Perturbative expansions of Rényi relative divergences and holography
PPerturbative expansions of R´enyi relative divergencesand holography
Tomonori Ugajin
Okinawa Institute of Science and Technology,Tancha, Kunigami gun, Onna son, Okinawa 1919-1
Abstract
In this paper, we develop a novel way to perturbatively calculate R´enyi relativedivergences D γ ( ρ || σ ) = tr ρ γ σ − γ and related quantities without using replica trick aswell as analytic continuation. We explicitly determine the form of the perturbativeterm at any order by an integral along the modular flow of the unperturbed state.By applying the prescription to a class of reduced density matrices in conformalfield theory, we find that the second order term of certain linear combination of thedivergences has a holographic expression in terms of bulk symplectic form, which is aone parameter generalization of the statement ”Fisher information = Bulk canonicalenergy”. a r X i v : . [ h e p - t h ] D ec ontents T (1) γ ( δρ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Second order term T (2) γ ( δρ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.1 Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 n = 2 term . . . . . . . . . 114.3 Contour choice: n ≥ T γ ( ρ ) . . . . . . . . . . . . . . . . . . . . . . 155.3 Bring n = 2 term to the standard form . . . . . . . . . . . . . . . . . . . . . 16 D γ ( ρ || σ ) X γ ( δρ ) and Y γ ( δρ ) by modular flow integrals . . . . . . . . . . . 186.2 Holographic expressions of X γ ( δρ ) and Y γ ( δρ ) . . . . . . . . . . . . . . . . . 196.2.1 Set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2.2 Holographic rewritings . . . . . . . . . . . . . . . . . . . . . . . . . . 20 K ( n ) γ ( s , · · · s n − )
23B Fixing the contour of n = 2 term 25C Simplifying T (2) γ ( δρ )
26D Direct Fourier transformation 28 Details of the holographic rewriting 29
The concept of entanglement is one of the keys to understand how holography works. Thisidea is supported by the Ryu Takayanagi formula [1, 2] and its covariant generalization [3],which relate the area of particular extremal surfaces in the bulk, to the entanglement en-tropies in the dual conformal field theory (CFT). As a concrete and quantitative applicationof this entanglement vs gravity program, recently it has been shown that bulk gravitationaldynamics can be read off from the entanglement structure of states in the dual CFT.In this line of developments, it was initially observed that so called first law of entangle-ment [4] is related to the linearized Einstein equations in the bulk [5, 6]. Consider startingfrom the vacuum reduced density matrix ρ and making it excited slightly ρ → ρ + δρ in a CFT. The change of the entanglement entropy δS obeys first law of entanglement, δS = tr [ Kδ ], where K = − log ρ is called modular Hamiltonian of ρ . For the subsys-tems of special type, the vacuum modular Hamiltonian has a local expression given by anintegral of energy density over the subsystem. There is a natural bulk counterpart of thevacuum modular Hamiltonian, namely, the generator of time translation of a topologicalblack hole with hyperbolic horizon, whose Bekenstein Hawking entropy gives the CFT vac-uum entanglement entropy [7]. The first law of entanglement is related to the first law ofthermodynamics applied to the topological black hole, and this enabled us to read off thelinearized equations of motion.Recently this nice story at the first order in the perturbation δρ has been generalizedto the quadratic order . It was noticed that, in CFT the second order change of the entan-glement entropy can be concisely summarized as an integral of correlation functions alongthe flow generated by the vacuum modular Hamiltonian K = − log ρ on the subsystem[8, 9, 10]. This was further extended to arbitrarily order in δρ and some technical issue waspointed out [11]. It was also recognized that by rewriting the CFT answer in terms of bulkvariables, we naturally identify it with bulk canonical energy[12], which was first foundholographically in [13]. This makes it possible to read off the bulk equations of motionbeyond the linearized level. 2iven these developments, it is now natural to generalize this story to other quan-tum information theoretic quantities. In particular we would like to find such a quantitywhich admits a nice perturbative expansion in CFT and has a dual holographic expression.Natural candidates having these properties are those involving powers of reduced densitymatrices, for example tr ρ γ which is related to R´enyi entropy.Conventionally, a R´enyi type quantity, like tr ρ γ has been computed by replica trick. Inthis trick, we first regard the R´enyi index to be a positive integer γ = n , and represent thequantity as a path integral on a branched space Σ n which is prepared by gluing n copiesof the original space with cuts along the subsystems. After the computation of the pathintegral, we then analytically continue the integer n to arbitrarily number γ . However,this trick has several disadvantages, even when we compute the quantity perturbatively.First of all, the analytic continuation is usually difficult to perform. For example, when weperturbatively expand tr ρ n for ρ = ρ + δρ , at quadratic order we encounter following sum (cid:88) k,m tr (cid:2) ρ k − δρρ m − k − δρρ n − m (cid:3) . (1)In order to analytically continue it in n we first need to perform this sum to get a closedexpression. Although for special cases we can do this, in general it is difficult. In additionto this, we do not know how to do analogous sums for the cubic term and higher. Second,there are ambiguities in the analytic continuations. According to the Carson’s theorem, weneed to correctly specify the behavior of tr ρ n on certain region of the complex n plane, inorder to fix the ambiguities.In order to overcome these difficulties, in this paper we would like to develop a new wayto perturbatively calculate R´enyi type quantities without using replica trick, and analyticcontinuation. The idea we employ is simple, namely writing tr ρ γ by a contour integral,tr ρ γ = (cid:90) C dz πi z γ tr 1 z − ρ , (2)where the contour C is chosen so that it includes all the poles of the integrand, but avoidthe contribution of the branch cut coming from z γ . We refer to [14, 15] for discussions onthe representation. By expanding the denominator of the integrand for perturbative states ρ = ρ + δρ , we can systematically write each term of the perturbative expansion by an3ntegral along the modular flow of the reference state ρ . If we apply this expansion for aclass of perturbative excited states from vacuum in a d dimensional CFT, we can write eachterm as an integral of a correlation function (cid:104)· · ·(cid:105) Σ γ on the branched space Σ γ = S γ × H d − along the modular flow generated by ρ . Here, S γ denotes the Euclidean time circle with2 πγ periodicity, and H d − is d − γ are difficult to calculate when d >
2, as the branched spaceΣ γ is not conformally related to d dimensional flat space, and even two point functions arehighly theory dependent ones.However, by the same trick, we can similarly expand the Petz’s quasi entropy [16] definedby, D γ ( ρ || σ ) = tr ρ γ σ − γ . (3)This quantity can be regarded as a one parameter generalization of relative entropy, ddγ D γ ( ρ || σ ) (cid:12)(cid:12) γ =1 = S ( ρ || σ ) = tr ρ log ρ − tr ρ log σ. (4)We also refer to recent studies on R´enyi generalizations of relative entropy [17, 18, 19, 20,21, 22] as well as perturbative calculations of relative entropy [23, 11, 24, 25, 26, 27, 28, 29].One notable feature of this R´enyi relative divergence is that, each term of its perturbativeexpansion involves a correlator on the regular space Σ which is conformally related to flatspace. This implies that the first few terms of the expansion are almost fixed by conformalsymmetry, and independent of the CFT we consider. In particular, this property enablesus to holographically write the quadratic terms of certain linear combinations of D γ ( ρ || σ )which we will denote by X γ ( δρ ) , Y γ ( δρ ), in terms of bulk symplectic form, without thedetails of the bulk to boundary dictionary. This generalizes the statement ” quantum fisherinformation = bulk canonical energy”. See also [30, 31] for recent discussions on bulksymplectic form.This paper is organized as follows. In section 2, we explain how to expand T γ ( ρ ) = tr ρ γ using the formula (2). We first derive expressions of the perturbative terms as integralswith respect to the entanglement spectrum of the unperturbed state. In section 3, wecheck these expressions against known results. In section 4 we express each term of theperturbative expansion as an integral along the modular flow of the unperturbed stateby Fourier transforming the spectral representation of the kernel derived in section 2. In4ection 5 we apply the formalism to reduced density matrices in conformal field theory, andwrite these perturbative terms in terms of correlation functions in CFT. In section 6, wediscuss a similar expansion of Petz’s quasi entropy and derive a holographic expression ofthe second order term. In the first few sections we focus on the R´enyi type quantity T γ ( ρ ) = tr ρ γ . (5)In the discussions we do not assume the index γ to be an positive integer γ ∈ Z + , whereone can use the replica trick. Although we will apply the prescription developing here toconformal field theory, the discussions in this section and the next few ones are applicablefor any density matrix of any theory.When the density matrix ρ is sufficiently close to the reference state ρ , ie ρ = ρ + δρ ,we can expand T γ ( ρ ) by a power series of δρ , T γ ( ρ ) = T γ ( ρ ) + ∞ (cid:88) n =0 T ( n ) γ ( δρ ) , (6)and decompose each term in the perturbative expansion by the spectra of the referencestate ρ . Let us first do this.We begin the discussion by first writing T γ ( ρ ) using the resolvent of ρ ,tr ρ γ = (cid:90) C dz πi z γ tr 1 z − ρ , (7)where the contour C is encircling the interval [ ρ min ,
1] in the z plane, but not z = 0,so that it picks up all contributions of eigenvalues of ρ . ρ min is the smallest eigenvalue ofthe density matrix ρ .(See figure 1.) When ρ is a reduced density matrix of a quantumfield theory, we need to put a UV cut off ε so that the density matrix ρ has a minimumeigenvalue, then after the calculation we send ε →
0. We will explicitly see that only theunperturbed term T γ ( ρ ) depends on the UV cutoff and rests do not. Therefore we canuniquely fix the the form of T ( n ) γ ( δρ ) , n ≥ C of the integral (7)(blue line). Red dots are the poles of themeromorphic function f ( z ) = tr z − ρ . These poles are in the segment 0 < Re z < ρ = ρ + δρ the resolvent can be easily expanded,1 z − ρ = ∞ (cid:88) n =0 R n ( δρ ) R n ( δρ ) = (cid:18) z − ρ ) δρ (cid:19) n z − ρ ) . (8)By inserting the complete set of eigenstates | ω i (cid:105) of the reference state ρ , (cid:90) dω i | ω i (cid:105)(cid:104) ω i | = 1 , ρ | ω i (cid:105) = e − πω i | ω i (cid:105) , (9)to the left of i -th term, taking trace, and evaluating 1 / ( z − ρ ) from the left, we have,tr [ R n ( δρ )] = (cid:90) n (cid:89) i =1 dω i n − (cid:89) i =1 z − e − πω i n − (cid:89) k =1 (cid:104) ω k | δρ | ω k +1 (cid:105)(cid:104) ω n | z − ρ ) δρ z − ρ ) | ω (cid:105) = (cid:90) n (cid:89) i =1 dω i z − e − πω ) n (cid:89) i =2 z − e − πω i n (cid:89) k =1 (cid:104) ω k | δρ | ω k +1 (cid:105) , (10)in the last term, ω n +1 ≡ ω is understood.In summary, here we expanded T γ ( ρ ) with respect to δρ , as in (6), and saw that the n th order term of the expansion T ( n ) γ ( δρ ) is given by T ( n ) γ ( δρ ) = (cid:90) n (cid:89) i =1 dω i (cid:34)(cid:90) C dz πi z γ z − e − πω ) n (cid:89) i =2 z − e − πω i (cid:35) n (cid:89) k =1 (cid:104) ω k | δρ | ω k +1 (cid:105) . (11)6y defining the kernel function, K ( n ) ( ω , · · · ω n ) ≡ (cid:90) C dz πi z γ ( z − e − πω ) n (cid:89) i =2 z − e − πω i , (12)we write, T ( n ) γ ( δρ ) = (cid:90) n (cid:89) i =1 dω i K ( n ) ( ω , · · · ω n ) n (cid:89) k =1 (cid:104) ω k | δρ | ω k +1 (cid:105) . (13) We have obtained the perturbative expansion using the spectrum of the reference state ρ . To get some insights, in this section we explicitly write down first few terms of theexpansion and check them against known results. T (1) γ ( δρ ) The first order term of the series is given by T (1) γ ( δρ ) = (cid:90) dω (cid:104) ω | δρ | ω (cid:105) (cid:90) C dz πi z γ ( z − e − πω ) = γ tr (cid:2) ρ γ − δρ (cid:3) . (14)as it should be. T (2) γ ( δρ ) Let us move on to the second order term T (2) γ ( δρ ). It is given by T (2) γ ( δρ ) = (cid:90) dωdω (cid:48) (cid:104) ω | δρ | ω (cid:48) (cid:105)(cid:104) ω (cid:48) | δρ | ω (cid:105) K ( ω, ω (cid:48) ) . (15)Precise form of K ( ω, ω (cid:48) ) can be derived by the contour integral, K ( ω, ω (cid:48) ) = (cid:90) C dz πi z γ ( z − e − πω ) ( z − e − πω (cid:48) )= 1( e − πω (cid:48) − e − πω ) (cid:104) ( γ − e − πγω + e − πγω (cid:48) − γe − π ( γ − ω e − πω (cid:48) (cid:105) . (16)7 .2.1 Checks γ = n ∈ Z + When the index γ is a positive integer, the kernel K γ ( ω, ω (cid:48) ) is decomposed into the sum, K ( ω, ω (cid:48) ) = (cid:34) γ − (cid:88) l =0 (( γ − − l ) (cid:0) e − πω (cid:1) γ − l (cid:16) e − πω (cid:48) (cid:17) l (cid:35) . (17)Plugging this into (15) and undoing the spectral decomposition, we recover the obviousexpansion (1) which we frequently encounter in replica calculations. The kernel avoidsthe difficulties of replica trick, by automatically doing the summation as well as analyticcontinuation in n .The von Neumann entropy limit T γ ( ρ ) is related to the von Neumann entropy S ( ρ ) by S ( ρ ) = − tr ρ log ρ = ∂∂γ T γ ( ρ ) (cid:12)(cid:12) γ =1 . (18)From (16) we derive the kernel for the quadratic part of the von Neumann entropy, ∂K γ ∂γ (cid:12)(cid:12) γ =1 = e πω (1 − e π ( ω − ω (cid:48) ) ) (cid:104) ( e − πω − e − πω (cid:48) ) + 2 π ( ω − ω (cid:48) ) e − πω (cid:48) (cid:105) . (19)In [8], a perturbative expansion of the von Neumann entropy S ( ρ + δρ ) was discussed,by expanding the modular Hamiltonian K ρ = − ρ + δρ using the identity,log ρ = (cid:90) ∞ dβ (cid:18) ρ + β − β + 1 (cid:19) , (20)the result of the quadratic order kernel in [8] agrees with (19). The ω integrals in the right hand side of (13) are of course hard to perform, as we do notknow precise form of the eigenvalue distribution of ρ . To proceed, we now express each8erm of the perturbative series T ( n ) γ ( δρ ) as an integral along the modular flow of ρ , byFourier transforming the kernel K ( n ) γ ( ω , · · · ω n ).This process is very analogous to the case of the von Neumann entropy perturbationdone in [8] for quadratic order term and generalized to higher order terms in [11]. It isconvenient to introduce the rescaled kernel, defined by K ( n ) γ ( ω , · · · ω n ) ≡ e πγω − π (cid:80) nk =1 ω k K ( n ) ( ω , · · · ω n ) , (21)= (cid:90) C dz πi z γ e π ( γ − ω ( z − e − πω ) n (cid:89) i =2 e − πω i z − e − πω i . (22)Using this function, we get T ( n ) γ ( δρ ) = (cid:90) n (cid:89) i =1 dω i K ( n ) ( ω , · · · ω n ) n (cid:89) k =1 (cid:104) ω k | δρ | ω k +1 (cid:105) = (cid:90) n (cid:89) i =1 dω i e − πγω +2 π (cid:80) nk =1 ω k K ( n ) γ ( ω , · · · ω n ) n (cid:89) k =1 (cid:104) ω k | δρ | ω k +1 (cid:105) = (cid:90) n (cid:89) i =1 dω i K γn ( ω , · · · ω n ) (cid:104) ω | e − πγK δ ˜ ρ | ω (cid:105) n (cid:89) k =1 (cid:104) ω k | ˜ δρ | ω k +1 (cid:105) , (23)where 2 πK = − log ρ is the modular Hamiltonian of ρ , and ˜ δρ = e πK δρ e πK . It can beeasily shown that the new kernel K ( n ) γ ( ω , · · · ω n ) is invariant under the shifts ω i → ω i + α , K ( n ) γ ( ω + α, · · · ω n + α ) = K ( n ) γ ( ω , · · · ω n ) . (24)So if we change the variables to { a i , b } , a i = ω i − ω i +1 , i = 1 · · · n − b = n (cid:88) i =1 ω i , (25) K ( n ) γ ( ω , · · · ω n ) only depends on n − { a i } i =1 ··· n − , K ( n ) γ ( ω , · · · ω n ) = K ( n ) γ ( a , a , · · · a n − ) . (26)Thanks to this property K ( n ) γ ( ω , · · · ω n ) has a nice Fourier transformation, K ( n ) γ ( ω , · · · ω n ) = (cid:90) C ds · · · ds n − e i (cid:80) n − k =1 s k a k K ( n ) γ ( s , · · · s n − ) , (27)9 s i } i =1 ··· n − are variables dual to the spectrum of ρ , therefore they have a geomet-ric interpretation, ie, they are parameterizing the modular flow of ρ . Also, as we willsee later, we need to properly choose the integration contours C in order for the Fouriertransformation (27) to correctly reproduce the kernel K ( n ) γ ( ω , · · · ω n ).Using this and undoing the spectral decompositions (9), we can write T ( n ) γ ( δρ ) as anintegral of real time { s i } variables, T ( n ) γ ( δρ ) = (cid:90) n (cid:89) i =1 dω i K ( n ) γ ( ω , · · · ω n ) (cid:104) ω | e − πγK δ ˜ ρ | ω (cid:105) n (cid:89) k =1 (cid:104) ω k | ˜ δρ | ω k +1 (cid:105) = (cid:90) C ds · · · ds n − K ( n ) γ ( s , · · · s n − ) tr (cid:34) e − πγK n − (cid:89) k =1 e iKs k ˜ δρ e − iKs k ˜ δρ (cid:35) . (28)In the actual CFT computations, this undoing is a bit tricky, and needed special cares.We will discuss on this in the latter sections. Let us first specify the form of the real time kernel K ( n ) γ ( s , · · · s n − ).The task is doing theinverse Fourier transformation, K ( n ) γ ( s , · · · s n − ) = (cid:90) da · · · da n − (2 π ) n − e − i (cid:80) n − k =1 s k a k K ( n ) γ ( a , · · · a n − ) . (29)The trick we use is very similar to the one developed in our previous paper [11]. Byinserting a delta function, δ ( q ) = 12 π (cid:90) dbe − iqb , (30)we can disentangle the multiple integral to a product of integrals of single variables { ω i } , δ ( q ) K ( n ) γ ( s , · · · s n − ) = 1(2 π ) n (cid:90) dbe − iqb (cid:90) da · · · da n − e − i (cid:80) n − k =1 s k a k K ( n ) γ ( a , · · · a n − )= n (2 π ) n (cid:90) dω · · · dω n e − iqb e − i (cid:80) n − k =1 s k a k K ( n ) γ ( ω , · · · ω n ) , (31)in the second line we used the relations (25).10ow the integral is δ ( q ) K ( n ) γ ( s , · · · s n − ) = n (2 π ) n (cid:90) dω · · · dω n e − iqb e − i (cid:80) n − k =1 s k a k K ( n ) γ ( ω , · · · ω n )= n (2 π ) n (cid:90) C dz πi z γ (cid:90) dω e − ω [ − π ( γ − i ( s + q )] ( z − e − πω ) × n − (cid:89) i =2 (cid:90) dω i e − ω i [2 π +( s k − s k − + q ) i ] z − e − πω i × (cid:90) dω n e − ω n [2 π − ( s n − − q ) i ] z − e − πω n ≡ n (2 π ) n (cid:90) C dz πi J ( z ) . (32)The strategy to compute this complicated integral is first compute each ω i integral, andexpress J ( z ) as a function of modular times { s i } i =1 ··· n − . We then perform the z integralby choosing the contour along the real axis, δ ( q ) K ( n ) γ ( s , · · · s n − ) = n (2 π ) n (cid:90) ∞ dβ πi ( J ( β − i(cid:15) ) − J ( β + i(cid:15) )) , (cid:15) → + . (33)The details of the calculation can be found in Appendix (A) and here we only presentthe final result for the kernel K ( n ) γ ( s , · · · s n − ) , K ( n ) γ ( s , · · · s n − ) = i π (cid:18) − i π (cid:19) n − ( s + 2 πiγ ) sin πγ sinh (cid:0) s +2 πiγ (cid:1) (cid:81) n − k =2 sinh (cid:0) s k − s k − (cid:1) sinh (cid:0) s n − (cid:1) (34) n = 2 term In the previous subsection we derived the expression (34) of the real time kernel K ( n ) γ ( s , · · · s n − ).In order to complete the discussion we need to properly fix the contour of the real timeintegrals C in (28). We can do so by demanding the Fourier transformation can be correctlyreversed, K ( n ) γ ( ω , · · · , ω n ) = (cid:90) C k n − (cid:89) k =1 ds k e i (cid:80) n − k =1 s k a k K ( n ) γ ( s , · · · s n − ) . (35)11igure 2: The contour C s of the integral (36)(blue line). Orange dots are the poles s k =2 πi ( k − γ ) of the kernel K (2) γ ( s ).We first consider the contour of quadratic n = 2 term, (cid:90) C s ds K (2) γ ( s ) e ias = i sin πγ π (cid:90) C s ds s + 2 πiγ sinh s sinh s +2 πiγ e ias , (36)which is a bit tricky compared to higher order terms. When a > K (2) γ ( s ) has two types of poles. s n = 2 πin, s k = 2 πi ( k − γ ) , n, k ∈ Z , k (cid:54) = 0 . (37)We can easily see that if one choose the contour C s which contains s n , n ≥
0, and s k , k ≥ K ( n ) γ ( a ) = (cid:90) C s ds K (2) γ ( s ) e ias . (38)Again we explicitly check this in Appendix B.It is useful to write the integral as follows. Since we can write the integrand,( s + 2 πiγ ) sin πγ sinh s sinh s +2 πiγ = s + 2 πiγ − e − s − s + 2 πiγ − e − ( s +2 πiγ ) (39)12hen, the contour integral is naturally split into two parts, (cid:90) C s ds ( s + 2 πiγ ) sin πγ sinh s sinh s +2 πiγ G ( s ) = (cid:90) ∞− i(cid:15) −∞− i(cid:15) ds (cid:20) s + 2 πiγ − e − s (cid:21) G ( s ) − (cid:90) ∞− πi ( γ − (cid:15) ) −∞− πi ( γ − (cid:15) ) ds (cid:20) s + 2 πiγ − e − ( s +2 πiγ ) (cid:21) G ( s )(40)for any function G ( s ) which is holomorphic on the strip − πγ < Im s < , when γ > − < γ <
1, the contour gets simplified, (cid:90) C s ds K (2) γ ( s ) G ( s ) = (cid:90) ∞− i(cid:15) −∞− i(cid:15) ds K (2) γ ( s ) G ( s ) (41) n ≥ terms Now we fix all the contours C k in the integral (35).In the above derivation we have used following formula, I ( ξ, β + i(cid:15) ) = (cid:90) ∞−∞ dω e − ωξ ( β + i(cid:15) ) − e − πω = β ( ξ π − ) (cid:32) e i ξ ξ (cid:33) . (42)Notice that ξ = p + it , and p was a real number. In order for the integral to have an inverse,we need to make sure the choice of the contour C t (cid:90) C t dt I ( p + it, β ) e iωt = e − ωp β − e − πω . (43)The integrand has poles at s n = ip + 2 πn . By an explicit calculation, we recognize that weneed pick up poles with n ≥
1, thus (cid:90) C dt ≡ (cid:90) ∞ + i ( p + (cid:15) ) −∞ + i ( p + (cid:15) ) dt. (44)This in particular means that K γn ( ω , · · · ω n ) = (cid:90) C dz πi z γ e π ( γ − ω ( z − e − πω ) n (cid:89) i =2 e − πω i z − e − πω i = n (cid:89) k =1 (cid:90) ∞ + i ( p k + (cid:15) ) −∞ + i ( p k + (cid:15) ) dt k e iω k t k (cid:90) C dz πi z γ n (cid:89) k =1 I ( it k + p k , z )= n − (cid:89) k =1 (cid:90) C k ds k e i (cid:80) n − k =1 s k a k K γn ( s , · · · s n − ) (45)13herefore we need to choose the following contours,Im s = − π ( γ − (cid:15) ) , Im s k − Im s k − = (cid:15), Im s n − = − (cid:15) (46)In particular when γ < n ≥ The discussion so far is quite general, applicable to any density matrices of any theories.From now on, we would like to apply the formula to a special type of reduced densitymatrices in conformal field theory(CFT). For this purpose, we first briefly summarize theconstruction of the reduced density matrices. For detailed discussions we refer to [11].
We start from a CFT on d dimensional cylinder R × S d − , ds = dt + dθ + sin θd Ω d − . (47)We consider a ball shaped subsystem A , which is given by A : [0 , θ ] × S d − , t = 0 , (48)and a reduced density matrix ρ V of a globally excited state | V (cid:105) on the region A , ρ V = tr A c | V (cid:105)(cid:104) V | . (49)The reduced density matrix has a path integral representation on the cylinder with a branchcut on A. The branched cylinder is mapped to S × H d − with the metric [7], ds = dτ + du + sinh ud Ω d − , τ ∼ τ + 2 π. (50)We find that in this frame ρ V has following expression [11], ρ V = e − πK V ( θ ) V ( − θ ) e − πK (cid:104) V ( θ ) V ( − θ ) (cid:105) (51)where K is the generator of the translation along τ direction, which can be identifiedwith the modular Hamiltonian of ρ and V ( ± θ ) are local operators corresponding to the14xcited states through state operator correspondence, located at τ = ± θ , u = 0. In thesmall subsystem limit θ → V ( θ ) → V ( − θ ).In this limit we can split the density matrix into the vacuum one ρ = e − πK and therest, ρ V = ρ + δρ . We do so by taking operator product expansion (OPE) of the two localoperators, ρ V = ρ + e − πK (cid:34) (cid:88) O :primaries C O V V B O ( θ , − θ ) (cid:35) e − πK (52)where the index O labels non identity primaries , and C O V V , B O ( θ , − θ ) are the OPEcoefficient and the OPE block of O respectively. T γ ( ρ ) Now we determine the perturbative expression of T γ ( δρ ) in CFT from (28). We write,tr ρ γ = tr ρ γ + (cid:88) T ( n ) γ ( δρ ) , (53)and for convenience we reproduce the expression of T ( n ) γ explicitly. T ( n ) γ ( δρ ) = (cid:90) ds · · · ds n − K ( n ) γ ( s , · · · s n − )tr (cid:34) e − πγK n (cid:89) k =1 e iKs k ˜ δρe − iKs k (cid:35) . (54)Since ˜ δρ = e πK δρe πK , in our case we have e iKs δ ˜ ρe − iKs = (cid:88) O :primaries C O V V B O ( is + θ , is − θ ) . (55)For our δρ , the trace in (54) can be regarded as a correlation function of the OPEblocks on the covering space Σ γ = S γ × H d − , with the metric (50) but the periodicity ofthe Euclidean time direction is changed τ ∼ τ + 2 πγ , (cid:104)· · ·(cid:105) Σ γ ≡ Z γ tr (cid:2) e − πγK · · · (cid:3) , (56)where Z γ is the CFT partition function on this space.15ombining these we can write each term of T γ ( δρ ) by an integral of correlation functionsof OPE blocks on Σ γ along modular flow of vacuum ρ .1 Z γ T ( n ) γ ( δρ ) = (cid:88) {O l } n (cid:89) l =1 C O l V V (cid:90) ds · · · ds n − K ( n ) γ ( s , · · · s n − ) (cid:104) n − (cid:89) k B O k ( is k + θ , is k − θ ) B O ( θ , − θ ) (cid:105) Σ γ (57) n = 2 term to the standard form We have seen n = 2 term is given by1 Z γ T (2) γ ( δρ ) = (cid:88) O :primaries (cid:90) C ds K (2) γ ( s ) (cid:104) B O ( is + θ , is − θ ) B O ( θ , − θ ) (cid:105) Σ γ (58)We can simplify this expression when 0 < γ <
1, and compare it with known results.In order to do so, let us focus on the contribution T (2) γ, O ( δρ ) of a particular primary O tothe n = 2 term. Since the OPE block B O is summing up descendants of the primary O ,wecan write it as B O ( θ , − θ ) = C ( θ , ∂ a ) O ( τ a ) (cid:12)(cid:12) τ a =0 (59)where C ( θ , ∂ a ) is a differential operator, and τ a is the coordinate of Euclidean timelikedirection. In the above we did not manifest the dependence of O on the coordinates ofhyperbolic space. The main ingredient of the formula is the integral of two point function, I ab = i π (cid:90) ∞− i(cid:15) −∞− i(cid:15) ds s + 2 πiγ sinh s sinh s +2 πiγ G ab ( s ) , G ab ( s ) = (cid:104)O ( is + τ a ) O ( τ b ) (cid:105) Σ γ . (60)and we can write, T (2) γ, O ( δρ ) = C ( θ , ∂ a ) C ( θ , ∂ b ) I ab (61)As we explain in Appendix C, we can obtain a simpler expression of T (2) γ, O ( δρ ), T (2) γ, O ( δρ ) = γ sin πγ π C ( θ , ∂ a ) C ( θ , ∂ b ) (cid:90) ∞−∞ ds sinh s − πiγ sinh s + πiγ G ab ( s − πiγ ) (62)Notice that in the γ → O to thesecond order term S (2) ( δρ ) of entanglement entropy [8], S (2) O ( δρ ) = C ( θ , ∂ a ) C ( θ , ∂ b ) (cid:90) ∞−∞ ds −
14 sinh (cid:0) s − i(cid:15) (cid:1) (cid:104)O ( is + τ a ) O ( τ b ) (cid:105) Σ . (63)16 Expansion of Petz’s quasi entropy D γ ( ρ || σ ) In this section, we consider a similar perturbative expansion for Petz’s quasi entropy [16],defined by D γ ( ρ || σ ) = tr ρ γ σ − γ , (64)In this section we consider the case where the one of the reduced density matrices isvacuum σ = ρ . We then write ρ = ρ + δρ , D γ ( ρ || ρ ) = ∞ (cid:88) n =2 D ( n ) γ ( δρ ) . (65)The derivation of the perturbative series is very similar to the one of T γ ( ρ ). We firstwrite D γ ( ρ || ρ ) = (cid:90) C dz πi z γ tr ρ − γ z − ρ , (66)then by expanding the denominator we obtain a similar perturbative series. One notabledifference is that in the power series of D γ ( ρ || ρ ), the ρ γ factor appears in the expansion(54) is canceled with the ρ − γ factor which appear in the definition (64). The explicitexpression of D ( n ) γ ( δρ ) is given by D ( n ) γ ( δρ ) = (cid:88) {O k } n (cid:89) k =1 C O k V V n − (cid:89) k =1 ds k (cid:90) ds k K ( n ) γ ( s , · · · s n − ) (cid:104) n − (cid:89) k B O k ( is k + θ , is k − θ ) B O n ( θ , − θ ) (cid:105) Σ (67)with the kernel K ( n ) γ ( s , · · · s n − ) defined in (34).One advantage of this quantity is that we can expand it in terms of correlation functionson the space without branch cut, Σ , on the contrary to R´enyi entropy itself, which isexpanded by correlators (cid:104)· · ·(cid:105) Σ γ on the space Σ γ with branch cuts, and they are highlytheory dependent quantities. This implies that first few terms of D γ ( ρ || σ ) are theoryindependent, and allows us to write them holographically.We also emphasize that the expressions (67) are only valid in some range of γ . Inparticular higher order terms D ( n ) γ ( δρ ) , n ≥
3, has an expression in terms of a modular flowintegral only in the range 0 < γ <
1. The limitation is again coming from the fact that17here is a consistent contour choice of the modular flow integrals (46) only in the range.However n = 2 term is still computable by the modular flow integral for any value of γ .Below, we will be focusing on following quantity, Z γ ( ρ || σ ) ≡ D − γ ( ρ || σ ) − D γ ( ρ || σ ) , (68)and its quadratic part, Y γ ( δρ ) ≡ d dt Z γ ( σ + tδρ || ρ ) (cid:12)(cid:12) t =0 , . (69)as well as its derivative with respect to the index γ , X γ ( δρ ) = ddγ Y γ ( δρ ) . (70)Notice that when γ = 0 ∂ γ Z γ ( ρ || σ ) reduces to the relative entropy ∂ γ Z γ ( ρ || σ ) (cid:12)(cid:12) γ =0 = 2 S ( σ || ρ ) , (71)in which the order of two density matrices is flipped ρ ↔ σ , and X γ ( δρ ) reduces to theFisher information, which is symmetric under the exchange. X γ ( δρ ) (cid:12)(cid:12) γ =0 = F ( ρ || σ ) . (72) X γ ( δρ ) and Y γ ( δρ ) by modular flow integrals Below we will focus on the range of the R´enyi index − < γ < D (2) γ ( δρ ), or equivalently0 < γ < X γ ( δρ ) and Y γ ( δρ ) .When the R´enyi index is in the window, Y γ ( δρ ) has following simple modular flowintegral representation, Y γ ( δρ ) = (cid:90) ∞− i(cid:15) −∞− i(cid:15) (cid:104) K (2) − γ ( s ) − K (2) γ ( s ) (cid:105) tr (cid:104) e − πK ˜ δρ ( s ) ˜ δρ (cid:105) ds. (73) K (2) γ ( s ) is given by K (2) γ ( s ) = i sin πγ π ( s + 2 πiγ )sinh s sinh s +2 πiγ . (74)18or the class of δρ we are interested in, we have Y γ ( δρ ) = C ( θ , ∂ a ) C ( θ , ∂ b ) (cid:90) ∞− i(cid:15) −∞− i(cid:15) (cid:104) K (2) − γ ( s ) − K (2) γ ( − s − πi ) (cid:105) (cid:104)O ( is + τ a ) O ( τ b ) (cid:105) Σ ds = C ( θ , ∂ a ) C ( θ , ∂ b ) (cid:90) ∞−∞ ds (cid:34) − (sin πγ ) / π sinh (cid:0) s − i(cid:15) (cid:1) sinh (cid:0) s − πiγ (cid:1) (cid:35) (cid:104)O ( is + τ a ) O ( τ b ) (cid:105) Σ . (75)In the second term of the first line, we used another expression of D γ ( δρ ) D γ ( δρ ) = C ( θ , ∂ a ) C ( θ , ∂ b ) I ba , I ba = (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) ds K (2) γ ( s − πi ) (cid:104)O ( is + τ b ) O ( τ a ) (cid:105) Σ , τ a > τ b , (76)and flipped the sign of the integration variable s → − s . The derivation of this expressionis the same with that of (129) in Appendix C.By taking derivative of (75) with respect to γ , we have an expression of X γ ( δρ ), X γ ( δρ ) = C ( θ , ∂ a ) C ( θ , ∂ b ) (cid:90) ∞−∞ ds −
14 sinh (cid:0) s − πiγ (cid:1) (cid:104)O ( is + τ a ) O ( τ b ) (cid:105) Σ . (77) X γ ( δρ ) and Y γ ( δρ ) So far we have obtained quadratic term Y γ ( δρ ) which is particular linear combination ofthe R´enyi relative divergence Z γ ( δ || ρ ), and its derivative X γ ( δρ ) in terms of modular flowintegral (75), (77).As we will see below, through AdS/CFT correspondence, they have simple bulk ex-pressions. The derivations are parallel to the argument of [10], where they obtained theholographic expression of quadratic term of the entanglement entropy S (2) ( δρ ). To explain this let us first recall the corresponding bulk set up. Our reference state is thevacuum reduced density matrix ρ , and since we take the subsystem A to be a ball shaperegion, corresponding Ryu Takayanagi surface can be regarded as the bifurcation surface r B = 1 of the topological black hole, ds = − ( r B − ds B + dr B ( r B −
1) + r B dH d − (78)19here dH d − denotes the metric of d − dH d − = du + sinh u d Ω d − . (79)In [10] it was shown that the CFT two point function in (75), (77) can be written interms of the bulk symplectic form ω φ of the bulk field φ dual to the CFT primary O , (cid:104)O ( is + τ a ) O ( τ b ) (cid:105) Σ = − (cid:90) dX B ω φ ( K E ( X B | τ ba ) , K R ( X B | s )) . (80)We evaluate the integral on fixed r B = r surface of the topological black hole (78), andcollectively denote the coordinates of the surface by X B . The bulk symplectic form is givenby ω φ ( δφ , δφ ) = n M ( δφ ∂ M δφ − δφ ∂ M δφ ) , (81)where n M is the normal vector of the r B = r surface. K E ( X B | τ ba ), K R ( X B | s ) are theEuclidean and Retarded bulk to boundary propagator of the bulk field φ , respectively. Theprimary operators in the CFT two point function are located at the origin of the hyperbolicspace u = 0. We omit this information in the bulk to boundary propagators. By plugging (80) into (75), and evaluating the remaining s integral by picking up poles ofthe kernel, we get Y γ ( δρ ) = i C ( θ , ∂ a ) C ( θ , ∂ b ) (cid:90) dX B ω φ ( K E ( X B | τ ba ) , K E ( X B | − πγ ) − K E ( X B | s B → s B + iτ a , and using the relation between theEuclidean bulk to boundary propagator and the expectation value of the bulk scalar fieldoperator φ ( X B ), C ( θ , ∂ a ) K E ( X B | τ a ) = (cid:104) V | φ ( X B ) | V (cid:105) ≡ (cid:104) φ ( X B ) (cid:105) V , (83)we get, Y γ ( δρ ) = i (cid:90) dX B ω φ ( (cid:104) φ (0) (cid:105) V , (cid:104) φ (2 πγ ) (cid:105) V − (cid:104) φ (0) (cid:105) V ) . (84) The argument here is very similar to the one in [10]. See Appendix E for the details. (cid:104) φ (2 πγ ) (cid:105) V is the expectation value of the bulk field rotated by 2 πγ along theEuclidean timelike direction, (cid:104) φ (2 πγ ) (cid:105) V ≡ tr (cid:2) ρ V e − πγK φ e πγK (cid:3) (85)In the argument of the bulk local field φ , we only manifested the Euclidean time likecoordinate, φ ( τ ) ≡ φ ( r B , τ + is B , u, Ω d − ) . (86)We can obtain a similar expression for X γ ( δρ ) just by taking a derivative of Y γ ( δρ ), X γ ( δρ ) = − π (cid:90) dX B ω φ ( (cid:104) φ (0) (cid:105) V , ∂ s (cid:104) φ (2 πγ ) (cid:105) V ) , (87)here we used the relation ∂ γ = − i∂ s . This integral is invariant under the deformation ofthe surface on which we are evaluating the integral. In particular we can choose the fixedtime slice s B = 0, then the integral can be written as, X γ ( δρ ) = − π (cid:90) Σ d Σ a ξ b T ab ( (cid:104) φ (0) (cid:105) V , (cid:104) φ (2 πγ ) (cid:105) V ) , (88)where Σ is the bulk region on the time slice s B = 0, which is enclosed by the boundarysubsystem A and the bifurcation surface of the topological black hole (ie, RT surface). Also d Σ a is the volume element of Σ, and ξ b is the timelike Killing vector of the black hole. T ab is a quadratic form of φ related to the stress energy tensor of the bulk field, T ab ( φ , φ ) = ∂ a φ ∂ b φ − m g ab φ φ (89)There is another way to derive this result. Let us come back to the CFT formula, X γ ( δρ ) = C ( θ , ∂ a ) C ( θ , ∂ b ) (cid:90) ∞−∞ ds −
14 sinh (cid:0) s − πiγ (cid:1) (cid:104)O ( is + τ a ) O ( τ b ) (cid:105) Σ (90)by changing the integration variable to t = s − πiγ and shifting the contour we get, X γ ( δρ ) = C ( θ , ∂ a ) C ( θ , ∂ b ) (cid:90) ∞−∞ dt −
14 sinh (cid:0) t − πi(cid:15) (cid:1) (cid:104)O ( i ( t + 2 πiγ ) + τ a ) O ( τ b ) (cid:105) Σ = (cid:90) ∞−∞ ds −
14 sinh (cid:0) s − πi(cid:15) (cid:1) tr (cid:104) ˜ δρ ( s ) e πγ ˜ δρ e − πγ (cid:105) (91)21n [11] it was shown that the excited state modular Hamiltonian K ρ of ρ , when expandedby δρ , the leading order correction to the vacuum modular Hamiltonian K is given by K ρ = K + (cid:90) ∞−∞ ds sinh s ˜ δρ ( s ) ≡ K + δK. (92)It was also shown that contribution of a primary operator O to the correction δK hasa bulk expression δK = 2 π (cid:90) Σ d Σ a ξ b T ab ( (cid:104) φ (0) (cid:105) V , ˆ φ ) , (93)where ˆ φ is the bulk field operator dual to O . By plugging this into (91), we recover theresult. In this paper we developed a novel way to perturbatively expand R´enyi tpe quantitiesinvolving powers of reduced density matrices. We then obtained a holographic expression ofthe quadratic parts of R´enyi relative divergences X γ ( δρ ) , Y γ ( δρ ) in terms of bulk symplecticform starting from the CFT calculations.It is interesting find a bulk derivation of this result. One difficulty in doing so is comingform the fact that in general there is no nice path integral representation of R´enyi relativedivergence. This is because even if reduced density matrices ρ, σ can be written by pathintegrals, ρ γ and σ − γ can not. If we could find such a representation, then we can map theCFT path integral calculationss to the bulk on shell action calculations. Indeed, in a specialcase where R´enyi relative divergence can be represented by a path integral, correspondingholographic calcuation is known[21]. However in order to derive a bulk formula for R´enyirelative divergence between two generic bulk configurations, we need to take a differentapproach. A possible approach would be first going back to replica trick [32], computetr ρ n σ m for positive integers n, m then analytically continue the result n → γ, m → − γ .Furthermore it would be nice if we could read off finer information of bulk geometriesusing R´enyi relative divergence. It has been shown that using relative entropy, we can readoff first non linear part of Einstein equations [8, 10] in particular. Since R´enyi relativedivergence is a one parameter generalization of relative entropy, and knows about details22f eigenvalue distribution of excited state reduced density matrices, it is natural to expectthis.Another interesting direction would be to calculate correlation functions with insertionsof modular flows of excited states, by using the technique developed in this paper. Forexample[33, 34, 35], two point function with an insertion of a modular flow (cid:104)O ( x )∆ it O ( y ) (cid:105) was considered. There, it was also argued that this is useful to extract information ofcorresponding bulk geometry. Naively speaking we can perturbatively compute them byWick rotating the R´enyi index γ to the imaginary value γ → it in our result. The taskwould be to check that there is no obstacle to do this. Acknowledgments
We thank Alex Belin, Tom Faulkner, Sudip Ghosh, Norihiro Iizuka, Robert Myers,TatsumaNishioka, Jonathan Oppenheim, G´abor S´arosi, Tadashi Takayanagi and Kotaro Tamaokafor discussions.
A The calculation of K ( n ) γ ( s , · · · s n − ) In this appendix, we explain the details of the calculation of the kernel K ( n ) γ ( s , · · · s n − ),starting from (32).In order to do this, we first decompose J ( z ) in (32) J ( z ) = z γ I ( ξ , z ) n − (cid:89) k =2 I ( ξ k , z ) I ( ξ n , z ) , (94)where ξ = − π ( γ −
1) + i ( s + q ) , ξ n = 2 π − ( s n − − q ) i, (95) ξ k = 2 π + ( s k − s k − + q ) i, ≤ k ≤ n − . (96)and I ( ξ, z ) = (cid:90) ∞−∞ dω e − ωξ z − e − πω , I ( ξ, z ) = (cid:90) ∞−∞ dω e − ωξ ( z − e − πω ) . (97)23or I ( ξ, z ), by carefully picking up the contributions of the relevant poles we have, I ( ξ, β + i(cid:15) ) = β ( ξ π − ) (cid:32) e − i ξ ξ (cid:33) , I ( ξ, β − i(cid:15) ) = β ( ξ π − ) (cid:32) e i ξ ξ (cid:33) . (98)One way to check these is using1 z + i(cid:15) − z − i(cid:15) = − πiδ ( z ) , (99)Then, Disc I = lim (cid:15) → + [ I ( z + i(cid:15) ) − I ( z − i(cid:15) )]= − πi (cid:90) ∞−∞ dωe − ξω δ ( β − e − πω ) = − iβ ( ξ π − ) . (100)This is consistent with (98).We can evaluate I ( ξ, z ) just by taking derivative of I ( ξ, z ) with respect to β , I ( ξ, β + i(cid:15) ) = − (cid:18) ξ π − (cid:19) β ( ξ π − ) (cid:32) e − i ξ ξ (cid:33) , I ( ξ, β − i(cid:15) ) = − (cid:18) ξ π − (cid:19) β ( ξ π − ) (cid:32) e i ξ ξ (cid:33) . (101)Combining these, we obtain the relevant expressions of J ( z ) J ( β + i(cid:15) ) = − β (cid:16) γ + (cid:80) nk =1 ξk π − ( n +1) (cid:17) (cid:0) ξ π − (cid:1)(cid:81) nk =1 ξ k e − i (cid:80) nk =1 ξ k , (102)and J ( β − i(cid:15) ) = − β (cid:16) γ + (cid:80) nk =1 ξk π − ( n +1) (cid:17) (cid:0) ξ π − (cid:1)(cid:81) nk =1 ξ k e i (cid:80) nk =1 ξ k . (103)Since γ + n (cid:88) k =1 ξ k π − ( n + 1) = − iqn π , (104)the β integral produces the delta function, (cid:90) ∞−∞ dβ πi β − iqn π = 2 πni δ ( q ) . (105)By picking up the discontinuity across the real line, we get24 ( q ) K ( n ) γ ( s , · · · s n − ) = n (2 π ) n (cid:90) ∞ dβ πi ( J ( β − i(cid:15) ) − J ( β + i(cid:15) )) , (cid:15) → + , = − n (2 π ) n (cid:18) πni δ ( q ) (cid:19) (cid:0) ξ π − (cid:1)(cid:81) nk =1 ξ k (cid:16) e i (cid:80) nk =1 ξ k − e − i (cid:80) nk =1 ξ k (cid:17) (106)Notice that e − i (cid:80) nk =1 ξ k − e + i (cid:80) nk =1 ξ k = e iπ ( γ − n ) − e − iπ ( γ − n ) (107)= 2 i ( − n sin πγ (108)and (cid:0) ξ π − (cid:1)(cid:81) nk =1 sin ξ k = − i n +1 π ( s + 2 πiγ )sinh (cid:0) s +2 πiγ (cid:1) (cid:81) n − k =2 sinh (cid:0) s k − s k − (cid:1) sinh (cid:0) s n − (cid:1) (109)From this we finally arrive at the expression of the kernel, K ( n ) γ ( s , · · · s n − ) = i π (cid:18) − i π (cid:19) n − ( s + 2 πiγ ) sin πγ sinh (cid:0) s +2 πiγ (cid:1) (cid:81) n − k =2 sinh (cid:0) s k − s k − (cid:1) sinh (cid:0) s n − (cid:1) (110) B Fixing the contour of n = 2 term In this appendix, we fix the correct contour C s of n = 2 real time integral (cid:90) C s ds K (2) γ ( s ) e ias = i sin πγ π (cid:90) C s ds s + 2 πiγ sinh s sinh s +2 πiγ e ias , (111)which reproduces the kernel in the frequency representation, (16) K (2) γ ( ω , ω ) = e πγω e − πω − πω K ( ω , ω )= e πγω e − πω − πω ( e − πω − e − πω ) (cid:2) ( γ − e − πγω + e − πγω − γe − π ( γ − ω e − πω (cid:3) . (112)Using a ≡ ω − ω , we have, K (2) γ ( a ) = e πa (1 − e πa ) (cid:2) ( γ −
1) + e πaγ − γe πa (cid:3) . (113)25et’s do the integral (111). There are two types of poles. s n = 2 πin, s k = 2 πi ( k − γ ) , (114)We choose a contour which contains s n , n ≥
0, and s k , k ≥
1. One way to manifest thecontour prescription is introducing an additional parameter x > K (2) γ ( s, x ) = (cid:18) i sin πγ π (cid:19) s + 2 πiγ sinh s + x sinh s +2 πiγ , (115)and finally send x → s n ] = i ( n + γ )2 π e − πan , Res[ s n ] = − ik π e πa ( γ − k ) . (116)By combining them, (cid:90) C s ds K (2) γ ( s ) e ias = 2 πi (cid:32)(cid:88) n Res[ s n ] + (cid:88) k Res[ s k ] (cid:33) = − (cid:34) (1 − e − πaγ ) (cid:88) k ke − πak + γ (cid:88) n e − πaγn (cid:35) = − e πa ( e πa − (cid:0) − γ ) + γe πa − e πaγ (cid:3) = K (2) γ ( a ) . (117)This is what we want. In the sum, we included n = 0 contribution. C Simplifying T (2) γ ( δρ ) In this section we simplify n = 2 term of T (2) γ ( δρ ). In section 5.3 we saw that the contributionof particular primary O to T (2) γ ( δρ ) can be written T (2) γ, O ( δρ ) = C ( θ , ∂ a ) C ( θ , ∂ b ) I ab (118)where I ab = i π (cid:90) ∞− i(cid:15) −∞− i(cid:15) ds s + 2 πiγ sinh s sinh s +2 πiγ G ab ( s ) , G ab ( s ) = (cid:104)O ( is + τ a ) O ( τ b ) (cid:105) Σ γ . (119)26nd C ( θ , ∂ a ) is a differential operator summing up all descendants.This expression only holds when τ a > τ b . This is because we started from the spectralrepresentation, I ab = (cid:90) dω dω K γ ( a ) e − πγω (cid:104) ω |O ( τ a ) | ω (cid:105)(cid:104) ω |O ( τ b ) | ω (cid:105) , (120)rewrote it in terms of the modular flow integral by K γ ( a ) = (cid:90) ∞− i(cid:15) −∞− i(cid:15) K γ ( s ) e ias , a = ω − ω . (121)and then undoing the spectral decomposition of the two point function G ab ( s ), (cid:90) dω dω e − πγω + ias (cid:104) ω |O ( τ a ) | ω (cid:105)(cid:104) ω |O ( τ b ) | ω (cid:105) = (cid:104)O ( is + τ a ) O ( τ b ) (cid:105) Σ γ . (122)The spectral integral only converges when τ a > τ b .When τ b > τ a , we instead write I ab = (cid:90) dω dω (cid:2) K γ ( a ) e − πγa (cid:3) e − πγω (cid:104) ω |O ( τ a ) | ω (cid:105)(cid:104) ω |O ( τ b ) | ω (cid:105) , (123) K γ ( a ) e − πγa = (cid:90) ∞− i(cid:15) −∞− i(cid:15) ds K γ ( s ) e ia ( s +2 πiγ ) = (cid:90) ∞ +2 πi ( γ − (cid:15) ) −∞ +2 πi ( γ − (cid:15) ) dt K γ ( t − πiγ ) e iat . (124)Since K γ ( t − πiγ ) = i sin πγ π t sinh t sinh t − πiγ (125)is regular on the strip 2 π ( γ − (cid:15) ) > Im t > < γ <
1, we deform the contour toIm t = (cid:15) K γ ( a ) e − πγa = (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) dt K γ ( t − πiγ ) e iat (126)Therefore for τ b > τ a we have I ab = (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) K γ ( s − πiγ ) (cid:104)O ( τ b ) O ( τ a + is ) (cid:105) Σ γ , τ b > τ a . (127)We have similar formule for I ba , just by flipping τ a ↔ τ b .27inally we combine these expressions to get a simpler form of T (2) γ ( δρ ). The two pointfunction in (119) is analytic in the strip region − πγ < Im s < τ ba . Since when 0 < γ < s → s − πiγ . Then the integral for τ a > τ b becomes I ab = i sin πγ π (cid:90) ∞− i(cid:15) −∞− i(cid:15) ds s + πiγ sinh s − πiγ sinh s + πiγ G ab ( s − πiγ ) , τ a > τ b (128)Now we do a similar thing for I ba , I ba = (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) K γ ( s − πiγ ) (cid:104)O ( τ a ) O ( τ b + is ) (cid:105) Σ γ , τ a > τ b . (129)By shifting the contour s → s + πiγ , and then flipping the sign s → − s we get I ba = i sin πγ π (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) ds − s + πiγ sinh s − πiγ sinh s + πiγ G ab ( s − πiγ ) (130)In the expressions (128) (130), we can take (cid:15) →
0. Finally we obtain I ab + I ba = γ sin πγ π (cid:90) ∞−∞ ds sinh s − πiγ sinh s + πiγ G ab ( s − πiγ ) (131) T (2) γ, O ( δρ ) is obtained by applying the differential operator, T (2) γ, O ( δρ ) = C ( θ , ∂ a ) C ( θ , ∂ b )( I ab + I ba ) (132)Notice that in the γ → S (2) ( δρ )entanglement entropy, S (2) O ( δρ ) = C ( θ , ∂ a ) C ( θ , ∂ b ) (cid:90) ∞−∞ ds −
14 sinh (cid:0) s − i(cid:15) (cid:1) (cid:104)O ( is + τ a ) O ( τ b ) (cid:105) Σ . (133) D Direct Fourier transformation
Here we would like to directly show that K γn ( s ) = (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) da π K γn ( ω ) e − ias = (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) da π e − ias sinh πa (cid:2) ( γ − − γe πa + e πγa (cid:3) (134)28he first piece is I = (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) da π e − ias sinh πa = s π (cid:18) − e − s (cid:19) (135)The second order term can be obtained by the shift s → s + 2 πi , therefore I = (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) da π e − ia ( s +2 πi ) sinh πa = ( s + 2 πi )4 π (cid:18) − e − s (cid:19) (136)Similarly, I = (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) da π e − ia ( s +2 πiγ ) sinh πa = ( s + 2 πiγ )4 π (cid:18) − e − ( s +2 πiγ ) (cid:19) (137)Then the total integral is( γ − I + γI + I = i π (cid:34) ( s + 2 πiγ ) sin πγ sinh s sinh s +2 πiγ (cid:35) (138)therefore we recover the first non trivial part. E Details of the holographic rewriting
In section 6.2.2, we used the result, Y γ ( δρ ) = (cid:90) dX B ω φ (cid:18) K E ( X B | τ ab , (cid:90) ∞−∞ ds Y ( s − i(cid:15) ) K R ( X B | s ) (cid:19) = i (cid:90) dX B ω φ ( K E ( X ,B | τ ba ) , K E ( X B | − πγ ) − K E ( X B | Y ( s − i(cid:15) ) = − (sin πγ ) / π sinh (cid:0) s − i(cid:15) (cid:1) sinh (cid:0) s − πiγ (cid:1) . (140)In this appendix, we prove this. The derivation is very similar to the one in [10].The retarded bulk to boundary propagator is given by K R ( X B | s ) = iθ ( s B − s ) lim (cid:15) → [ K E ( X B | is − (cid:15) ) − K E ( X B | is + (cid:15) )] . (141)29n particular, as a function of s , the retarded propagator is non vanishing only in thewindow −∞ < s < s ∗ . The value of s ∗ is fixed by demanding that the boundary point isnull separated from the bulk point X B . 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