Entanglement entropy from one-point functions in holographic states

Journal of High Energy Physics, Jun 2016

For holographic CFT states near the vacuum, entanglement entropies for spatial subsystems can be expressed perturbatively as an expansion in the one-point functions of local operators dual to light bulk fields. Using the connection between quantum Fisher information for CFT states and canonical energy for the dual spacetimes, we describe a general formula for this expansion up to second-order in the one-point functions, for an arbitrary ball-shaped region, extending the first-order result given by the entanglement first law. For two-dimensional CFTs, we use this to derive a completely explicit formula for the second-order contribution to the entanglement entropy from the stress tensor. We show that this stress tensor formula can be reproduced by a direct CFT calculation for states related to the vacuum by a local conformal transformation. This result can also be reproduced via the perturbative solution to a non-linear scalar wave equation on an auxiliary de Sitter spacetime, extending the first-order result in arXiv:​1509.​00113.

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Entanglement entropy from one-point functions in holographic states

Received: May Entanglement entropy from one-point functions in Matthew J.S. Beach 0 1 Jaehoon Lee 0 1 Charles Rabideau 0 1 Mark Van Raamsdonk 0 1 0 6224 Agricultural Road , Vancouver, BC, V6T 1W9 , Canada 1 Department of Physics and Astronomy, University of British Columbia For holographic CFT states near the vacuum, entanglement entropies for spatial subsystems can be expressed perturbatively as an expansion in the one-point functions of local operators dual to light bulk elds. Using the connection between quantum Fisher information for CFT states and canonical energy for the dual spacetimes, we describe a general formula for this expansion up to second-order in the one-point functions, for an arbitrary ball-shaped region, extending the rst-order result given by the entanglement rst law. For two-dimensional CFTs, we use this to derive a completely explicit formula for the second-order contribution to the entanglement entropy from the stress tensor. We show that this stress tensor formula can be reproduced by a direct CFT calculation for states related to the vacuum by a local conformal transformation. This result can also be reproduced via the perturbative solution to a non-linear scalar wave equation on an auxiliary de Sitter spacetime, extending the rst-order result in arXiv:1509.00113. AdS-CFT Correspondence; Gauge-gravity correspondence 1 Introduction Background 2 3 4 2.1 2.2 3.1 3.2 4.1 4.2 4.3 4.4 Conformal transformations of the vacuum state Entanglement entropy of excited states Perturbative expansion Excited states around thermal background Second-order contribution to entanglement entropy Stress tensor contribution: direct calculation for CFT2 5 Auxiliary de Sitter space interpretation A Direct proof of the positivity B Rindler reconstruction for scalar operators in CFT2 1 Introduction In holographic conformal eld theories, states with a simple classical gravity dual interpretation have a remarkable structure of entanglement: according to the holographic entanglement entropy formula [1{3], their entanglement entropies for arbitrary regions (at leading order in large N ) are completely encoded in the extremal surface areas of an asymptotically AdS spacetime. In general, the space of possible entanglement entropies (functions on a space of subsets of the AdS boundary) is far larger than the space of possible asymptotically AdS metrics (functions of a few spacetime coordinates), so this property of geometrically-encodable entanglement entropy should be present in only a tiny fraction of all quantum eld theory states [4]. It is an interesting question to understand better which CFT states have this property,1 and which properties of a CFT will guarantee that families of low-energy states with geometric entanglement exist. 1Even in holographic CFTs, it is clear that not all states will have this property. For example, if j 1i and j 2i are two such states, corresponding to di erent spacetimes M 1 and M 2 , the superposition j 1i+j 2i is not expected to correspond to any single classical spacetime but rather to a superposition of M 1 and M 2 . Thus, the set of \holographic states" is not a subspace, but some general subset. { 1 { For a hint towards characterizing these holographic states, consider the gravity perspective. A spacetime M dual to a holographic state j i is a solution to the bulk equations of motion. Such a solution can be characterized by a set of initial data on a bulk Cauchy surface (and appropriate boundary conditions at the AdS boundary). The solution away from the Cauchy surface is determined by evolving this initial data forwards (or backwards) in time using the bulk equations. Alternatively, we can think of the bulk solution as being determined by evolution in the holographic radial direction, with \initial data" speci ed at the timelike boundary of AdS. In this case, the existence and uniqueness of a solution is more subtle, but the asymptotic behavior of the elds determines the metric at least in a perturbative sense (e.g. perturbatively in deviations from pure AdS, or order-by-order in behavior of the various elds. According to the AdS/CFT dictionary, this boundary behavior is determined by the one-point functions of low-dimension local operators associated with the light bulk elds. On the other hand, the bulk spacetime itself allows us to calculated entanglement entropies (and many other non-local quantities). Thus, the assumption that a state is holographic allows us (via gravity calculations) to determine the entanglement entropies and other nonlocal properties of the state (again, at least perturbatively) from the local data provided by the one-point functions: j i ! hO (x )i ! asymptotics ! (x ; z) ! entanglement entropies S(A) (1.1) where here indicates all light elds including the metric.2; 3 The recipe (1.1) could be applied to any state, but for states that are not holographic, the results will be inconsistent with the actual CFT answers. Thus, we have a stringent test for whether a CFT state has a dual description well-described by a classical spacetime: carry out the procedure in (1.1) and compare the results with a direct CFT calculation of the entanglement entropies; if there is a mismatch for any region, the state is not holographic.4 In this paper, our goal is to present some more explicit results for the gravity prediction Sgrav(hO i) in cases where the gravitational equations are Einstein gravity with matter and A the region is taken to be a ball-shaped region B. We will work perturbatively around the vacuum state to obtain an expression as a power series in the one-point functions of CFT 2Here, the region A should be small enough so that the bulk extremal surface associated with A should be contained in the part of the spacetime determined through the equations of motion by the boundary values; we do not need this restriction if we are working perturbatively. 3Results along these lines in the limit of small boundary regions or constant one-point functions appeared in [5{9]. 4Another interesting possibility is that the one-point functions could give boundary data that is not for the radial evolution problem obeys certain constraints. entanglement (1)SB = hHBi, where HB log vBac = 2 Z B d the second-order answer; in this case, it is less clear whether the gravity results from (1.1) should hold for any CFT or whether they represent a constraint from holography. To obtain explicit formulae at this order, we begin by writing SB(j i) = SBvac + hHBi S( Bjj vBac) which follows immediately from the de nition of relative entropy S( Bjj vBac) reviewed in section 2 below. We then make use of a recent result in [11]: to second-order in perturbations from the vacuum state, the relative entropy for a ball-shaped region in a holographic state5 is equal to the \canonical energy" associated with a corresponding wedge of the bulk spacetime. We provide a brief review of this in section 2 below. On shell, the latter quantity can be expressed as a quadratic form on the space of rst-order perturbations to pure AdS spacetime, so we have hHBi SB = 1 2 E ( ; ) + O( 3) : Rearranging this, we have a second-order version of (1.2): ) + O( 3 ) 1 2 E ( Z ; ) + O( 3) : (1.2) (1.3) (1.4) (1.5) (1.6) (1.7) SB(j i) = SBvac + (1)SB + (2)SB + O( 3 ) = SBvac + = SBvac + 2 hHBi Z B 1 2 E ( d d 1 R2 x ; r 2 2R hT00i { 3 { As we review in section 2 below, the last term can be written more explicitly as Z E ( ; ) = !( g; $ g) aTa(b2) b ; where is a bulk spatial region between B and the bulk extremal surface B~ with the same boundary, ! is the \presymplectic form" whose integral de nes the symplectic form 5This second-order relative entropy is known as quantum Fisher information. matter elds, and perturbations on gravitational phase space, Ta(b2) is the matter stress tensor at second-order in the bulk is a bulk Killing vector which vanishes on B~. The rst-order bulk (including the metric perturbation) may be expressed in terms of the boundary one-point functions via bulk-to-boundary propagators Z DB (x; z) = K (x; z; x0)hO (x0)i ; (1.8) where DB is the domain of dependence of the ball B. Given the one-point functions within DB, we can use (1.8) to determine the linearized bulk perturbation in and evaluate (1.7). The expression (1.6), (1.7), and (1.8) together provide a formal result for the ball entanglement entropy of a holographic state, expanded to second-order in the boundary one-point functions. Explicit results for 1+1 dimensional CFTs. In order to check the general formula and provide more explicit results, we focus in section 3 on the case of 1+1 dimensional CFTs, carrying out an explicit calculation of the gravitational contributions to (1.7) starting from a general boundary stress tensor. We nd the result with integrals over the entire domain of dependence region. Recently, in [12] it has been pointed out that the rst-order result (1)S(x ; R) for the entanglement entropy of a ball with radius R and center x can be obtained as the solution to the equation of motion for a free scalar eld on an auxiliary de Sitter space ds2 = LR2d2S ( dR2 + dx dx ) with the CFT energy density hT00(x )i acting as a source term at R = 0. In section 5, we show that in the 1+1 dimensional case, the stress tensor term (1.10) for the entanglement entropy at secondorder can also results from solving a scalar eld equation on the auxiliary de Sitter space if we add a simple cubic interaction term. In an upcoming paper [13], it is shown that this agreement extends to all orders for a suitable choice of the scalar eld potential. The resulting nonlinear wave equation also reproduces the second-order entanglement entropy HJEP06(21)85 near a thermal state in the auxiliary kinematic space recently described in [14]. Including the contributions from matter elds or moving to higher dimensions, the expression for entanglement entropy involves one-point functions on the entire causal diamond DB, so reproducing these results via some local di erential equation will require a more complicated auxiliary space that takes into account the time directions in the CFT. This direction is pursued further in [13, 15]. Discussion. While the explicit two-dimensional stress tensor contribution (1.10) can be obtained by a direct CFT calculation for a special class of states, we emphasize that in general the holographic predictions from (1.1) are expected to hold only for holographic states in CFTs with gravity duals. It would be interesting to understand better whether all of the second order contributions we considered here are universal for all CFTs or whether they represent genuine constraints/predictions from holography.6 In the latter case, and for the results at higher order in perturbation theory, it is an interesting question to understand better which CFT states and/or which CFT properties are required to reproduce the results through direct CFT calculations. This should help us understand better which theories and which states in these theories are holographic. 2 Background Our holographic calculation of entanglement entropy to second-order in the boundary onepoint functions makes use of the direct connection between CFT quantum Fisher information and canonical energy on the gravity side, pointed out recently in [11]. We begin with a brief review of these results. 2.1 Relative entropy and quantum Fisher information Our focus will be on ball-shaped subsystems B of the CFTd, for which the the vacuum density matrix is known explicitly through (1.3). More generally, we can write it as vac = e HB ; B HB = Z B0 B T ; (2.1) 6There is evidence in [16{18] that at least some of the contributions at this order can be reproduced by CFT calculations in general dimensions, since they arise from CFT two and three-point functions, though is the volume form on the surface perpendicular to a unit vector n , and B is a conformal Killing vector de ned in the domain of dependence region DB, with B = 0 on @B. For the ball B with radius R and center x0 in the t = t0 slice, we have R [R2 (t t0)2 (~x By the conservation of the current B T associated with this conformal Killing vector, the HJEP06(21)85 integral in (2.1) can be taken over any spatial surface B0 in DB with the same boundary For excited states, the density matrix measure of this di erence is the relative entropy B will generally be di erent than vBac. One S( Bjj vBac) = tr( B log B) tr( B log vBac) = hHBi SB ; where HB is the vacuum modular Hamiltonian given in (2.1), SB = tr( B log B) is the entanglement entropy for the region B and indicates the di erence with the vacuum B( ) = vac + B hHBi) so the leading contribution to relative entropy appears at second-order in . This quadratic in 1 with no contribution from 2 , S( B( )jj vBac) = 2 h 1 ; 1i vBac + O( 3) ; h ; i 1 2 tr d d log( + ) =0 : This quadratic form, which is positive by virtue of the positivity of relative entropy, de nes a positive-(semi)de nite metric on the space of perturbations to a general density matrix . This is known as the quantum Fisher information metric. Rearranging (2.4) and making use of (2.6), we have SB = SBvac + Z B0 BhT i 2 h 1 ; 1i vBac + O( 3) : This general expression is valid for any CFT, but the O( 2) term involving the quantum Fisher information metric generally has no simple expression in terms of local operator expectation values. However, for holographic states we can convert this term into an expression quadratic in the CFT one-point functions by using the connection between quantum Fisher information and canonical energy. For a one-parameter family of states near the vacuum, we can expand B as (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) Consider now a holographic state, which by de nition is associated with some dual asymptotically AdS spacetime M . Near the boundary, we can describe M using a metric in Fe erman-Graham coordinates as z2 ds2 = 2 `AdS dz2 + dx dx + zd (x; z)dx dx (2.9) where (z; x) has a nite limit as z ! 0 and = 0 for pure AdS. The relative entropy S( Bjj vBac) can be computed at leading order in large N by making use of the holographic entanglement entropy formula, which relates the entanglement entropy for a region A to the area of the minimal-area extremal surface A~ in M with SA Area(A~) hHBi term in relative entropy to a gravitational quantity, since it implies that the expectation value of the CFT stress tensor is related to the asymptotic behaviour of the metric through [19] (2.10) (2.11) (2.12) (2.13) h T i = d`dAd1S Thus, for holographic states, we can write For a one-parameter family of holographic states j ( )i near the CFT vacuum, the dual spacetimes M ( ) can be described via a metric and matter elds = (g; matter) with some perturbative expansion g = gAdS + g1 + 2 g2 + O( 3) ; matter = matter + 2 1 matter + O( 3) : 2 By the result (2.8) from the previous section, the second-order contribution to entanglement entropy is equal to the leading order contribution to relative entropy. This is related to a gravitational quantity via (2.12). The main result in [11] is that this second-order quantity can be expressed directly as a bulk integral over the spatial region the integrand is a quadratic form on the linearized bulk perturbations g1 and To describe the general result, consider the region between B and B~ in pure AdS spacetime, and de ne RB as the domain of dependence of this region, as shown in gure 1. Alternatively, RB is the intersection of the causal past and the causal future of DB; it can be thought of as a Rindler wedge of AdS associated with B. On RB, there exists a Killing vector which vanishes at B~ and approaches the conformal Killing vector B at the matter. 1 between B and B~ where boundary. In Fe erman-Graham coordinates, this is [R2 z 2 (t t0)2 (~x (2.14) R { 7 { t B HJEP06(21)85 extremal surface B~. blue lines indicate the ow of B, and the red lines B. The surface lies between B and the The vector B is timelike hence de nes a notion of time evolution within the region RB; the \Rindler time" associated with this Rindler wedge. The \canonical energy", dual to relative entropy at second-order, can be understood as the perturbative energy associated with this time, as explained in [20]. This is quadratic in the perturbative bulk elds including the graviton, and given explicitly by E ( g1; 1) = W ( 1; $ B 1) = = = Z Z Z !full ( 1; $ B ! ( g1; $ B g1) + !( g1; $ B g1) 1) Z Z !matter ( 1; $ B 1) In the rst line, W is the symplectic form associated with the phase space of gravitational solutions on , and $ B 1 is the Lie derivative with respect to on 1, the rst-order perturbation in metric and matter elds. The symplectic form is equal to the integral over of a \presymplectic" form !full which splits into a gravitational part and a matter part as in the third line. The matter part can be written explicitly in terms of Ta(b2), the matter stress tensor at quadratic order in the elds, while the gravitational part ! is given explicitly by !( 1; 2) = 1 In deriving (2.15) it has been assumed that the metric perturbation has been expressed in a gauge for which the coordinate location of the extremal surface B~ does not change (2.15) (2.16) { 8 { (so that B continues to vanish there), and the vector B continues to satisfy the Killing equation at B~. Thus, we require that BjB~( ) = 0; $ B g( )jB~( ) = 0: (2.17) (2.18) As shown in [20], it is always possible to satisfy these conditions; we will see an explicit example below. 3 Second-order contribution to entanglement entropy Using the result (1.7), we can now write down a general expression for the ball entanglement entropy of a general holographic state up to second-order in perturbations to the vacuum state, in terms of the CFT one-point functions. According to (2.8) and (2.15), the secondorder term in the entanglement entropy for a ball B can be expressed as an integral over the bulk spatial region between B and the corresponding extremal surface B~, where the integrand is quadratic in rst-order bulk perturbations. These linearized perturbations are determined by the boundary behavior of the elds via the linearized bulk equations. In general, to determine the linearized perturbations in the region (or more generally in the Rindler wedge RB), we only need to know the boundary behavior in the domain of dependence region DB, as discussed in detail in [21]. The relevant boundary behaviour of each bulk eld is captured by the one-point function of the corresponding operator. We can express the results as Z DB ( 1) (x; z)j = ddx0K (x; z; x0)hO (x0)iCF T (3.1) where K (x; z; x0) is the relevant bulk-to-boundary propagator. As discussed in [21{23], K should generally be understood as a distribution to be integrated against consistent CFT one-point functions, rather than a function. Since the expression (3.1) is linear in the CFT expectation values, the result (1.7) is quadratic in these one-point functions and represents our desired second-order result. To summarize, for a holographic state, the second-order contribution to entanglement entropy in the expansion (2.8) is the leading order contribution to the relative entropy S( Bjj vBac). This is dual to canonical energy, given explicitly by: (2)SB = h 1 ; 1i vBac = E ( 1 ; 1) = !( g1; $ B g1)+ BaTa(b2) b : (3.2) 1 2 and these can be expressed in terms of the CFT one-point functions on DB as (3.1). Example: CFT2 stress tensor contribution In this section, as a sample application of the general formula, we provide an explicit calculation of the quadratic stress tensor contribution to the entanglement entropy for { 9 { In two dimensions, these constraints can be expressed most simply using light-cone coordinates x = x t, where we have h T i = 0 : Thus, a general CFT stress tensor can be described by the two functions, hT++(x+)i and h T (x ) . i Assuming that the state is holographic, there will be some dual geometry of the form (2.9). According to (2.11), the stress tensor expectation values determine the asymptotic form of the metric as ++(x; 0) = 8 GN `AdS hT++(x+) i (x; 0) = 8 GN `AdS h T (x ) i Now, suppose that our state represents a small perturbation to the CFT vacuum, so that the stress tensor expectation values and the asymptotic metric perturbations are governed by a small parameter : ++(x; 0) h+(x+) holographic states in two-dimensional conformal eld theories. This arises from the rst term in (1.7). For a general CFT state, the stress tensor is traceless and conserved, ab = hab + ($V g)ab = hab + raVb + rbVa : (3.10) Then the metric perturbation throughout the spacetime is determined by this asymptotic behavior by the Einstein equations linearized about AdS. Here, we need only the components in the eld theory directions, which give 1 The solution in our Fe erman-Graham coordinates with boundary behaviour (3.6) is (+1+)(x; z) = h+(x+) (1) (x; z) = h (x ) with the linearized perturbation (1) independent of z. Satisfying the gauge conditions. We would now like to evaluate the metric contribution to (3.2) This formula assumes the gauge conditions (2.17) which di er from the Fe erman-Graham gauge conditions we have been using so far. Thus, we must nd a gauge transformation to bring our metric perturbation to the appropriate form. In general, we can write (3.3) (3.4) is the desired metric perturbation satisfying the gauge condition, and h is the perturbation in Fe erman-Graham coordinates (equivalent to for d = 2). The procedure for nding an appropriate V and evaluating (3.9) is described in detail in [11], but we review the main points here. De ning coordinates (XA; Xi) so that the extremal surface lies at some xed value of XA with Xi describing coordinates along the surface, the gauge condition (2.17) (equivalent to requiring that the coordinate location of the extremal surface remains xed) gives ririVA + [ri; rA]V i + rihiA 1 2 rAhii while the condition (2.18) that B continues to satisfy the Killing equation at B~ gives (hiA + riVA + rAVi) jB~ = 0 ; A h D 12 ADhCC + rAVD + rDV A D DrC V CC = 0 : To solve these, we rst expand our general metric perturbation in a Fourier basis. h (t; x; z) = Z h + +h^+(k)eikx+ + h^ (k)eikx i dk ; with a gauge choice hza(t; x; z) = 0. For each of the basis elements, we use the equations (3.11), (3.12) and (3.13) to determine V and its rst derivatives at the surface V . For these calculations, it is useful to de ne polar coordinates (z; x) = (r cos ; r sin ). Since the gauge conditions are linear in V , the conditions on V for a general perturbation are obtained from these by taking linear combinations as in (3.14), Va(t; x; z) = Z hV^a+(k)eikx+ + V^a (k)eikx i dk : (3.11) (3.12) (3.13) (3.14) (3.15) After requiring Va remain nite at = 2 , we nd e ikt e ikt k3r2 cos2 k3r2 cos2 e ikt 2 k2 r cos e ikt k3r2 cos i cos(kr) + sin sin(kr) i sin(kr) i sin cos(kr) (k2r2 cos2 1)eikr sin 2 (k2r2 cos2 sin + ikr cos2 + 2i sin )eikr sin ! 2 (2 + k2r2 cos2 2 ikr sin )eikr sin 2 sin(kr) k3r2 2i cos(kr) + 2kr sin +r3k3 sin cos2 +i r2k2 cos2 kr2 + 2 eikr sin (3.16) using where identity where Thus, we have where [ ; V ]a = b !(g; ; $ g) = d ( ; X) !(g; ; $ ) = !(g; h; $ h) + d = (h + $V g; [ ; V ]) ($ h; V ) : E = Z !(g; h; $ h) + (h; V ) : Z and we have used that $ g = 0. We can simplify this expression using the gravitational ( ; X) = 1 V^t+(k; t; r; ). The results here give the behavior of V and its derivatives only at the surface B~ (r = R in polar coordinates). Elsewhere, V can be chosen arbitrarily, but we will see that our calculation only requires the behavior at B~. Evaluating the canonical energy. Given the appropriate V , we can evaluate (3.9) !(g; ; $ ) = !(h + $V g; $ B (h + $V g)) = !(g; h; $ h) + !(g; h + $V g; $[ ;V ]g) !(g; $ h; $V g) Finally, choosing V so that it vanishes at B, we can rewrite (3.9) as In this nal expression, we only need V and its derivatives at the surface B~. Thus, we can now calculate the result explicitly for a general perturbation. In the Fourier basis, the nal result in terms of the boundary stress tensor is where the kernel is K^2(k1; k2) = `AdSK3(K k2). We note in particular that the result splits into a left-moving part and a right-moving part with no cross term. (3.17) (3.18) (3.19) (3.20) (3.21) (3.22) (3.23) (3.24) Transforming back to position space g ; (3.26) where the kernel K2 is symmetric under exchange of x1 and x2 , and has support only on x i 2 [ R; R]. Focusing only on the domain of support, we have Using the relation c = 3`AdS=(2GN ) between the CFT central charge and the gravity parameters, we recover the result (1.10) from the introduction. Like the leading order result in (2.8), the integrals can be taken over any surface B0 with boundary @B. The fact that we only need the stress tensor on a Cauchy surface for DB is special to the stress tensor in two dimensions, since the conservation relations allow us to nd the stress tensor expectation value everywhere in DB from its value on a Cauchy surface. For other operators, or in higher dimensions, the result will involve integrals over the full domain of dependence. We will see an explicit example in the next subsection. Positivity of relative entropy requires E to be positive which requires the kernel to be positive semi-de nite. As we show in appendix A, one can demonstrate that the positivity explicitly, providing a check of our results. An alternative proof of positivity is given in section 5. As a more complete check, we will show in section 4 that this result can be reproduced by a direct CFT calculation for the special class of states that can be obtained from the vacuum state by a local conformal transformation. 3.2 Example: scalar operator contribution We now consider an explicit example making use of the bulk matter eld term in (1.7) in order to calculate the terms in the entanglement entropy formula quadratic in the scalar operator expectation values. The discussion for other matter elds would be entirely parallel. This example is more representative, since the formula will involve scalar eld expectation values in the entire domain of dependence DB, i.e. a boundary spacetime region rather than just a spatial slice. The results here are similar to the recent work in [16{18], but we present them here to show that they follow directly from the canonical energy formula. We suppose that the CFT has a scalar operator of dimension with expectation value with mass m2 = ( d) and asymptotic behavior hO(x)i. According to the usual AdS/CFT dictionary, this corresponds to a bulk scalar eld where is a constant depending on the normalization of the operator O. The leading e ects of the bulk scalar eld on the entanglement entropy (3.2) come from the matter term in the canonical energy (x; z) ! z hO(x)i ; (3.28) (3.29) Using the explicit form of B from (2.14) and from (2.2), this gives (for a ball centered at the origin) This expression is valid for a general bulk matter eld. For a scalar eld, we have 1 where gab is the background AdS metric and 1 represents the solution of the linearized scalar eld equation on AdS, n d 1 o with boundary behavior as in (3.28). This solution is given most simply in Fourier space, where we have 1(k; z) = 2 ( + 1) Z (2 )d k02>~k2 ddk d =2 z 2 J qk2 0 ~k2z hO(k)i ; (3.33) (3.30) (3.31) (3.32) (3.34) (3.35) (3.36) (3.37) k2 0 Z eik x ~k2 This reproduces previous results in the literature [5, 17]. where = d=2, but we can formally write a position-space expression using a bulk-toboundary propagator K(x; z; x0) as [24, 25] 1(x; z) = dx0K(x; z; x0)hO(x0)i : The integral here is over the boundary spacetime, however it has been argued (see, for example [21, 22]) that to reconstruct the bulk eld throughout the Rindler wedge RB (and speci cally on ), we need only the boundary values on the domain of dependence region. We recall some explicit formulae for this \Rindler bulk reconstruction" in appendix B. Combining these results, we have a general expression for the scalar eld contribution to entanglement entropy at second-order in the scalar one-point functions, (2)SBscalar = `dAd1S Z R dz Z 2 0 zd 1 x2<R2 z2 d d 1 x (R2 R z 2 x2) m2 z2 1 2 where 1 is given in (3.33) or (3.34) . As a simple example, consider the case where the scalar eld expectation value is constant. In this case it is simple to solve (3.32) everywhere to nd that Inserting this into the general formula (3.35), and performing the integrals, we obtain 1(x; z) = hOiz : (2)SBscalar = `dAd1S 2 4 hOi2R2 d 2 ( d 2 1 2 ( ) ( + 32 ) d2 + 1) : In section 3.1, we used the equivalence between quantum Fisher information and canonical energy to obtain an explicit expression for the second-order stress tensor contribution to the entanglement entropy for holographic states in two-dimensional CFTs. This is applicable for general holographic states, whether or not other matter elds are present in the dual spacetime (in which case there are additional terms in the expression for entanglement entropy). In special cases where there are no matter elds, the spacetime is locally AdS and we can understand the dual CFT state as being related to the vacuum state by a local conformal transformation. We show in this section that in this special case, we can reproduce the holographic result (3.27) through a direct CFT calculation, providing a strong consistency check. We note that the result does not rely on taking the large N limit or on any special properties of the CFT, so the formula holds universally for this simple class of states. Our approach will be to develop an iterative procedure to express the entanglement entropy as an expansion in the stress tensor expectation value for this special class of states. We evaluate the entanglement entropy for these states from a correlation function of twist operators obtained by transforming the result for the vacuum state.7 Similarly, the stress tensor expectation values follow directly from the form of the conformal transformation. Inverting the relationship between the required conformal transformation and the stress tensor expectation value allows us to express the entanglement entropy as a perturbative expansion in the expectation value of the stress tensor. Similar CFT calculations have also been used recently in [13]. 4.1 Conformal transformations of the vacuum state In two-dimensional CFT, under a conformal transformation w = f (z), the stress tensor (4.1) (4.2) (4.3) (4.4) 2 000(z) 0(z) + 3 00(z)2 + 3 0(z)2 000(z) + 3 0(z) 00(z)2 + The CFT vacuum is invariant under the SL(2; C) subgroup of global conformal transformations. However, for transformations which are not part of this subgroup, the vacuum state transforms into excited states. The action of the full conformal group includes the full Virasoro algebra which involves arbitrary products and derivatives of the stress tensor Id transforms as derivative expanded as fz + (z); zg = 000(z) 2 2 Here c is the central charge of the CFT and the inhomogeneous part is the Schwarzian T 0(w) = dw dz T (z) + c 12 ff (z); zg : ff (z); zg f 000(z) f 0(z) 3f 00(z)2 2f 0(z)2 : For an in nitesimal transformation f (z) = z + (z), the Schwarzian derivative can be These states capture the gravitational sector of the gravity dual. Other excited states can be obtained by the action of other primary operators and their descendants. However we restrict ourselves to the class states that are related to `pure gravity' excitations, which are the states obtained by conformal transformation of the vacuum state. We denote the excited state as jf i = Uf j0i where Uf is the action of a conformal transformation on the vacuum j0i. The expectation value of the stress tensor for the state perturbed state jf i is hf jT (z)jf i = h0jUfy T (z) Uf j0i = h0jT 0(w)j0i = df dz where we used that h0jT (z)j0i = 0. The anti-holomorphic component of the stress tensor T (z) is similarly related to the anti-holomophic part of the conformal transformation f . To leading order in a conformal transformation near the identity, this equation relates the conformal transformation to hT (z)i by a third-order ordinary di erential equation. The three integration constants correspond to the invariance of hT (z)i under the global conformal transformations. Thus we have an invertible relationship between the conformal transformations modulo their global part and hT (z)i, at least near the identity. 4.2 Entanglement entropy of excited states In a two-dimensional CFT, the entanglement entropy can be explicitly computed using the replica method [27, 28]. The computation can be reduced to a correlation function of twist operators , which are conformal primaries with weight (hn; hn) = 2c4 (n 1=n; n 1=n). The Renyi entropy is exp (1 n)S(n) = h +(z1) (z2)i = (z2 z1) 2hn (z2 z1) 2hn : (4.6) The entanglement entropy is obtained by taking the n ! 1 limit of S(n). Svac = lim S(n) = lim (1 n) 1 log(z2 z1) 2hn (z2 z1) 2hn n!1 n!1 c 12 = log (z2 z1)2(z2 z1)2 For the excited states obtained by conformal transformations z ! w = f (z) the Renyi entropy is exp (1 n)Se(xn) = hf j +(z1) (z2)jf i (4.7) (4.8) = df dz z1 hn df hn df hn df hn dz z2 dz z1 dz z2 h0j +(z1) (z2)j0i : (4.9) Here z1; z2 are the points f (z1) = f (z1) = R, f (z2) = f (z2) = R. The entanglement entropy of the excited state is Sex = lim Se(xn) = n!1 c 12 log f 0(z1)f 0(z2)f 0(z1)f 0(z2)(z2 z1)2(z2 z1)2 2 2 : (4.10) Therefore the change in entanglement entropy respect to the vacuum state is S Sex Svac = c 12 + log c 12 log f 0(f 1(R))f 0(f 1 ( R))(f 1(R) By inverting (4.5), the conformal transformation required to reach the state jf i can be expressed as a function of the expectation value of the stress tensor. Plugging this f into (4.11), allows us to express the entanglement entropy as a function of the expectation value of the stress tensor alone, as we set out to do. In practice, we will invert (4.5) order by order in a small conformal transformation and express the entanglement entropy as an expansion in the resulting small stress tensor. The second-order term in this expansion will be the Fisher information metric. In the following, we will focus on the holomorphic term in (4.5), noting that the antiHJEP06(21)85 holomorphic part follows identically.8 Perturbative expansion Consider a conformal transformation perturbation near the identity transformation w = f (z) = z + f1(z) + 2f2(z) + 3f3(z) + ; (4.12) where is a small expansion parameter. In this expansion, 1c2 hT (w)i = f1000(w)+ 2 32 f100(w)2 and the entanglement entropy is 12 c Sex = log f 0(z1)f 0(z2)(z2 z1)2 2 3f10 (w)f1000(w) + f2000(w) f1(w)f10000(w) +O( 3) ; (4.13) = log (2R)2 2 + + 2 + O( 3) : R (f10 ( R) + f10 (R)) + f1( R) f1(R) R (f1(R) f1( R))2 4R2 + f1( R)f10 ( R) + f1(R)f10 (R) + f2( R) f2(R) R Linear order. To rst-order in , the stress tensor is given by 8Note that the potential cross-term between left and right moving contributions vanished in the gravitational computation of (2)S. (4.15) so that change in the expectation value of the modular Hamiltonian becomes hHBi = dz (R2 z2)f1000(z) z2)f100(z) + 2 zf10 (z) f1(z) R R R(f10 (R) + f10 ( R)) (f1(R) f1( R)) : From (4.11) we also have that the rst-order change in entanglement entropy is Comparing with (4.16) we see that the rst law of entanglement holds (1)S = hHBi : Second-order. The second-order change in entanglement entropy gives the second-order relative entropy as the modular Hamiltonian is linear in the expectation value of the stress tensor. This is the quantum Fisher metric in the state space, which is dual to the canonical energy in gravity [11]. In this section, we obtain the expression for canonical energy from the CFT side and nd an exact match to the results of section 3.1. Our procedure so far yields the entanglement entropy of a subregion in terms of a perturbative expansion in small stress tensor expectation value S = K1(z)hT (z)i Z dz B 2 + fz $ zg : To obtain K2(z1; z2), we need to invert the relationship in (4.13) order by order, the lower order solutions fi 1; fi 2; f1 becoming sources for the i-th order solution. Taking the explicit expression for hT (z)i to simplify solving the di erential equations, is su cient to extract the Fourier transformed kernel. The rst-order solution is f1(z) = F1 + F2z + F3z2 + 12i c eik1z ; where Fi are constants that corresponds to the global part of the conformal transformation and do not e ect the nal result. We take these constants to be zero for simplicity. The second-order solution is f2(z) = (4.16) (4.17) (4.18) (4.19) (4.20) (4.21) (4.22) With these solutions, we obtain K~1(k) = k2 2 sin (kR) kR cos (kR) kR ; as well as K~2(k1; k2) = c where is a window function with support x 2 [ R; R]. The second-order position space kernel is Z K1(z) = dk K~1(k)e ikz = R2 z 2 R W (R; z) W (R; x) (sgn (R + x) + sgn (R x)) 2 K2(z1; z2) = 6 2 ((R cR2 z1)2(R + z2)2 (R + z1)2(R z2)2 R R z2 Taking the inverse Fourier transformation of K~1(k) This can be obtained by a conformal transformation from the vacuum with h jT j i = 2 c this reproduces the kernel for canonical energy in (3.27). t = 0 slice, z = z = x and our result becomes This result holds for regions de ned on any spatial slice of the CFT. If we choose the Changing variables using x1 = x r, x2 = x + r, the kernel is simply SE(2E) = Z B dx1 dx2 K2(x1; x2) [hT++(x1)ihT++(x2)i + hT (x1)ihT (x2)i] : The anti-holomorphic part is the same with z ! z, and the cross term vanishes. With the relation c = 3`AdS 2GN K2(x; r) = K2(x; r) = jrj)2 Excited states around thermal background A similar analysis can be applied to perturbations around a thermal state with temperature T = 1. If we denote homogeneous thermal state j i, the stress tensor one-point function is On top of this transformation, one could also apply an in nitesimal conformal transformation to obtain non-homogeneous perturbation around thermal state. A similar computation as the previous section leads to the rst-order kernel 2 K1 (z) = sinh( 2 R ) sinh (R z) sinh (R + z) ; (4.32) which is the modular hamiltonian of thermal state in 2d CFT. Furthermore, the second-order kernel is K2 (z1; z2) = Consistency check: homogeneous BTZ perturbation As a check, consider the homogeneous perturbation example, where hT i = hT i = 8GN .9 In AdS3 this is a perturbation towards the planar BTZ geometry ds2 = 1 z2 dz2 + (1 + z2=2)2dx2 (1 z2=2)2dt2 in Fe erman-Graham coordinates. Holographic renormalization (2.11) tells us the stress tensor expectation value of the dual CFT is 1 2 hTtti = hT i + hT i = As the black hole corresponds to the thermal state in CFT, the dual state be obtained by (4.34) (4.35) (4.36) (4.37) (4.38) the conformal transformation (4.31). entropy with respect to the vacuum is First, applying (4.11) for this conformal transformation, the change in entanglement S = R2 6G which matches the previous known results [5, 11]. The linear order equals hHBi as expected from the entanglement rst law. The second-order term gives the quantum Fisher information or the canonical energy E = d 2 d 2 ( E S) =0 = R4 45GN : Using the formula using the second-order kernel (4.19) and (4.27), we obtain the same canonical energy E = 2 d 2 1 Z dz1 Z dz2 K2(x1; x2)hT ihT i =0 = R4 45GN : 9 = 2 2 sets the temperature. S+ since S follows identically. iary de Sitter space with metric In [ 12 ], it was pointed out that the leading order perturbative expression (1.2) for entanglement entropy, expressed as a function of the center point x and radius R of the ball B, is a solution to the wave equation for a free scalar eld on an auxiliary de Sitter space, with hT00(x)i acting as a source. It was conjectured that higher order contributions might be accounted for by local propagation in this auxiliary space with the addition of self-interactions for scalar eld. In this section, we show that for two-dimensional CFTs, the second-order result (1.10) can indeed be reproduced by moving to a non-linear wave equation with a simple cubic interaction to this scalar eld. A slight complication is that we actually require two-scalar elds; one sourced by the holomorphic stress tensor T++, and the other sourced by the anti-holomorphic part T ; the perturbation to the entanglement entropy is then the sum of these two scalars, S = S+ + S , reproducing both terms in (1.10). We will focus on To reproduce the second-order results for entanglement entropy, we consider an auxiland consider a scalar eld S+ with mass m2L2dS = 2 and action ds2dS = L2dS R2 The equation of motion is As shown in [ 12 ], the rst-order perturbation (1.2) obeys the linearized wave equation 2 rdS m2 S+(R; x) = 12 cL2dS ( S+(R; x))2 : 2 rdS m2 (1)S+(R; x) = 0 : We can immediately check that the second-order perturbation (1.10) is consistent with the nonlinear equation by acting with the dS wave equation on the second-order kernel (4.27) 2 rdS m2 K2(x1 x; x2 x) = 24 cL2dS K1(x1 x)K1(x2 x) : Integration against the CFT stress tensor then gives (5.3). Alternatively, introducing the retarded10 bulk-to-bulk propagator [29] GdS( ; x; 0; x 0) = 2 + 0 2 (x x 0)2 4 0 10These propagators are de ned to be non-zero only within the future directed light-cone. This is important in reproducing both the support and the exact form of K2(x1; x2). (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) and bulk-to-boundary propagator !0 KdS( ; x; x 0) = lim 4 lim GdS( ; x; 0; x 0) = we can show directly that the solution with boundary behavior 2 (x x0)2 ; δS+ x1 0! 4 Z (R, x) δS+ g3 δS+ x2 (5.7) (5.8) (5.9) #2 : (5.11) x − R x + R cubic interaction given by (5.2). The bold (red) line is the conformal boundary of de Sitter which is identi ed with a time slice of the CFT. S+ is sourced by the CFT stress tensor on this boundary. where pjgdSj and the squared expression are manifestly positive and the bulk-to-bulk propagator (5.6) is positive over the range of integration where (y x0)2 (R )2. That this expression is negative is required by the positivity of relative entropy, since we showed above that (2)S represents the leading order perturbative expression for the relative entropy. Recently, it has been realized that the modular Hamiltonian in certain non-vacuum states in two dimensional CFTs can be described by propagation in a dual geometry [14] For R ! 0 gives at rst-order and (2)S+(R; x0) = 12 Z cL2dS dS S+ = 3 hT++iR2 + O(R3) : (1)S+(R; x0) = dx KdS(R; x0; x)hT++(x)i d 0dx0pjgdSj GdS(R; x0; 0; x0) dx KdS( 0; x0; x)hT++(x)i Z 2 ; (5.10) at second-order, where the latter term comes from the diagram shown in gure 2. The integrals can be performed directly to show that these results match with the expressions (1.2) and (1.10) respectively. manifestly negative. More explicitly, we have A useful advantage of writing the second-order result in the form (5.10) is that it is (2)S+(R; x0) = cL2dS d dypjgdSj R2 + 2 (x0 y)2 "Z R By dx KdS ( ; y; x)hT++(x)i matching the kinematic space found previously in [30{33]. We nd that the results of section 4.4 can be explained by the same interacting theory (5.2) on this kinematic space. The kinematic space dual to the thermal state is11 ds2 = 4 2L2dS 2 sinh2 2 R (5.12) The second-order perturbation to the entanglement entropy from (4.33) obeys the wave equation (5.3) with the same interactions in this kinematic space. We could imagine adding additional elds propagating in de Sitter to capture the contributions to the entanglement entropy from scalar operators discussed in section 3.2. However, unlike the contribution from the stress tensor, this contribution involves integration of the one-point functions over the full domain of dependence DB. In higher-dimensions, this will also be true for the stress tensor contribution. The R = 0 boundary of the auxiliary de Sitter space does not include the time direction of the CFT, so any extension of these results to contributions of other operators or higher dimensional cases will require a more sophisticated auxiliary space. Promising work in this direction is discussed in [13, 15]. Acknowledgments We thank Michal Heller, Ali Izadi Rad, Nima Lashkari, Don Marolf, Robert Myers, and Philippe Sabella-Garnier for helpful discussions. This research is supported in part by the Natural Sciences and Engineering Research Council of Canada, and by grant 376206 from the Simons Foundation. A Direct proof of the positivity Consider the left moving part of perturbation h+(x+) / T++(x+). The real space h+(x) must be real valued functions for a perturbation of AdS3. We can expand h+(x) in a Taylor series h+(x) = P1 n=0 anxn so that the canonical energy is given by E X n X anam m Z Z B B dx1dx2 x1nx2mK2(x1; x2) X n m X anamRn+m+4 An;m : (A.1) where the proportionality factor is up to a positive constant and An;m = 1 (n + m + 3)(n + m + 1) > 8 > > < 0 1 (n+1)(m+1) > : nmnm(n++n2+)(mm++32) if n + m odd if n; m even if n; m odd (A.2) which is clearly non-negative and symmetric in n; m. 11The kinematic space dual to the BTZ black hole was rst described in [30, 31]. The explicit form of the metric in the coordinates we are using can be found in [14]. To show that the canonical energy is positive, we need to show the matrix M with entries given by An;m = An 1;m 1 12 is positive de nite. To do so, we will use proof by induction and Sylvester's criterion which states that a square matrix M is positive de nite if and only it has a positive determinant and all the upper-left submatrices also have a positive determinant. Proof by induction Suppose that the N N matrix MN whose components are given by An;m is positive de nite. Then consider the block matrix constructed as HJEP06(21)85 MN+1 = AN+1;N+1 MN B ! BT 1 (A.3) (A.4) (A.5) where B is a N -column vector with entries given by Ai;N+1. Since MN is positive-de nite, it has a positive determinant and all the upper-left submatrices of MN also have a positive determinant by Sylvester's criterion. To show that MN+1 is positive-de nite, we need only show it has a positive determinant since all the upper-left submatrices are already known . The determinant of MN+1 can be evaluated using the formula det(MN+1) = AN+1;N+1 2 det(MN ) det(MN + BT B) so it is su cient to show det(MN + BT B) < 2 det(MN ) : We denote the eigenvalues of MN + BT B by i M+Bwhere they are ordered from largest to smallest 1 M+B 2 M+B : : : non-zero eigenvalue is given by N M+B: Since BT B is a rank-one matrix, the sole = Tr(BT B) = PiN=1 Ai;N+1 0. Since BT B is positive semi-de nite, there exists an upper bound on det(MN +BT B) given by the Weyl inequality M+B i 1 2 iM + iwhere i M are the eigenvalues of MN in order from largest to smallest : : : N . We then expand the determinant as det(MN + BT B) = Y 1 M+B N M Y i=1 iM = 1 + M 1 det(MN ) : (A.6) So it remains to show that 1 0 to complete the proof. The maximum eigenvalue 1M is bound from below by the minimum sum of a column of MN through the PerronFrobenius theorem (equivalently Gershgorin circle theorem). For the matrix MN , the minimum sum of a column vector is simply the sum of the N -th column PiN=1 Ai;N since Ai;j decreases with i and j. Therefore it remains to show N i=1 M i M+B B N X i=1 Ai;N 2 Ai;N+1 0 : (A.7) 12The inelegant notation change is due to conventional matrix notation starting at n = 1, while the Taylor series starts at n = 0. X A2i;N (N+1)=2 X i=1 (N+1)=2 X i=1 N=2 i=1 We split this sum up into two cases. The rst case is if N is even. Then we have since the nal term in Pi(=N1 1)=2 A22i 1;N+1 is zero. Explicitly analyzing the coe cients, we see that A2i;N A22i 1;N+1 is always positive for all i 2 f1 : : : N=2g, so clearly the entire sum is positive. In the case of odd N , the sum becomes Each term in this sum is also positive, so we have shown M 1 B 0. The expressions in (A.8) and (A.9) are not obviously positive, but they reduce to some polynomial equations which can be shown to be positive. Therefore we've shown det(MN + BT B) < 2 det(MN ), thus MN+1 is positive-de nite given that MN is. Since M1 is positive-de nite we completed the proof by induction. The kernel for canonical energy is explicitly positive-semide nite as required by the positivity of relative entropy. B Rindler reconstruction for scalar operators in CFT2 In this appendix we nd an expression for the matter contribution to the second-order perturbation to the entanglement entropy of a ball B using Rindler reconstruction so as to only use the one-point functions of the scalar operator in the domain of dependence DB. We specialize to two dimensional CFTs in order to obtain a more explicit expression which can be compared to the gravitational contribution (1.10). Further discussions of Rindler reconstruction can be found in the literature [21, 22, 24, 25, 34]. Coordinates on the Rindler wedge RB of radius R can be given by (r; ; ) which map back into Poincare coordinates by where 1 < r < 1. using [21] (r; ; ) = Z The scalar eld dual to an operator O can be reconstructed in this Rindler wedge f!;k(r) = r 1 d!dk e i! ik f!;k(r)O!;k ; 1 r2 i ! 2 2F1 2 i(! + k) 2 2 i(! + k) 2 ; ; r 2 z = t = x = r cosh r cosh r cosh Rpr2 1 sinh R + pr2 + pr2 Rr sinh + pr2 1 cosh 1 cosh 1 cosh ; ; ; (B.1) (B.2) (B.3) (B.4) (B.5) where O!;k is the Fourier transform of the CFT expectation value of the operator This can be expressed in terms of the operator in the original coordinates O!;k = Z d d ei! +ik hO( ; )i : dtdx h(R + x + t)i k+2! (R x t) i k+2! 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Matthew J. S. Beach, Jaehoon Lee, Charles Rabideau. Entanglement entropy from one-point functions in holographic states, Journal of High Energy Physics, 2016, 85, DOI: 10.1007/JHEP06(2016)085