z , ¯ z ) O ( z , ¯ z ) = X p C p X k, ¯ k ( z − z ) h p − h − h + | k | (¯ z − ¯ z ) ¯ h p − ¯ h − ¯ h + | ¯ k | β pk β p ¯ k O { k, ¯ k } p ( z , ¯ z )(2.4)into the correlation function which yields terms containing hO { k, ¯ k } p Ψ O O i . At large centralcharge, these terms can be approximated by hO { k, ¯ k } p Ψ O O i ≈ Ψ p hO { k, ¯ k } p O O i , (2.5)where Ψ p is defined by Ψ p = hO p Ψ O O ihO p O O i . (2.6)This can be shown by employing the form of hO { k, ¯ k } p Ψ O O i in terms of a string of differentialoperators L k i , ¯ L k i acting on hO p Ψ O O i , where L Ψ − k i = − X j =3 , , Ψ (1 − k i ) h j ( z j − z ) k i + 1( z j − z ) k i − ∂ z j ! . (2.7)– 5 –ow hO p Ψ O O i scales as e − c/ S cl. in the semiclassical limit where S cl. ∼ O ( c ) while Ψ p ∼O ( c ) and h Ψ ∼ O ( c ). Hence, we can neglect the derivatives acting on Ψ p and on the h Ψ term to obtain (2.5) in the leading order in c . Now, use a conformal transformation to send z → z → z → ∞ and z to the cross ratio x . This implies hO O Ψ O O i ≈ X p Ψ p ( z, x, ¯ x ) C p C p F p , ( x ) ¯ F p , (¯ x ) , (2.8)where F p , ( x ) is the desired conformal block which in the semiclassical limit scales as F p , ( x ) ∼ e − c f cl. ( x ) as was conjectured in [26] and recently shown in [28]. The semiclassicalconformal block f cl. depends only on the cross ratio x and on b h i , b h p . The decouplingequation (2.3) then yields at leading order in c " ∂ z + X i b h i ( z − z i ) − ∂ z i f cl. ( x ) z − z i ! Ψ p = 0 . (2.9)There is one separate decoupling equation for each term in the sum over p since generically,each term has a different monodromy and thus must vanish separately. All terms involvingderivatives of Ψ p vanish to leading order due to Ψ p ∼ O ( c ). From the expression for thecross ratio x = ( z − z )( z − z )( z − z )( z − z ) , we obtain linear relations among the ∂ z i f cl. X i ∂ z i f cl. = X i ( z i ∂ z i f cl. − b h i ) = X i ( z i ∂ z i f cl. − z i b h i ) = 0 . (2.10)These follow from ∂ z i f cl. = ( ∂x∂z i ) ∂ x f cl. and P i ∂x∂z i = P i ∂x∂z i z i = P i ∂x∂z i z i = 0 as can easilybe shown from the definition of x and the conformal transformation properties of correlationfunctions of primary operators. This yields the final form of the decoupling equation, " ∂ z + b h z + b h ( z − x ) + b h ( z − − b ( h + h + h − h ) z ( z −
1) + x (1 − x ) ∂ x f cl. z ( z − x )( z − Ψ p = 0 . (2.11)To obtain f cl. from this equation, we use the fact that the solutions Ψ p must have a certainmonodromy when z is taken in a loop around 0 , x . This monodromy can be derived from thedecoupling equation of hO p Ψ O O i , (cid:20) b ∂ z + (cid:18) h p ( z − z ) + 1 z − z ∂ z (cid:19) + X i =3 , (cid:18) h i ( z − z i ) + 1 z − z i ∂ z i (cid:19)(cid:21) hO p Ψ O O i = 0 . (2.12)As z → z , the leading coefficient of the OPE between Ψ and O p is given by ( z − z ) κ O p ( z )where κ can be determined by inserting this coefficient into (2.12), (cid:20) b κ ( κ − z − z ) κ − + X i =3 , (cid:18) h i ( z − z i ) + 1 z − z i ∂ z i (cid:19) ( z − z ) κ + h p ( z − z ) κ − − κ ( z − z ) κ − (cid:21) hO p O O i = 0 . (2.13)– 6 –he leading contribution in z → z is given by (cid:20) b κ ( κ −
1) + h p − κ (cid:21) ( z − z ) κ − = 0 . (2.14)Thus as b → z → z , κ = 12 (cid:18) ± q − h p b (cid:19) and Ψ p ∼ ( z − z ) (1 ± √ − h p b ) . (2.15)Therefore, the monodromy matrix around 0 , x is given by M ,x = e iπ (1+ √ − h p b ) e iπ (1 − √ − h p b ) . (2.16)The trace of the monodromy matrix, which is independent of the basis in which the twosolutions of (2.11) are decomposed, is given byTr M ,x = − (cid:18) π q − h p b (cid:19) . (2.17)Thus, the torus conformal block can be extracted from (2.11) by choosing ∂ x f cl. such thatthe monodromy of the solution Ψ p around a loop enclosing z and z is given by (2.17) .Finally, the conformal block is obtained by integrating the chosen ∂ x f cl. .The four point conformal block in other channels is obtained from the same decouplingequation by imposing different monodromy conditions. For example, for the t -channel blockwe impose the monodromy condition around the insertion points of O and O , Tr M ,x = − (cid:16) π q − h p b (cid:17) . Higher point conformal blocks on the plane are computed from similarmonodromy methods derived analogously to the four-point case. For n point blocks, thedecoupling contains n − n − We now continue with the derivation of a monodromy method for conformal blocks on thetorus. The derivation of this monodromy method proceeds in a very similar way to the oneon the plane. We illustrate the derivation using the two-point function on the torus hO ( z ) O ( z ) i τ = Tr[ e πiτ ( L − c/ e − πi ¯ τ (¯ L − c/ O ( z ) O ( z )] , (2.18)however conformal blocks for other correlation functions on the torus are obtained in a similarfashion, as we briefly discuss as the end of this section. The modular parameter of the torusis denoted by τ , related to the inverse temperature by β = − πiτ . We also introduce theparameter Q = e − β = e πiτ , (2.19) The loop needs to enclose both z and z in order for the OPE between O ( z ) and O ( z ) to converge. – 7 –ritten with an uppercase Q instead of the standard lowercase q to distinguish it from theconformal dimension h q of the internal index of the conformal block which we are about toderive.As on the plane, we insert the degenerate operator Ψ( z, ¯ z ) into (2.18). To derive thecorresponding decoupling equation, we use the conformal Ward identity on the torus [29] , h T ( z ) Y i O i ( z i ) i τ = "X i (cid:0) h i ( ℘ ( z − z i ) + 2 η ) + ( ζ ( z − z i ) + 2 η z i ) ∂ z i (cid:1) + 2 πi∂ τ h Y i O i ( z i ) i τ . (2.20)Here, z ∼ z + 1 ∼ z + τ are the coordinates on the torus with modular parameter τ and ℘ ( z ) , ζ ( z ) denote Weierstraß elliptic functions with associated η parameter (see App. A formore details on the Weierstraß functions). Using the definition of the Virasoro generators,ˆ L − n Ψ( z ) = Z dw πi w − z ) n − T ( w )Ψ( z ) . (2.21)and the conformal Ward identity (2.20), we see that h Y i O i ( z i ) ˆ L − Ψ( z ) i τ = (cid:20) X i ( h i ( p ( z − z i ) + 2 η ) + ( ζ ( z − z i ) + 2 η z i ) ∂ z i )+ 2 η z∂ z + 2 h Ψ η + 2 πi∂ τ (cid:21) h Y i O i ( z i )Ψ( z ) i τ (2.22)and h Y i O i ( z i ) ˆ L − Ψ( z ) i τ = ∂ z h Y i O i ( z i )Ψ( z ) i τ . (2.23)Thus h Q i O i ( z i )Ψ( z ) i τ obeys the decoupling equation (cid:20) b ∂ z + X i ( h i ( p ( z − z i ) + 2 η ) + ( ζ ( z − z i ) + 2 η z i ) ∂ z i )+ 2 η z∂ z + 2 h Ψ η + 2 πi∂ τ (cid:21) h Y i O i ( z i )Ψ( z ) i τ = 0 . (2.24)To relate hO ( z ) O ( z )Ψ( z ) i τ to a conformal block, we decompose the trace over statesinto contributions from a primary O q and its descendants and insert the appropriate OPEcontractions. For the two-point function, there are two possible channels (see fig. 1). The projection block is obtained by OPE contracting O and O q . On the other hand, for the OPEblock we contract O and O , hO ( z ) O ( z )Ψ( z ) i τ = X q X l Q h q − c/ | l | hO { l } q ( z ) O ( z ) O ( z )Ψ( z ) O { l } q ( z ∞ ) i (c.c.)= X p,q C p X k,l ( z − z ) h p − h − h + | k | Q h q − c/ | l | β pk hO { l } q ( z ) O { k } p ( z )Ψ( z ) O { l } q ( z ∞ ) i (c.c.) , (2.25) Note that [29] uses a convention where correlation functions on the torus are normalized by the inverseof the partition function Z ( τ ) and thus the expression for the conformal Ward identity there contains anadditional term (2 πi∂ τ Z ( τ )) h Q i O i ( z i ) i τ . – 8 –2 p q p q Figure 1 . Conformal blocks for the two point function on the torus. Left: OPE channel, right:projection channel. where (c.c.) denotes schematically the antiholomorphic parts of the expression and z → − i ∞ while z ∞ → + i ∞ . We defineΨ pq = hO q ( z )Ψ( z ) O p ( z ) O q ( z ∞ ) ihO q ( z ) O p ( z ) O q ( z ∞ ) i . (2.26)As on the plane, in the large c limit hO { l } q ( z ) O { k } p ( z )Ψ( z ) O { l } q ( z ∞ ) i (c.c.) ≈ Ψ pq hO { l } q ( z ) O { k } p ( z ) O { l } q ( z ∞ ) i (c.c.) (2.27)This yields hO O Ψ i τ ≈ X p,q C p C qpq Ψ pq F ,pq ¯ F ,pq , (2.28)where F ,pq is the conformal block which we want to compute. Assuming that exponentiationof the conformal blocks in the semiclassical limit holds, F ,pq ∼ e − c/ f cl. , and using that ∂ z f cl. = − ∂ z f cl. , we obtain (cid:20) ∂ z + X i =1 , (cid:16) b h i ( ℘ ( z − z i ) + 2 η ) + ∂ z f cl. ( − i +1 ( ζ ( z − z i ) + 2 η z i ) (cid:17) − πi∂ τ f cl. (cid:21) Ψ pq = 0 . (2.29)From the definition of Ψ pq we derive the monodromy conditions in the same way as on theplane. For the OPE block, these areTr M z ,z = − π q − h p b ) , Tr M z = − π q − h q b ) . (2.30)The subscripts of the monodromy matrices show around which cycles the monodromy is taken.The derivation for the projection block works analogously. Here we contract O q ( z ) O ( z ): hO ( z ) O ( z )Ψ( z ) i τ = X p,q C p q X k,l ( z − z ) h p − h q − h + | k | Q h q − c/ | l | β pk q hO { k } p ( z ) O ( z )Ψ( z ) O { l } q ( z ∞ ) i (c.c.) . (2.31)Using Ψ pq defined by Ψ pq = hO p ( z )Ψ( z ) O ( z ) O q ( z ∞ ) ihO p ( z ) O ( z ) O q ( z ∞ ) i (2.32)– 9 –nd related to the two point correlator by hO O Ψ i τ ≈ X p,q C p q C q p Ψ pq F q, p ¯ F q, p (2.33)we find the same decoupling equation (2.29). However, the monodromy conditions differ.They are given byTr M z ,z = − π q − h p b ) , Tr M z ∞ = − π q − h q b ) . (2.34)To solve the decoupling equation, it is useful to perform a coordinate transformation u = e − πiz . Using the transformation of primary operators under conformal transformationsas well as the series representations of the Weierstrass elliptic functions from app. A, thedecoupling equation becomes (cid:20) ∂ u + y ( h − (1 + y ) ∂ y f cl. ) ∞ X m = −∞ Q m u ( u − Q m )( u − Q m (1 + y )) + 1 / − Q∂ Q f cl. u + h ∞ X m = −∞ Q m u ( u − Q m ) + h ∞ X m = −∞ Q m (1 + y ) u ( u − Q m (1 + y )) (cid:21) Ψ pq = 0 , (2.35)where we have chosen w.l.o.g. z = 0 and e − πiz = 1+ y . In these coordinates, the monodromyconditions becomeTr M , y = − π q − h p b ) , Tr M = − π q − h q b ) (OPE block)Tr M , y = − π q − h p b ) , Tr M ∞ = − π q − h q b ) (projection block)(2.36)This representation of the decoupling equation is immediately applicable for the calculationof the OPE block, which is defined through a series expansion in y and Q . Using this seriesexpansion as well as a WKB approximation for large h p , h q , (2.35) can be solved order byorder. For example, to first order in y and Q we get f OPEcl. = − b ( h p − h − h ) log y − ( b h q − /
4) log Q + 12 b ( h p + h − h ) y − b h p h q Q + ... (2.37)The projection block, on the other hand, can be expanded in a series in q = Q/ (1 + y ) and q = 1 + y . The decoupling equation can then be solved in the same way as for the OPEblock order by order in q and q . For example, to first order in q and q we obtain f projectioncl. = − ( b ( h p − h ) − /
4) log q − ( b h q − /
4) log q − b ( h − h p + h q )( h − h p + h q )2 h q q − b ( h + h p − h q )( h + h p − h q )2 h p q + ... (2.38)We have checked that the results for both the OPE and the projection block are in agreementwith the recursion formulas derived in [30] (see app. B for detailed expressions) as well asexplicit calculations up to third order. – 10 –t is clear that the above derivation can be easily generalized to other conformal blocks onthe torus. The simplest case is the zero-point block on the torus, i.e. the Virasoro character.Performing a similar derivation as above or equivalently taking the limit h , ,p → (cid:20) ∂ u + 1 / − Q∂ Q f cl. u (cid:21) Ψ q = 0 , (2.39)together with the monodromy condition Tr M = − π q − h q b ). In this case, we cangive the full solution. The decoupling equation is solved by Ψ q = u / ± √ Q∂ Q f cl. , from whichwe obtain f cl. = (1 / − b h q ) log Q which correctly reproduces the leading order contribution in c of the Virasoro character χ q = η ( τ ) Q h q − ( c − / ∼ e − c/ f cl. . For a general n -point conformalblock, the decoupling equation is given by (cid:20) ∂ z + n X i =1 (cid:16) b h i ( ℘ ( z − z i ) + 2 η ) + ∂ z i f cl. ( ζ ( z − z i ) + 2 η z i ) (cid:17) − πi∂ τ f cl. (cid:21) Ψ = 0 , (2.40)and there are n monodromy conditions around non-trivial cycles determined by the OPEcontractions. By conformal transformations, the insertion point of one of the operators canbe fixed, for example to z →
0. Then, there are n independent accessory parameters ∂ τ f cl. and ∂ z i f cl. for i ≥ We now turn to the computation of the partition function Z n on the replica surface R n .In general, the partition function on any higher genus Riemann surface can be expandedin zero-point conformal blocks, which can again be calculated via a monodromy method.This monodromy method can be derived by inserting the degenerate operator directly onthe higher genus Riemann surface – in contrast to the last section, where we inserted it ina correlation function on the torus – and inserting projection operators in the appropriateplaces. The difficulty of this approach is of course that deriving the decoupling equationfor an arbitrary Riemann surface is quite hard. However, we will see that things simplifyfor the special higher genus surface that we are interested in, that is the replica surface R n . Assuming that the dominant contribution to the partition function depends only onthe temperature and size of the entangling interval and not on any other moduli of R n , wefind the same decoupling equation as for the twist operator correlator on the torus. Thedifference to the last section lies in the monodromy conditions. The zero-point block on R n admits more general monodromy conditions (corresponding to different channels) than theconformal block on the torus. One of these more general monodromy conditions will give thedominant contribution to the entanglement entropy for large intervals.Before deriving the decoupling equation on R n , we collect some facts about the topologyand moduli of R n . For simplicity, we specialize again to the single interval case. The replicasurface is given by n copies of a torus with modular parameter τ , joined at a branch cut along– 11 –he entangling interval A . We use coordinates z, ¯ z to parametrize R n with identifications z ∼ z + 1 and z ∼ z + τ . In these coordinates, R n is described by a branched cover of thetorus with branch points located at z = z , + k + lτ for k, l ∈ Z . Near the branch points,the covering map is given by y n ∝ ( z − z − k − lτ ) and y n ∝ / ( z − z − k − lτ ). The genusof R n is then obtained by the Riemann-Hurwitz theorem. The ramification index at eachbranch point is equal to n , yielding g = n . Since the Euler characteristic is χ <
0, there areno conformal Killing vectors. This implies by the Riemann-Roch theorem that there exist3( n −
1) holomorphic quadratic differentials ω ( i ) zz parametrizing deformations of the complexstructure of the Riemann surface. The ω ( i ) zz are meromorphic doubly periodic functions thatare regular everywhere on the covering surface, i.e. ω ( i ) yy dy = ω ( i ) zz (cid:0) dzdy (cid:1) dz is non-singular forall y . Simple examples include ω (1) zz = const. which is trivially regular and doubly periodic aswell as ω (2) zz = ζ ( z − z ) − ζ ( z − z ) + 2 η ( z − z ). ω (2) zz is regular since near z = z + k + lτ we have ω (2) yy ∝ y n − which is regular at y = 0 for n ≥
2. Near z = z + k + lτ , regularitycan be shown in an analogous way. In fact, these two examples are the only ones relevant forthe following arguments since they are the only ones that respect the Z n replica symmetrypermuting the different copies of the torus with each other .The derivation of the decoupling equation on the replica surface then proceeds in a similarfashion as in the previous section. Assuming exponentiation of the zero-point block in thesemiclassical limit, the conformal Ward identities for a general Riemann surface [29] imply adecoupling equation of the form " ∂ z + h T zz i + n X i =1 ω ( i ) zz ∂ w i f cl. Ψ( z ) = 0 , (2.41)where w i are the modular parameters associated to ω ( i ) zz . h T zz i is the expectation value of theenergy momentum tensor. It can be derived along the lines of [15]: h T zz i transforms with aSchwarzian derivative, h T yy i = (cid:18) ∂z∂y (cid:19) h T zz i + nc { z, y } , (2.42)and h T yy i must be regular. The Schwarzian derivative term comes with a nc/
12 prefactorsince the total central charge of the theory on the replica surface is n times the central chargeof the theory on the torus that we started with. Together with the requirement that h T zz i bedoubly periodic, regularity of h T yy i implies h T zz i = c (cid:18) n − n (cid:19) X i ( ℘ ( z − z i ) + 2 η ) . (2.43) This can be seen as follows. The replica symmetry acts as y → ye πi/n . Therefore, only the ω zz ∼ ( z − z i ) α i with α i ∈ Z , i = 1 , α i < − z = z i . α i > ω yy . This leaves only α i = 0 , − – 12 – ( P p O O P p ) O O i ∼ p ∼ P p O O O O hO ( P p O O P p ) O i ∼ p ∼ O P p O O O Figure 2 . Inserting projection operators P p onto the Verma module of a primary O p into a correlatoryields the conformal block with internal weight h p . The 1 / ( z − z i ) poles in ℘ ( z − z i ) give a 1 /y contribution to h T yy i that cancels with theSchwarzian derivative term . Letting the sum over i in (2.41) run only over i = 1 ,
2, werecover the decoupling equation (2.29) for the twist correlator. Restricting the sum to thisrange means that we assume ∂ w i f cl. = 0 for i >
2, i.e. we assume that the result for thepartition function on the replica surface does not depend on other moduli of the replicasurface than the size of the torus τ and the length of the entangling interval z − z .However, as mentioned in the beginning of this section, the admitted monodromy con-ditions for the decoupling equation (2.41) are more general than those of (2.29) for the twistcorrelator. To see this, recall that conformal blocks of any correlation function can be ob-tained in two equivalent ways. Either we can perform OPE contractions between two or moreoperators and then keep only terms of particular primaries and their descendants in the OPEor equivalently we can insert projection operators onto the Verma modules of these primariesin the correlation function at appropriate places. The projectors of the latter approach canbe thought of as non-local operators acting in a closed line around the operators whose OPEcontractions are performed in the former approach (see fig. 2). For the zero-point block onan arbitrary higher genus Riemann surface, there are in general 3( n −
1) projectors to beinserted corresponding to 3( n −
1) monodromy conditions. However, as mentioned above weassume that the partition function on the higher genus Riemann surface depends only on twoof the moduli and thus we consider only two of the 3( n −
1) monodromy conditions.Which monodromy conditions are appropriate for the calculation of the entanglemententropy? For the conformal block, the monodromy conditions must be taken around thespatial circle and around z , z . On the other hand, for the zero-point block on the replicasurface the prescription described in this section still leaves open the question of where to putthe monodromy conditions – i.e. which channel to choose – in order to obtain the dominant For ease of comparison with the previous section we have also added a constant term c (cid:0) n − n (cid:1) η to h T zz i which is not strictly necessary for regularity and could be absorbed into the prefactor of ω (1) zz . It is also possible to calculate the modular transformed block for the twist correlator, which we expect tobe the dominant contribution at large temperature and small intervals. In this case, the monodromy conditionthat fixes the time dependence is taken around the time circle. – 13 – A c Figure 3 . Branch cut structure for large intervals. We can decompose the branch cut along A (denoted in red on the left) into a branch cut along the full spatial circle (denoted in red on the right)and a branch cut in the opposite direction along A c (denoted in blue on the right). contribution to the partition function from the vacuum block. Taking the limits of high andlow temperature, it is clear that for small intervals the monodromy condition which fixesthe time dependence must be taken around the spatial circle for small temperatures and thetime circle for large temperatures. It is also clear that the other monodromy condition fixingthe dependence on the size of the entangling interval must be taken around the entanglinginterval A between z and z for small intervals. For large intervals, the correct monodromycondition is obtained by reformulating the problem along the lines of [7, 22]. We separatethe branch cut on the torus along A yielding the replica surface into a branch cut along thefull spatial circle and a branch cut in the opposite direction along A c (see fig. 3). We thenimpose trivial monodromy around A c to fix the dependence on the size of the entanglinginterval. For small temperatures, the monodromy condition around the spatial circle remainsunchanged. However, for high temperatures the monodromy condition around the time circleis now transformed into a monodromy condition around a time circle of size nτ , since thebranch cut along the full spatial circle connects all n replica copies together to effectivelycreate a torus with modular parameter nτ .Note again that it is perfectly valid to use any of the above monodromy conditions forall values of the temperature and entangling interval size. However, outside of the regimes ofvalidity of the monodromy conditions described above, we don’t expect the vacuum block togive the dominant contribution in the semiclassical limit and thus the partition function inthis case would be obtained by summing up all of the conformal blocks for different values ofthe dimensions of the exchanged operators. The cross-over point between the regimes must bedetermined by an analysis of these contributions from the exchange of non-identity operators.Let us also note that explicit calculations for the free fermion case support the argumentspresented in this section with regards to the differences between twist correlators and partitionfunctions on R n and with regards to the monodromy condition for large intervals. Namely,in [9] it was observed that the twist correlator on the torus does not give the correct answerfor the entanglement entropy and in particular violates Bose-Fermi equivalence and modularcovariance. This was traced back in [10] to the way in which different spin structures ofthe replica surface R n combine to give the total answer for the partition function Z n . Thereplica surface, being composed of n copies of a torus, has 2 n nontrivial cycles around which– 14 –he fermions have either periodic or antiperiodic boundary conditions. To calculate the totalpartition function Z n on the replica surface, it is necessary to sum over all possible spinstructures each of which corresponds to a particular choice of boundary conditions aroundthe nontrivial cycles of the replica surface. For small intervals, the replica surface essentiallyfactorizes into n unconnected torus copies. In this limit, it was shown in [10] that Z n is givenby an “uncorrelated” sum where the summation over spin structures is performed for eachof the n tori separately. For large intervals, Z n is given by a “correlated” sum where onlyone sum over spin structures is performed, i.e. we take equal boundary conditions around thetime resp. space circles of each of the n replica copies [10]. In this case, the partition function Z n of the replica surface is essentially given by the partition function on a torus with modularparameter nτ . In this section, we present the calculation of the entanglement entropy on the torus at largecentral charge. As explained in the previous section, the entanglement entropy can be ob-tained from the partition function Z n on the replica surface R n which decomposes into zero-point conformal blocks. The claim we want to investigate is that at large central charge c → ∞ and for n → Z n comes from the vacuum block with h p = h q = 0. The derivation of this statement proceeds as follows. First, it is necessaryto show that the semiclassical limit is well-defined not only for h p,q = O ( c ) but also for h p,q = O ( c ). This means that for h p,q = γc the limits lim c →∞ and lim γ → of the conformalblock commute. A discussion of this point starting from the recursion relation for torus con-formal blocks is relegated to app. B. In the next step we solve the decoupling equation (2.29)perturbatively in ε = n − In the low temperature limit β → ∞ the torus degenerates into a cylinder with periodicspace direction. For the cylinder, the entanglement entropy of a single interval can be ob-– 15 –ained directly by mapping this cylinder to the plane and using the known formula for theentanglement entropy on the plane [3], S A = c π ( z − z ))) + const. (3.1)To obtain the same result from the monodromy method, we series expand Ψ pq and f cl. in n −
1: Ψ pq = P k Ψ ( n ) pq ( n − k and f cl. = P k f n ( n − k . The decoupling equation (2.29)at zeroth order in n − ∂ z − πi∂ τ f ]Ψ (0) pq ( z ) = 0 . (3.2)This is solved by Ψ (0) pq ( z ) = exp( ± p πi∂ τ f z ) . (3.3)By a coordinate transformation u = exp( − πiz ), we obtain ˜Ψ (0) pq ( u ) = u h Ψ Ψ (0) pq ( z = i log( u ) / (2 π )).Imposing trivial monodromy of ˜Ψ (0) pq ( u ) around u = 0 is equivalent to antiperiodic monodromyconditions for Ψ (0) pq ( z ) around the spatial circle of the torus, Ψ (0) pq ( z + 1) = − Ψ (0) pq ( z ). As ex-pected, this implies that f is equal to the leading order in c of the vacuum character on thetorus, f = πiτ / − β/ ⇔ e − c/ f = e c/ β = χ h =0 ( β ) | c →∞ . (3.4)At first order in n −
1, the decoupling equation is given by[ ∂ z − πi∂ τ f ]Ψ (1) pq ( z ) + m ( z )Ψ (0) pq ( z ) = 0 , (3.5)yielding Ψ (1) pq ( z ) = e − iπz πi Z z dx m ( x ) e iπx Ψ (0) pq ( x ) − e iπz πi Z z dx m ( x ) e − iπx Ψ (0) pq ( x ) , (3.6)where m ( z ) is given by m ( z ) = X i (cid:18)
12 ( ℘ ( z − z i ) + 2 η ) + ( − i +1 ( ζ ( z − z i ) + 2 η z i ) ∂ z f (cid:19) − πi∂ τ f . (3.7)To compute the conformal block we impose trivial monodromy around z , z which is equiv-alent to the vanishing of I z ,z dx m ( x ) e ± iπx Ψ (0) pq ( x ) . (3.8)This gives the z , z dependence of f , f = log(sin( π ( z − z ))) + C ( τ ) . (3.9)From trivial monodromy around u = 0 we find the τ dependence to be ∂ τ f = 0, whichimplies that C ( τ ) = const. is independent of τ . The antiholomorphic conformal block ¯ f gives the same result as the holomorphic one. Then the entanglement entropy is given by S A = c ( f + ¯ f ), in agreement with (3.1). In this limit, the OPE vacuum block of the twistcorrelator gives the same results, since it is given by the same monodromy method as thezero-point vacuum block of the replica partition function computed in this section.– 16 – .1.2 Low temperature and large intervals In this limit, we demand trivial monodromy around the spatial circle (i.e. around u = e − πiz =0) as well as trivial monodromy around A c (i.e. around z , z − In the high temperature limit β → S A = c (cid:18) τiπ sinh (cid:18) iπτ ( z − z ) (cid:19)(cid:19) + const. (3.10)In the monodromy method, we impose trivial monodromy around the time circle andaround z , z . As in the low temperature limit, we solve the decoupling equation (2.29) in aseries expansion around n −
1. An analogous calculation as above yields f cl. = − πi τ + (cid:15) log (cid:18) τ sinh (cid:18) πiτ ( z − z ) (cid:19)(cid:19) + const. = f + (cid:15)f (3.11)At zeroth order in (cid:15) we recover the leading order in c of the vacuum character χ h =0 ( β ) = e c
24 4 π β = e − c ( − πi τ ) . The entanglement entropy given by the first order contribution, S A = c ( f + ¯ f ), agrees with the result from the cylinder (3.10).From the twist correlator point of view, we obtain the high temperature limit by amodular transformation τ → − /τ from the low temperature result. The two point functionon the torus transforms covariantly, hO ( − z /τ, − ¯ z / ¯ τ ) O ( − z /τ, − ¯ z / ¯ τ ) i − /τ = ( − τ ) h + h ( − ¯ τ ) ¯ h +¯ h hO ( z , ¯ z ) O ( z , ¯ z ) i τ . (3.12)If the two point function of twist operators is dominated by a single conformal block, theconformal block for the modular transformed τ is obtained from the modular transformationproperties of the correlation function as f − /τ cl. ( − z /τ, − z /τ ) = − b ( h + h ) log( − τ ) + f τ cl. ( z , z ) . (3.13)Therefore, we can immediately read off the high temperature behavior of the twist correlatorfrom f τ cl. ( z , z ) | τ →∞ = h f − /τ cl. ( − z /τ, − z /τ ) + b ( h + h ) log( − τ ) i τ →∞ (3.14)For h = h = 1 / (cid:15) , f τ cl. ( z , z ) | τ → = πiτ + (cid:15) log(sin( π ( z − z ))) + const. we again obtain(3.11). Thus the twist correlator agrees with the replica partition in this limit.Let us note that is also possible to determine the high temperature expansion of twistcorrelator by applying the modular transformation to the monodromy conditions. At low– 17 –emperatures, we demand trivial monodromy around the spatial circle of the torus at time.Since the modular transformation τ → − /τ exchanges the time and space directions of thetorus, we can obtain the high temperature behavior of the twist correlator by demanding triv-ial monodromy around the time circle of the torus, showing directly the equivalence betweenthe twist correlator result and the replica partition function in the limit of high temperatureand small entangling intervals. As explained in sec. 2.3, the correct monodromy condition of the zero-point block on thereplica surface for large intervals imposes trivial monodromy around A c and z → z + nτ . Tozeroth order in n − (0) pq ( z ) = exp( ±√ πi∂ τ f ).Imposing Ψ (0) pq ( z + nτ ) = − Ψ (0) pq ( z ) yields f = − πi n τ . (3.15)To first order, the solution readsΨ (1) pq ( z ) = nτ e − iπznτ πi Z z dx m ( x ) e iπxnτ Ψ (0) pq ( x ) − nτ e iπznτ πi Z z dx m ( x ) e − iπxnτ Ψ (0) pq ( x ) . (3.16)Imposing trivial monodromy around A c and expanding in n − f cl. = − πi τ + ( n − (cid:18) iπτ + log (cid:18) τ sinh (cid:18) iπτ (1 − ( z − z )) (cid:19)(cid:19)(cid:19) + const. (3.17)giving S A = c (cid:18) iπτ + log (cid:18) τiπ sinh (cid:18) iπτ (1 − ( z − z )) (cid:19)(cid:19)(cid:19) + const. (3.18)As expected from general arguments [5, 7], the difference between the entanglement entropyfor A and for the complement A c in the limit of large interval size is given by the thermalentropy S ( β ) = β ( − β − ∂ β log Z ( β )) = c π β = c iπτ , using the partition function Z ( β →
0) = exp( c
12 4 π β ) from the Cardy formula. The twist correlator, on the other hand, cannotreproduce this feature as is easy to see by applying the modular transformation argumentfrom the last section which gives (3.11) in disagreement with (3.17). We now argue that the results in the limits considered in the previous section are valid forall temperatures and interval sizes in holographic CFTs. This statement holds if the vacuumblock computed in the previous section gives the dominant contribution to the partitionfunction on the replica manifold R n in the large central charge limit.Let us first consider the case n = 1, i.e. the zeroth order in the n − c . If thisvacuum character dominates the partition function, Z ( β ) takes on the universal form Z ( β ) = ( exp (cid:0) c β (cid:1) , β > π exp (cid:0) c
12 4 π β (cid:1) , β < π (3.19)– 18 –or all temperatures. For consistency of the computation method, dominance of the conformalblock to first order in n − n −
1. It has beenargued in [31] that a partition function of the form (3.19) is characteristic of a holographicCFT. Thus it is a necessary condition that the CFT in question be holographic in order forthe results of sec. 3.1 to hold at arbitrary temperatures.More explicit conditions on the CFT in question can be given following an argument fordominance of the vacuum block in the zero temperature case which goes as follows [4]. Thearguments below hold for CFTs with OPE coefficients C p that grow at most exponentiallywith c and a density of states for light operators that does not grow with c . Here lightoperators mean operators with conformal weight h, ¯ h < h gap where h gap is of the order ofthe central charge. In the large central charge limit, the conformal block expansion in the s -channel of the four-point twist correlator on the plane takes on the form h σ n (0) σ n ( x ) σ n (1) σ n ( ∞ ) i = X p C p exp (cid:18) − c (cid:0) f cl. ( h σ n /c, h p /c, x ) + ¯ f cl. (¯ h σ n /c, ¯ h p /c, ¯ x ) (cid:1)(cid:19) . (3.20)On account of the cluster decomposition principle, the vacuum exchange is the leading con-tribution in the s -channel around x = 0 and in the t -channel around x = 1. This implies thatthe contribution of the heavy operators with h, ¯ h > h gap in the s -channel in some finite regionaround x = 0 is suppressed exponentially and similarly for the t -channel around x = 1. Thesparse spectrum of the light operators allows ignoring the multiplicity factors for h, ¯ h < h gap .Consider first the scenario that the OPE coefficients grow subexponentially with c . In thiscase, we can also ignore the coefficient C p and the sum over the light operators in (3.20) isdominated exponentially in e − c by its largest term. If the semiclassical conformal block f cl. is a monotonically increasing function with h p /c , this implies that the vacuum block with thelowest possible h p /c = 0 dominates. In the case that the OPE coefficients grow exponentiallywith c , the sum over the light operators gives Z h gap dh p d ¯ h p exp (cid:18) c (cid:0) g ( h p /c, ¯ h p /c ) − f cl. ( h σ n /c, h p /c, x ) − ¯ f cl. (¯ h σ n /c, ¯ h p /c, ¯ x ) (cid:1)(cid:19) , (3.21)where g ( h p /c, ¯ h p /c ) contains the contribution of the OPE coefficients and multiplicities. Thisintegral is dominated either by the endpoints of the integration or by a saddle point. Nearcoincident points, the leading universal term in any correlation function is given by thevacuum exchange. Therefore, in a finite region around x = 0 and x = 1, the integral in (3.21)is dominated by the endpoint at h p = 0. However, now – unlike for subexponentially growingOPE coefficients – it is not possible to exclude saddle points dominating the integral (3.21)in some finite subset of x ∈ [0 , f cl. increases monotonically with This is equivalent to the requirement that correlation functions obey the cluster decomposition principleand are smooth in a neighborhood of the point where multiple operator insertion points coincide [4]. – 19 –he internal conformal weights h p,q . In sec. 3.4, we present numerical evidence that this isindeed the case. Therefore, the results of sec. 3.1 are valid for all temperatures and intervalsizes assuming subexponentially growing OPE coefficients and a sparse spectrum of lightoperators. For exponentially growing OPE coefficients, the results are still valid in a finiteregion around the respective limiting points. The points at which the different limits exchangedominance are obtained as follows. From the requirement of consistency of our results at n = 1with the partition function (3.19) we see that the low and high temperature regimes exchangedominance at the Hawking-Page phase transition point β = 2 π . The transition between smalland large interval behavior in the high temperature regime is estimated by equating the resultsfor the conformal block in the small and large interval regime. The dominant contributioncomes from the smaller conformal block.Assuming the aforementioned restrictions on the CFT spectrum and OPE coefficients,the results match perfectly with the predictions from the RT formula [5, 17]. There is also adirect way to implement the monodromy computation for the calculation of the entanglemententropy in the dual gravity theory, as was shown in [15] at zero temperature. This wasgeneralized to the finite temperature case in [21, 22]. We obtain the same monodromy methodas [21, 22] from the CFT side. This clearly shows that our results are valid for holographicCFTs.Let us also note that the universal form of the partition function (3.19) in holographictheories can be derived from the same vacuum block dominance argument as the universalform of the entanglement entropy. The partition function can be expanded in zero-pointblocks on the torus either in a low temperature expansion (zero-point blocks are Virasorocharacters) or in a high temperature expansion (zero-point blocks are modular transformedVirasoro characters). From the known form of the Virasoro characters, the leading ordercontribution in the central charge of the characters is given by χ ∼ Q h q − c/ where Q = e πiτ resp. Q = e − πi/τ in the low and high temperature expansions. Since ( − /c ) log χ is anincreasing function of h q , the same arguments as for the entanglement entropy given aboveapply to the partition function which is dominated by the vacuum character with h q = 0. The generalization to an entangling interval consisting of the union of multiple intervals, A = [ z , z ] ∪ [ z , z ] ... [ z N − , z N ], is straightforward. The decoupling equation is given by " ∂ z + N X i =1 (cid:18)
14 ( n − /n )( ℘ ( z − z i ) + 2 η ) − ∂ z i f cl. ( ζ ( z − z i ) + 2 η z i ) (cid:19) − πi∂ τ f cl. Ψ( z ) = 0 . (3.22)We impose trivial monodromy around N pairs ( i, j ) of interval endpoints z i , z j to fix the ∂ z i f cl. . The temperature dependence is fixed by demanding trivial monodromy around eitherthe spatial circle, a time circle of size τ or a time circle of size nτ depending on the temperatureand total entangling interval size | A | = P i | z i − z i − | . This yields– 20 – low temperature: trivial monodromy for z → z + 1 S A = c X ( i,j ) log (sin( π ( z i − z j ))) + const. (3.23)• high temperature and small total interval size: trivial monodromy for z → z + τS A = c X ( i,j ) log (cid:18) τiπ sinh (cid:18) iπτ ( z i − z j ) (cid:19)(cid:19) + const. (3.24)• high temperature and large total interval size: trivial monodromy for z → z + nτS A = c iπτ + X ( i,j ) log (cid:18) τiπ sinh (cid:18) iπτ ( z i − z j ) (cid:19)(cid:19) + const. (3.25)Which monodromy condition and which combination of pairs ( i, j ) to take, i.e. in whichchannel to expand the conformal block, depends on the interval size. We expect the dominantcontribution to come from the channel with the smallest f cl. , in which case we find agreementwith the RT formula. However, we caution that this argument depends on the vacuum blockdominating the partition function, which we have checked only for a single interval.One particular interesting special case of the above calculation is the time dependence ofthe entanglement entropy between two intervals on opposite boundaries of a two-sided BTZblack hole. The two-sided BTZ black hole is dual to the thermofield double state. On theCFT side, the entanglement entropy is obtained by positioning the two intervals at a distance τ / z = ¯ z = 0 z = ¯ z = Lz = 2 t + L + τ / , ¯ z = − t + L + ¯ τ / z = 2 t + τ / , ¯ z = − t + ¯ τ / , (3.26)where L is the size of the entangling interval taken to be equal on both boundaries and t is thetime coordinate at which both parts of the entangling interval are placed on the asymptoticboundaries of the wormhole (see [33] for more details on the setup). For small interval size L , there are two conformal blocks to consider. At early times, i.e. for small t , the dominantcontribution comes from imposing trivial monodromy around z , z and z , z , while at latetimes the vacuum block with trivial monodromy around z , z and z , z dominates. Takinginto account that due to the analytic continuation f cl. = ¯ f cl. , the corresponding entanglemententropy is given by S A = 2 c (cid:18) τiπ cosh (cid:18) πiτ t (cid:19)(cid:19) + const. (3.27) Note that this time coordinate has nothing to do with the euclidean time coordinate on the torus on whichwe calculate the finite temperature correlator in the euclidean CFT. – 21 –t early times and by S A = 2 c (cid:18) τiπ sinh (cid:18) iπτ L (cid:19)(cid:19) + const. (3.28)at late times. This reproduces the phase transition in the dual RT surfaces from geodesicsthat connect the two boundaries through the interior of the two-sided black hole at early timesto disconnected geodesics on opposite boundaries that do not enter the black hole interior atlate times [33]. In this section, we provide numerical evidence that the vacuum block exponentially dominatesthe twist correlator in the large c limit. For simplicity, we restrict to the single interval case.Assuming the same conditions on the spectrum and OPE coefficients of the CFT detailed inthe last section, we need to show that the conformal block monotonically increases with theweight of the internal operators h p,q .For the zero temperature case, this was done numerically in [4] for arbitrary n , givingevidence that the Rényi entropies are given by the vacuum conformal block contributiononly. However, the calculation is much simpler if we restrict to n close to one which implies h p /c (cid:29) h i /c → i = 1 , , ,
4. In this limit, the conformal block can be obtained in closedform from the monodromy calculation by a WKB expansion in 1 / ( h p /c ) [14], c f cl. (0 , h p /c, x ) = h p (cid:18) π K (1 − x ) K ( x ) − log 16 (cid:19) , (3.29)where K ( x ) is the complete elliptic integral of the first kind. Thus, f cl. is an increasingfunction of h p /c if πK (1 − x ) /K ( x ) − log 16 > x < /
2. In fact, f cl. (0 , h p /c, x ) − f cl. (0 , h p /c, − x ) reaches its crossover point exactly at x = 1 /
2, confirming that at this point dominance is exchanged from the s to the t -channelblock.Applying the same arguments as on the plane to the case of the torus, it is clear thatthe vacuum block dominates if the semiclassical block is an increasing function of h p and h q .Without loss of generality, we take z = 0 in the following. Restricting again to n ≈
1, weneed to find the semiclassical block in the limit h p,q /c (cid:29) h , /c →
0. Unlike on the plane,however, f cl. can not easily be obtained in a closed form expression from the monodromycalculation in this limit . Thus, we apply a series expansion in y = e − πiz − Q = e πiτ on top of the WKB approximation in 1 / ( h p,q /c ). While this yields a very precise numericalapproximation to the true value of the conformal block if enough terms are included in theseries, the expansion has a restricted domain of validity in the y, Q plane. In particular, theseries expansion in the cross-ratio x of the four-point block on the plane converges for | x | < The reason for this is that the solution of the decoupling equation on the torus takes on the schematicform of an integral over p A∂ z f cl. + B∂ τ f cl. for some functions A and B , from which it is not easily possibleto extract ∂ z f cl. and ∂ τ f cl. . The solution of the decoupling equation on the plane, on the other hand, is givenby an integral over √ A∂ x f cl. from which ∂ x f cl. can be factored out immediately. – 22 – q c h q c h q c h p c h p c h p c h p c h q c Figure 4 . Derivative of the semiclassical conformal block f cl. w.r.t. h p (left) and h q (right) for z = 0 . β = 4 π and different values of h p and h q in the range [0 , c/ h p and h q . Theseries expansion used in this figure was truncated at order 10 in both y and Q . [14], therefore we expect the series expansion of the two-point block on the torus to have aconvergence radius of | y | = 1 (the torus block reduces to the block on the plane in the limit Q → | y | >
1, we observe large fluctuationsin the value of the conformal block as we include more terms in the series expansion. Thenumerics for the convergence radius in Q is less clear, but also in this variable we observelarge fluctuations close to Q = e − π . Thus, we can check the vacuum block dominance onlyin a restricted region around the origin in y , corresponding to small intervals. The restrictedconvergence radius in Q is not limiting since above the Hawking-Page transition temperaturegiven by Q = e − π , we expect the block in the high temperature expansion to dominate. Thehigh temperature expansion of the conformal block is given by a series expansion in e πiz /τ − e − πi/τ . In the h p,q /c (cid:29) h , /c limit, the series coefficients are equal to those of the lowtemperature expansion. Moreover, in the same limit at high temperatures and for n → nτ ) is equivalent to the conformal block in the small interval limit withthe replacement z → − z .We show some plots of ∂f cl. ∂h p and ∂f cl. ∂h q in fig. 4 for small temperatures and values of y and Q inside the convergence radius. The plots for the conformal block in the high temperaturelimit show no significant differences from the ones in the low temperature limit. We find in allcases that inside the convergence radius of the series expansion ∂f cl. ∂h p,q >
0, i.e. the assumptionof vacuum block dominance is fulfilledWhile it is not possible to find an analytic continuation for the conformal block from atruncated series expansion, we can use a Padé approximant to get a heuristic approximationof the series outside its convergence radius. The Padé approximation works by replacing thetruncated series expansion by a rational function whose Taylor expansion agrees with theseries expansion up to the order in which the truncation was performed [34]. In many cases,this approximation has a better radius of convergence than the original series expansion dueto poles in the function limiting the radius of convergence of its Taylor expansion being taken– 23 – .0 0.2 0.4 0.6 0.8 1.0 z Series expansionPade approximant 1st orderPade approximant 2nd orderPade approximant 3rd orderPade approximant 4th order z - Series expansionPade approximant 1st orderPade approximant 2nd orderPade approximant 3rd orderPade approximant 4th order
Figure 5 . Derivative of the Padé approximation of the semiclassical conformal block w.r.t. the internalconformal weight plotted against z . The upper plot shows the conformal block at β = 4 π in the lowtemperature expansion and the lower plot the block at β = π in the high temperature expansion.For plotting convenience we set h p = h q which gives a derivative of f cl. that is constant for all h p .As we increase the order of the denominator polynomial in the Padé approximation, the fluctuationsin f cl. outside of the convergence radius of the series expansion decrease. Moreover, multiple Padéapproximants converge to the same value outside of the convergence radius. The convergence radius | y | = 1 is reached at z = 1 / z = log(2)4 π = 0 . y and Q . into account in the Padé approximation. We plot the Padé approximant of f cl. in the specialcase h p = h q for different orders of the denominator polynomial in fig. 5 depending on z (thesize of the entangling interval). We find that different Padé approximants for f cl. converge tothe same value and yield ∂f cl. ∂h p,q > | y | = 1of the series expansion of f cl. . While this is not a formal proof, it does indicate that vacuumblock dominance holds also outside of the convergence radius. Let us briefly summarize the main points investigated in this publication. In the first partin sec. 2, we took a detailed look at monodromy methods for the computation of arbitraryconformal blocks on the torus as well as zero-point conformal blocks on the special highergenus Riemann surface relevant to the calculation of the entanglement entropy via the replicatrick. We then applied the derived monodromy methods to the calculation of the entanglement– 24 –ntropy in sec. 3. We found that for holographic CFTs, the vacuum conformal block dominatesthe partition function on the replica surface and therefore the entanglement entropy takes ona universal form in agreement with the RT formula.The main advancement compared to previous work on entanglement entropy for finitesize and finite temperature systems at large central charge is as follows.First of all, we clarified the role of the correlation function of twist operators on the toruscompared to the partition function Z n on the replica surface R n . The twist correlator onlygives the correct results for small intervals or low temperature, while the replica partitionfunction Z n must give the entanglement entropy in all cases by construction. The reason whythe twist correlator agrees with Z n for small intervals or low temperature is evident from themonodromy method. The conformal blocks of the twist correlator obey the same monodromymethod as the zero-point conformal blocks for Z n apart from the set of allowed monodromyconditions, which for the twist correlator blocks is a subset of those of the replica partitionfunction.Secondly, our derivation of the monodromy method is based solely on CFT techniquesand thus provides a non-trivial check of both the RT formula as well as previous calculations of[21, 22] using the monodromy method on the gravity side. Moreover, the monodromy methodpresented here is derived from first principles, further strengthening the heuristic argumentsthat were used to justify the monodromy method in [21, 22]. The derivation from the CFTside furthermore allows for a determination of the restrictions on the operator content andOPE coefficients that must be obeyed by the CFT in question in order for the vacuum blockto give the dominant contribution and the RT results for the entanglement entropy to bevalid. Perhaps unsurprisingly, these restrictions are equivalent to those imposed in [4] forthe entanglement entropy at zero temperature. That is, the entanglement entropy for anyCFT with large central charge, sparse spectrum of light operators and at most exponentiallygrowing OPE coefficients agrees with the RT formula not only at zero temperature as wasalready known from the results of [4], but also at finite temperature and finite size. Inaddition, the same arguments that imply the universal form of the entanglement entropy inthese CFTs also imply a universal form of the partition function at all temperatures, anotherfeature of holographic CFT investigated also in [31].We close with a short outlook on possible future directions. One interesting direction is aninvestigation in full generality of the issue under which conditions the twist correlator yieldsthe same results as the replica partition function Z n . Previous work in the free fermion case[9, 10] indicates that for short intervals the twist correlator is equivalent to the replica partitionfunction, a result which we can corroborate for large c CFTs. Does this hold in general?Other directions can be found in further applications of the monodromy method. Sinceconformal blocks are the basic building blocks of any correlation function, the derivation of themonodromy methods from first principles lays the groundwork for a number of applications. Inparticular, the monodromy methods found in this paper may prove useful in deriving recursionrelations in the dimensions of the exchanged operators for torus conformal blocks with betterconvergence properties than the known recursion relations in the central charge found in [30].– 25 –oreover, the monodromy methods also pave the way to the study of other informationtheoretic quantities closely connected to the entanglement entropy such as entanglementnegativity [35–37], symmetry resolved entanglement [38–42] or entwinement [43–47] for finitetemperature states, providing opportunities for new insights into the physics of the dual AdSblack holes. We hope to be able to report on further results in this direction in the future.
Acknowledgements
I would like to thank Johanna Erdmenger and Christian Northefor useful discussions. I acknowledge financial support by the Deutsche Forschungsgemein-schaft (DFG, German Research Foundation) under Germany’s Excellence Strategy throughWürzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter -ct.qmat (EXC 2147, project-id 390858490).
A Conventions for elliptic functions
This appendix contains an overview over conventions and useful identities for the Weierstraßelliptic functions used in the rest of the publication (see e.g. [48] for more details about thesefunctions). All elliptic functions are defined for a lattice Λ generated by the identifications z ∼ z + 1 and z ∼ z + τ . The Weierstraß elliptic functions ℘ ( z ) , ζ ( z ) and σ ( z ) are defined by ℘ ( z ) = − ζ ( z ) = 1 z + X ( m,n ) =(0 , (cid:18) z + n + mτ ) − n + mτ ) (cid:19) ,ζ ( z ) = σ ( z ) σ ( z ) = 1 z + X ( m,n ) =(0 , (cid:18) z + n + mτ ) − n + mτ ) + z ( n + mτ ) (cid:19) ,σ ( z ) = z Y ( m,n ) =(0 , (cid:18) − zn + mτ (cid:19) exp zn + mτ + 12 z ( n + mτ ) ! . (A.1) ℘ ( z ) is a true elliptic (i.e. doubly periodic) function while ζ ( z ) and σ ( z ) are quasiperiodic: ℘ ( z + 1) = ℘ ( z + τ ) = ℘ ( z ) ζ ( z + 1) = ζ ( z ) + 2 η , ζ ( z + τ ) = ζ ( z ) + 2 η σ ( z + 1) = − exp(2 η ( z + 1 / σ ( z ) , σ ( z + τ ) = − exp(2 η ( z + τ / σ ( z ) , (A.2)where η = ζ (1 /
2) and η = ζ ( τ /
2) = τ η − πi . Another useful definition of ℘ ( z ) and ζ ( z ) isgiven by ℘ ( z ) + 2 η = (2 πi ) ∞ X m = −∞ Q m u ( u − Q m ) = (cid:18) πiτ (cid:19) ∞ X m = −∞ ˜ Q m ˜ u (˜ u − ˜ Q m ) ζ ( z ) − η z = iπ ∞ X m = −∞ Q m + uQ m − u = − iπτ ∞ X m = −∞ ˜ Q m + ˜ u ˜ Q m − ˜ u , (A.3)where u = e − πiz , Q = e πiτ and ˜ u = e πiz/τ , ˜ Q = e − πi/τ .– 26 – Recursion relations for torus conformal blocks
For completeness, this appendix shows the recursion formulas for the two-point conformalblocks on the torus following as a special case from the general method derived in [30]. Fordetails of the derivation, see [30] and also [49] for the one-point torus block. For the OPEblock, the conformal block is given by the following recursion scheme F OPE21 ,pq ( h p , h q , c ) = U OPE ( h p , h q , c ) − X r ≥ ,s ≥ ∂c rs ( h q ) ∂h q Q rs A c rs ( h q ) rs P rsc rs ( h q ) (cid:20) h p h q + rs (cid:21) P rsc rs ( h q ) (cid:20) h p h q (cid:21) c − c rs ( h q ) F OPE21 ,pq ( h q , h q + rs, c rs ( h q )) − X r ≥ ,s ≥ ∂c rs ( h p ) ∂h p y rs A c rs ( h p ) rs P rsc rs ( h p ) (cid:20) h q h q (cid:21) P rsc rs ( h p ) (cid:20) h h (cid:21) c − c rs ( h p ) F OPE21 ,pq ( h p + rs, h q , c rs ( h p )) (B.1)where the fusion polynomials are given by P rsc (cid:20) h h (cid:21) = r − Y m =1 − r,m ∈ − r +2 N s − Y n =1 − s,n ∈ − s +2 N λ + λ + mb + nb − λ − λ + mb + nb − λ i = p ( b + b − ) − h i while the prefactor is A crs = 12 ( r,s ) Y ( m,n )=(1 − r, − s ) ( m,n ) =(0 , , ( r,s ) ( mb + nb − ) − . (B.3) c rs denotes the value of the central charge where degenerate representations of the Virasoroalgebra appear, c rs ( h ) = 1 + 6( b rs ( h ) + b − rs ( h )) ,b rs ( h ) = rs − h + p ( r − s ) + 4( rs − h + 4 h − r . (B.4)The c -regular part U is given by U OPE ( h p , h q , c ) = " ∞ Y n =2 − Q n i,j ≥ Q i y j s ij ( h q , h p , h q )(1 − h p − j ) j ( h p + h − h ) j i !(2 h q ) i j !(2 h p ) j , (B.5)where we define the rising and falling Pochhammer symbols by( a ) n = n − Y k =0 ( a + k ) ( a ) ( n ) = n − Y k =0 ( a − k ) (B.6)– 27 –nd s ij ( h , h , h ) = h h | ( L i − ) † O h (1) L j − | h i = i X p =0 (cid:18) ip (cid:19) ( j ) ( p ) (2 h + j − ( p ) ( h + h − h − ( j − p )) i − p ( h + h − h ) j − p , j ≥ i j X p =0 (cid:18) jp (cid:19) ( i ) ( p ) (2 h + i − ( p ) ( h + h − h − ( i − p )) j − p ( h + h − h ) i − p , i ≥ j (B.7)The recursion formula for the projection block is given by F projection2 p, q ( h p , h q , c ) = U projection ( h p , h q , c ) − X r ≥ ,s ≥ ∂c rs ( h q ) ∂h q q rs A c rs ( h q ) rs P rsc rs ( h q ) (cid:20) h p h (cid:21) P rsc rs ( h q ) (cid:20) h p h (cid:21) c − c rs ( h q ) F projection2 p, q ( h p , h q + rs, c rs ( h q )) − X r ≥ ,s ≥ ∂c rs ( h p ) ∂h p q rs A c rs ( h p ) rs P rsc rs ( h p ) (cid:20) h q h (cid:21) P rsc rs ( h p ) (cid:20) h q h (cid:21) c − c rs ( h p ) F projection2 p, q ( h p + rs, h q , c rs ( h p )) , (B.8)where the c -regular part is given by U projection ( h p , h q , c ) = " ∞ Y n =2 − Q n i,j ≥ q i q j s ij ( h q , h , h q ) s ji ( h p , h , h q ) i !(2 h q ) i j !(2 h p ) j . (B.9)By explicit calculation, it is easy to check in the first few orders of the series expansion thatfor h = h and h p,q = γc , the limits lim γ → and lim c →∞ of the OPE block commute. We havechecked this up to fourth order in y, Q . A more convenient proof is possible with a recursionrelation in the conformal weights of the exchanged operators h p,q , as derived for the conformalblock on the plane in [14]. In fact, the singular parts proportional to ∼ / ( h p,q − h rs ) of sucha recursion relation are proportional to the singular parts ∼ / ( c − c rs ( h p,q )) of the aboverecursion relations in c [30]. Using these known singular parts, one can show that the limits γ → c → ∞ commute for the singular parts of this recursion relation to all orders,assuming the above conditions on h , ,p,q . Together, these calculations provide some evidencethat the semiclassical limit is well-defined for the vacuum block on the torus derived in sec. 3.Since the vacuum block on the torus in most limits is equivalent to a zero-point block on thereplica surface R n , this also provides evidence that the semiclassical limit for the zero-pointvacuum block on R n is well-defined. The authors of [30] also claim to have found a different recursion formula in the conformal dimensions ofthe exchanged primaries h p,q for a class of conformal blocks termed the “necklace blocks” which include theprojection block considered here as a special case. We cannot confirm this claim since the formula presentedin [30] agrees neither with the recursion relation in the central charge derived in [30] nor with an explicitcalculation in the first few orders of the series expansion in q , q for the projection block. – 28 – eferences [1] L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A Quantum Source of Entropy for Black Holes , Phys. Rev. D (1986) 373.[2] M. Srednicki, Entropy and area , Phys. Rev. Lett. (1993) 666 [ hep-th/9303048 ].[3] P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory , J. Stat. Mech. (2004) P06002 [ hep-th/0405152 ].[4] T. Hartman,
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