Holographic Mutual Information and Critical Exponents of the Strongly Coupled Plasma
IIPM/P-2020/002
Prepared for submission to JHEP
Holographic Mutual Information and CriticalExponents of the Strongly Coupled Plasma
Hajar Ebrahim , and Gol-Mohammad Nafisi Department of Physics, University of Tehran, North Karegar Ave., Tehran 14395-547, Iran School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531,Tehran, Iran
E-mail: [email protected] , [email protected] Abstract:
This note contains discussions on the entanglement entropy and mutual infor-mation of a strongly coupled field theory with a critical point which has a holographic dual.We investigate analytically, in the specific regimes of parameters, how these non-local op-erators behave near the critical point. Interestingly, we observe that although the mutualinformation is constant at the critical point, its slope shows a power-law divergence in thevicinity of the critical point. We show that the leading behavior of mutual information atand near the critical point could yield a set of critical exponents if we regard it as an orderparameter. Our result for this set of static critical exponents is (1 / , / , / ,
2) which isidentical to the one calculated via the thermodynamic quantities. Hence it suggests thatbeside the numerous merits of mutual information, this quantity also captures the criticalbehavior of the underlying field theory and it could be used as a proper measure to probethe phase structure associated with the strongly coupled systems. a r X i v : . [ h e p - t h ] F e b ontents Following the recent advances in theoretical physics, one could observe that the quantuminformation theory and quantum gravity have become the front-runners of current the-oretical research programs. Due to the developments in studying black hole physics viaholography in recent years, it has become evident that the concept of entanglement notonly plays a crucial role in bringing together these two seemingly unrelated areas, resultingin fruitful insights toward understanding the important properties of underlying physicalsystems, it might also shed lights on our current view of quantum gravity [1]. For a given– 1 –ipartitioned system in general, entanglement entropy measures the amount of quantumentanglement between its two sub-systems. In the context of quantum field theory, onecould also calculate the entanglement entropy between two spacetime regions using thereplica trick method [2]. Following up the seminal work of Cardy and Calabrese, in whichthey obtaind the entanglement entropy of a two dimensional conformal field theory, gen-eralizations of their results for the higher dimensional field theories have been an activeline of research [3–5]. It was also shown that the entanglement entropy in field theo-ries suffers from short-distance divergence obeying an area-law behavior which makes it ascheme-dependent quantity in the UV limit [6, 7].In the context of AdS/CFT correspondence [8–10], quantum entanglement has becomeone of the main research interests as well. Ryu and Takayanagi (and later Hubeny, Ranga-mani and Takayanagi) proposed a general recipe for calculating the entanglement entropyof d -dimensional large-N CFTs which admit holographic dual [11, 12]. Their proposal hassuccessfully satisfied the necessary conditions required for the entanglement entropy of fieldtheories and matched with the prior known results obtained for the two-dimensional CFT[13–15]. The remarkable success of this proposal stimulated numerous works which gaveus more insights toward better understanding of this topic [16–22].In order to overcome the scheme-dependent measure of entanglement, one can use aspecific linear combinations of entanglement entropies called mutual information which isdefined by I ( A : B ) ≡ S ( A ) + S ( B ) − S ( A ∪ B ) , where S denotes the entanglement entropyof its associated spacetime region. Mutual information is a finite and positive semi-definitequantity which measures the total correlations between the two disjoint regions A and B [23–25]. We will show that in our background the dominant term in mutual informationfeatures an area-law behavior in high temperature limit, in contrast to the entanglemententropy which has a volume-law behavior within the same thermal limit. Therefore mutualinformation would be a more reliable quantity to be used in order to investigate the physicalproperties of systems described by QFTs.In this paper we consider N = 4 super Yang-Mills theory at finite temperature, T ,charged under a U (1) subgroup of its SU (4) R R-symmetry group which includes one chemi-cal potential, µ , and it is dual to the well-known 1RCBH background [26–30]. More detaileddiscussions regarding this background can be found in section 2. Due to the fact that un-derlying theory is conformal, its phase diagram is one-dimensional and it is characterized bythe ratio µ/T . This one-dimensional line ends in a critical point denoted by µ c /T c = π/ √ δ = 2 and γ = 12 , (1.1)– 2 –nd by using the well-known scaling relations we determine the four static critical exponentsto be ( α, β, γ, δ ) = (cid:18) , , , (cid:19) , (1.2)which are in full agreement with the ones obtained previously in the literature using ther-modynamic quantities [30, 32, 33]. As we mentioned in the introduction, we are interested in studying the critical phenomenaof a strongly coupled plasma using the framework of holography. Therefore we start witha holographic geometry in five dimensions dual to the aforementioned 4-dimensional fieldtheory with critical point, which is known as the 1RCBH background [26–30].
We consider a gravitational theory on a five dimensional manifold with metric g µν , consist-ing of a gauge field, A µ , and a scalar field (dilaton), φ , which is described by the followingEinstein-Maxwell-Dilaton action S EMD = 116 πG (5) N (cid:90) d x √− g (cid:20) R − f ( φ )4 F µν F µν −
12 ( ∂ µ φ )( ∂ µ φ ) − V ( φ ) (cid:21) , (2.1)where G (5) N is the 5-dimensional Newton constant. The coupling function between the gaugefield and the dilaton , f ( φ ), and the dilaton potential, V ( φ ), are given by f ( φ ) = e − (cid:113) φ ,V ( φ ) = − R (cid:18) e φ √ + 4 e − (cid:113) φ (cid:19) , (2.2)where R is the asymptotic AdS radius. The 1RCBH background is the solution to theequations of motion of the EDM action in eq. (2.1) and it is described by ds = e A ( z ) (cid:16) − h ( z ) dt + d(cid:126)x (cid:17) + e B ( z ) h ( z ) R z dz , (2.3)where A ( z ) = ln (cid:18) Rz (cid:19) + 16 ln (cid:18) Q z R (cid:19) ,B ( z ) = − ln (cid:18) Rz (cid:19) −
13 ln (cid:18) Q z R (cid:19) ,h ( z ) = 1 − M z R (cid:16) Q z R (cid:17) ,φ ( z ) = − (cid:114)
23 ln (cid:18) Q z R (cid:19) , Φ( z ) = M Qz h R (cid:16) Q z h R (cid:17) − M Qz R (cid:16) Q z R (cid:17) , (2.4)– 3 –n which Φ( z ) is the electric potential given by the temporal component of the gauge fieldand it is chosen such that it is zero on the horizon and regular on the boundary [31, 32].Note that we are working in the Poincare patch coordinates by defining z = R /r suchthat z is the radial bulk coordinate and the boundary lies at z →
0. The black hole mass isdenoted by M while Q denotes its charge. By using the fact that h ( z h ) = 0 , one can obtaina relation for the mass which then gives us the following expression for the blackening factor h ( z ) = 1 − (cid:18) zz h (cid:19) (cid:16) Qz h R (cid:17) (cid:16) QzR (cid:17) . (2.5)The location of the black brane horizon, z h , could be expressed in terms of M and Q as z h = R (cid:115) Q + (cid:112) Q + 4 M R M . (2.6) The field theory dual to the geometry background discussed in the last subsection is char-acterized by the temperature, T , and the chemical potential, µ . Following the usual recipefor obtaining the temperature—Wick rotating the temporal coordinate of the metric, per-forming a Taylor expansion of the metric coefficients near the horizon and imposing theperiodicity condition—we obtain the Hawking temperature as T = 14 πR (cid:12)(cid:12)(cid:12) e A ( z h ) − B ( z h ) h (cid:48) ( z h ) z h (cid:12)(cid:12)(cid:12) , (2.7)hence T = 12 πz h (cid:16) Qz h R (cid:17) (cid:114) (cid:16) Qz h R (cid:17) , (2.8)where the prime symbol in eq. (2.7) denotes the derivative with respect to z coordinate.The chemical potential is given by µ = 1 R lim z → Φ( z ) , (2.9)therefore µ = QR (cid:114) (cid:16) Qz h R (cid:17) . (2.10)By using eqs. (2.8) and (2.10) we obtain the following useful non-negative dimensionlessquantity Qz h R = √ λ (cid:16) ± (cid:112) − λ (cid:17) s.t. λ ≡ (cid:18) µ/Tπ/ √ (cid:19) . (2.11)For our future use, we rewrite temperature in terms of the dimensionless quantity Qz h /R as T = ˆ T (cid:32) ξ √ ξ (cid:33) , (2.12)– 4 –here we have defined ˆ T ≡ /πz h and ξ ≡ Q z h /R . In order to see which sign of the eq.(2.11) relates to a thermodynamically stable phase, one needs to obtain the entropy andcharge density in terms of µ/T first. One can show that the entropy density, s , and U (1)charge density, ρ , are given by s = R G (5) N z h (cid:112) ξ , ρ = QR πG (5) N z h (cid:112) ξ . (2.13)Now suppose that the thermodynamic potential of a system is given by Φ( x , ..., x r ) de-pending on some set of variables { x , ..., x r } . Then for a stable phase, the Hessian matrix, H , of the associated potential defined by H ij ≡ (cid:20) ∂ Φ ∂x i ∂x j (cid:21) , (2.14)should be positive-definite . Here we can choose the free energy density f which satisfies − df = sdT + ρdµ , as our relevant thermodynamic potential. Hence by evaluating itsHessian matrix which then reduces to H = [ ∂ ( s, ρ ) /∂ ( T, µ )] , we find out that if we choosethe minus sign in eq. (2.11), both principal minors of H become strictly positive for µ/T ∈ [0 , π/ √
2] or λ ∈ [0 ,
1] . Therefore H is positive-definite and the local thermodynamicstability of the field theory dual to 1RCBH background is guaranteed. Note that since λ ∈ [0 ,
1] then the parameter Qz h /R ∈ [0 , √
2] , therefore ξ , would be a number of the oneorder of magnitude.In order to classify the phase transitions in this model, we observe that for the secondderivatives of the free energy density with respect to T and µ we have − (cid:18) ∂ f∂T (cid:19) µ = (cid:18) ∂s∂T (cid:19) µ ≡ C µ T and − (cid:18) ∂ f∂µ (cid:19) T = (cid:18) ∂ρ∂µ (cid:19) T ≡ χ , (2.15)where C µ is the specific heat at constant chemical potential and χ is the 2nd order R-charge susceptibility. One could see that both diverge at µ/T = π/ √ µ c /T c = π/ √ Qz h /R = √ µ/T as expected, since the underlying theory is conformal. Suppose that a CFT exists on a Cauchy surface C of a d -dimensional Lorentzian manifold B d . We define region A to be a subset of C such that A ∪ A c = C where A c is its comple-ment. This region has a boundary ∂A (the entangling surface) which is a co-dimension 2hypersurface in B d . We then assume that the Hilbert space H of the CFT can be factorizedinto H A ⊗H A c and we let ρ to be a density operator (matrix) associated to a state | ψ (cid:105) ∈ H . Note that the converse does not necessarily imply the global stability since the positive-definiteness ofHessian matrix for a convex function indicates a local minima, therefore the stability should be considereda local one instead. – 5 – igure 1 . A simplified sketch of a strip A on the Cauchy surface C with characteristic length l which has a unique minimal surface γ A in the bulk anchored on its boundary. Now by defining the reduced density operator for region A to be ρ A ≡ tr A c ( ρ ) where tr A c denotes the partial trace over A c , one can measure the entanglement between regions A and A c using the von Neumann entropy which is a non-local quantity defined by S ( A ) ≡ − tr( ρ A log ρ A ) . (3.1)In the framework of the AdS/CFT correspondence where we have a d -dimensional CFT dualto a ( d + 1)-dimensional asymptotically AdS spacetime M d +1 , one can use the holographicentanglement entropy (Ryu-Takayanagi and Hubeny-Rangamani-Takayanagi prescriptions)which is given by [11, 12] S ( A ) = A ( γ A )4 G ( d +1) N , (3.2)where γ A is a co-dimension 2 extremal surface in M d +1 with the area A ( γ A ) such that ∂γ A = ∂A and G ( d +1) N is the ( d + 1)-dimensional Newton constant. This recipe has alreadypassed the tests one expects for the entanglement entropy. Also the quantities derived fromthis relation such as holographic mutual information satisfy all the necessary conditions—as well as an extra feature called monogamy— required for any entanglement measure inthe context of quantum information theory [13, 14, 25, 34]. In the holographic set up, we choose our boundary system to be an infinite rectangularstrip of characteristic length l (Fig.1) and we parameterize the boundary coordinate x interms of the bulk coordinate z . We specify this strip by x (1) ≡ x ∈ [ − l , l , x ( i ) ∈ [ − L , L , i = 2 , , (3.3)such that L → ∞ . By assuming that this measure is mathematically well-defined in QFT. – 6 – .2 Area and Characteristic Length
For a general bulk manifold M d +1 with the metric g µν , the extremal surface γ A is aco-dimension 2 hypersurface in M d +1 whose area functional is given by A ( γ A ) = (cid:90) d d − x (cid:112) det ( g MN ) , (3.4)where g MN is the induced metric on γ A . For the geometric background of eq. (2.4) on theconstant time slice, we parameterize x ≡ x ( z ) and obtain the area as A = 2 L (cid:90) z c dz e A ( z ) (cid:115) x (cid:48) ( z ) + R z h ( z ) e B ( z ) − A ( z )) . (3.5)Since the integrand of eq. (3.5) does not have an explicit dependence on x , if we constructits Hamiltonian we get the following differential equation x (cid:48) ( z ) ≡ dxdz = R z e A ( z c ) e B ( z ) − A ( z ) (cid:112) h ( z ) (cid:112) e A ( z ) − e A ( z c ) , (3.6)where z = z c is the extrema of the minimal surface where z (cid:48) ( x ) = 0. By substituting eq.(3.6) in eq. (3.5) we obtain A = 2 L R (cid:90) z c dz z c z (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) ξ (cid:16) zz h (cid:17) ξ (cid:16) z c z h (cid:17) − (cid:18) zz h (cid:19) ξ ξ (cid:16) zz h (cid:17) − × (cid:16) z c z (cid:17) ξ (cid:16) zz h (cid:17) ξ (cid:16) z c z h (cid:17) − − , (3.7)where we have used the definition ξ ≡ (cid:0) Qz h /R (cid:1) which we introduced previously insection 2.2. By integrating the differential equation of eq. (3.6) and imposing the bound-ary conditions x ( z c ) = 0 and x (0) = ± l/ l (cid:90) z c dz (cid:34) ξ (cid:18) zz h (cid:19) (cid:35) − − (cid:18) zz h (cid:19) ξ ξ (cid:16) zz h (cid:17) − × (cid:16) z c z (cid:17) ξ (cid:16) zz h (cid:17) ξ (cid:16) z c z h (cid:17) − − . (3.8)Since it is not easy to calculate this integral analytically, by the help of the generalizedmultinomial expansions given in the appendix A we show that the eq. (3.8) can be repre-– 7 –ented by the following series l z c ∞ (cid:88) k =0 k (cid:88) n =0 ∞ (cid:88) m =0 ∞ (cid:88) j =0 G knmj F knmj (cid:18) z c z h (cid:19) k + n + m ) , (3.9)where G knmj ≡ Γ (cid:0) k + (cid:1) Γ (cid:0) j + m + (cid:1) Γ (2 + 3 j + k + n )2 π Γ ( n + 1) Γ ( k − n + 1) Γ ( j + 1) Γ (3 + 3 j + k + n + m ) ,F knmj ≡ ( − k + n ξ k − n + m (1 + ξ ) n (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) − m . (3.10)Note that in order to make use of the binomial expansions for negative powers we madesure that the following relations are satisfied for the whole range of ξ ∈ [0 ,
2] and for z c between boundary and the horizon ξ (cid:16) z c z h (cid:17) ξ (cid:16) z c z h (cid:17) (cid:18) − z z c (cid:19) < ξ (cid:18) zz h (cid:19) − (1 + ξ ) (cid:18) zz h (cid:19) < . (3.11)These expansions can also be used to represent the area in eq. (3.7) by A = 2 L R π ∞ (cid:88) k =0 k (cid:88) n =0 ∞ (cid:88) m =0 ∞ (cid:88) j =0 Γ (cid:0) k + (cid:1) Γ (cid:0) j + m + (cid:1) Γ ( n + 1) Γ ( k − n + 1) Γ ( j + 1) Γ ( m + 1) × ( − k + n ξ k − n + m (1 + ξ ) n (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) − m − (cid:18) z c z h (cid:19) m × (cid:90) z c dz (cid:34) ξ (cid:18) zz h (cid:19) (cid:35) (cid:34) − (cid:18) zz c (cid:19) (cid:35) m z − (cid:18) zz c (cid:19) j (cid:18) zz h (cid:19) k + n ) . (3.12)As one would expect in general, the area enclosed by the extremal surface is divergentdue to its near boundary behavior. Here one could show that the last integral (hence thearea) remains finite if the condition k + n + 3 j > k = n = j = 0) and ( k = 1 , n = j = 0) terms together and perform their sum over m toget the part of the area in which the divergent term is contained. By doing so, we obtain A ≡ L R (cid:15) + 3 ξ z h − z c (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) , (3.13)where z = (cid:15) , such that (cid:15) →
0, is the cut-off surface in the bulk geometry related to theUV regulator of the field theory. We see that the divergent term in eq. (3.13) has anarea-law behavior which appears in the corresponding holographic entanglement entropy This method of calculating the entanglement entropy and mutual information has been initially usedin [35–37]. – 8 –s well. This result is indeed expected in a d -dimensional field theory side where the leadingdivergence in the UV limit (cid:15) → /(cid:15) term . It is given by A fin = L R z c ξ (cid:18) z c z h (cid:19) − (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) + 1 + ξ ξ (cid:18) z c z h (cid:19) (cid:32) ξ (cid:18) z c z h (cid:19) (cid:33) − + L R z c (cid:40) ∞ (cid:88) k =2 k (cid:88) n =0 ∞ (cid:88) m =0 Λ knm Γ (cid:0) m + (cid:1) Γ ( k + n − k + n + m + 1) (cid:18) z c z h (cid:19) k + n + m ) × (cid:34) ( m + 1) + ( k + n − (cid:32) ξ (cid:18) z c z h (cid:19) (cid:33)(cid:35)(cid:41) + L R z c (cid:40) ∞ (cid:88) k =0 k (cid:88) n =0 ∞ (cid:88) m =0 ∞ (cid:88) j =1 Λ knm Γ (cid:0) m + j + (cid:1) Γ ( k + n + 3 j − j + 1) Γ ( k + n + m + 3 j + 1) (cid:18) z c z h (cid:19) k + n + m ) × (cid:34) ( m + 1) + ( k + n + 3 j − (cid:32) ξ (cid:18) z c z h (cid:19) (cid:33)(cid:35)(cid:41) , (3.14)whereΛ knm ≡ ( − k + n Γ (cid:0) k + (cid:1) π Γ ( n + 1) Γ ( k − n + 1) ξ k − n + m (1 + ξ ) n (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) − m − . (3.15)We should point out that although this result for the area is lengthy and hard to workwith, it gives us the vantage point of investigating the behaviors of entanglement entropyand mutual information near the critical point analytically, which we will discuss in theforthcoming sections. As one could observe in eq. (3.14), the area of the minimal surface would be characterizedby its two dimensionless parameters ξ and z c /z h . In this section we investigate the holo-graphic entanglement entropy with respect to z c /z h which introduces two thermal limits,while we leave its analysis with regard to the parameter ξ which controls the critical be-havior to section 6. Now given the ratio of the extremal surface location to the horizonlocation, i.e. z c /z h , one could expect to see two different cases for the area obtained in theprevious section (hence for the entanglement entropy) namely when z c /z h (cid:28) z c /z h ∼ Note that our preferred cut-off independent measure of entanglement would be the mutual informationinstead, as we discuss it in section 5. – 9 –hile never penetrating it. This is due to the fact that in a static asymptotically AdSspacetime, the minimal surface does not pass beyond the horizon of an existing black hole[38] . For the field theory side with the introduced scale l , we can immediately translatethe aforementioned cases into the two inequivalent thermal limits; ˆ T l (cid:28)
T l (cid:29) T is defined in eq. (2.12). Hence one could identify the z c /z h (cid:28) z c /z h ∼ One of the main concerns while dealing with the infinite series representation of functions isthe issue of their convergence, since depending on their growth, they might simply divergeas well. In the low temperature limit where z c /z h (cid:28) z c /z h )obtaining l = z c (cid:40) a − a ξ (cid:18) z c z h (cid:19) + (cid:20) a (1 + ξ )2 + a ξ (cid:21) (cid:18) z c z h (cid:19) + O (cid:18) z c z h (cid:19) (cid:41) , (4.1)where we performed the sum over j and the numerical constants a , a and a are givenin the appendix B. By solving eq. (4.1) perturbatively for z c at 4th order in ( l/z h ) we get z c = la (cid:40) ξ a (cid:18) lz h (cid:19) + 12 a (cid:20) ξ (cid:18) − a a (cid:19) − a a (1 + ξ ) (cid:21) (cid:18) lz h (cid:19) + O (cid:18) lz h (cid:19) (cid:41) . (4.2)Now if we expand the finite part of the area in eq. (3.14) to the lowest orders, we obtain A finitelow = L R z c (cid:34) ξ (cid:18) z c z h (cid:19) − (cid:35) + L R z c ∞ (cid:88) j =1 Γ (cid:0) j + (cid:1) √ π Γ ( j + 1) (3 j − (cid:34) ξ (cid:18) z c z h (cid:19) + (cid:32) (cid:0) − ξ + 9 ξ + 9 (cid:1) j − ξ + 1)18 j + 6 (cid:33) (cid:18) z c z h (cid:19) (cid:35) . (4.3)Finally, by performing the sum and substituting for z c from eq. (4.2) in the last expressionand then using eq. (3.2), we obtain the entanglement entropy in the low temperature limit We will comment on this point in appendix D where we show how close could minimal surface get tothe horizon in the high temperature limit. – 10 –s S finitelow = R G (5) N (cid:18) Ll (cid:19) (cid:40) a ( w −
1) + ξ (cid:18) lz h (cid:19) + 12 a (cid:34) (1 + ξ ) (cid:0) − w + 3 w + 2 ( w − a a (cid:1) + ξ (cid:0) ( w − a a − − w (cid:1)(cid:35) (cid:18) lz h (cid:19) (cid:41) , (4.4)where the numerical constants w , w and w are given in the appendix B. We note that inthe limit where Q → z h = 1 /πT and the subleading terms become 2nd and 4thorder in T l as expected from the AdS-RN results. To make this relation more transparentwe define c ≡ a ( w − ≈ − . ,f ( ξ ) ≡ (1 + ξ ) (cid:0) − w + 3 w + 2 ( w − a a ) (cid:1) a + ξ (cid:0) ( w − a a − − w (cid:1) a ≈ .
13 (1 + ξ ) − .
43 ( ξ . (4.5)The first term in eq. (4.4) which we denoted by c in the last expression, does not dependon temperature and it is the contribution of the AdS boundary. Another consistency checkfor our result would be the case in which we set the chemical potential to zero. The metricof the 1RCBH background then reduces to the AdS-Schwarzschild metric and it is easyto see that we recover the result which was obtained previously in the literature for thisparticular background [35].By using the reparametrization of eq. (2.12) we can rewrite the low temperature limitof entanglement entropy as S finitelow = R G (5) N (cid:18) Ll (cid:19) (cid:40) c + ξ π ˆ T l ) + 12 f ( ξ ) ( π ˆ T l ) (cid:41) , (4.6)where ˆ T would be equal to T in the limit Q → ξ , which wouldappear in the mutual information as well, will be utilized later in order to investigate itsbehavior near the critical point. As we mentioned in the previous section, infinite series do not always converge. Fortunately,for a given divergent series some methods of summability or regularization are available toapply in order to overcome the issue of divergence. We observe that in the high temperaturelimit where z c ∼ z h , the infinite sum of eq. (3.14) does not converge . By making useof the mentioned methods however, we can regularize this series and make it convergentby rearranging it in such a way that we could recover a term proportional to l . We This divergence is due to the growth of series for z c = z h and it is not related to UV divergence. We will show in appendix D that the sum for l in eq. (3.9) converges for z c ∼ z h after regularization. – 11 –ave included the full expression of the resulted regularized series in the appendix C.Therefore we can take the limit z c → z h of eq. (C.1) and by using eq. (3.2), we obtain theentanglement entropy in the high temperature regime as S finitehigh = R G (5) N (cid:18) Lz h (cid:19) (cid:40)(cid:112) ξ (cid:18) lz h (cid:19) + ( S + S + S ) (cid:41) , (4.7)where we defined S ≡ ξ − − ξ − ξ − ξ − ξ + (cid:112) ξ + 1 (cid:18) − ξ − ξ + 214105 ξ + 2435 ξ + 1635 ξ (cid:19) , S ≡ ∞ (cid:88) k =2 k (cid:88) n =0 ∞ (cid:88) m =0 Γ (cid:0) k + (cid:1) Γ (cid:0) m + (cid:1) Γ ( k + n + 2) ( − k + n ξ k − n + m (1 + ξ ) n − m − π Γ ( n + 1) Γ ( k − n + 1) Γ ( k + n + m + 3) × (cid:26) m + 1 k + n − (cid:20) m + 1 k + n (cid:18) mk + n + 1 (cid:19)(cid:21) + (1 + ξ )( m + 1) k + n (cid:18) mk + n + 1 (cid:19)(cid:27) , S ≡ ∞ (cid:88) k =2 k (cid:88) n =0 ∞ (cid:88) m =0 ∞ (cid:88) j =1 Γ (cid:0) k + (cid:1) Γ (cid:0) j + m + (cid:1) Γ ( k + n + 3 j + 2) π Γ ( n + 1) Γ ( j + 1) Γ ( k − n + 1) Γ ( k + n + m + 3 j + 3) × ( − k + n ξ k − n + m (1 + ξ ) n − m − × (cid:40) m + 1 k + n + 3 j − (cid:20) m + 1 k + n + 3 j (cid:18) mk + n + 3 j + 1 (cid:19)(cid:21) + (1 + ξ )( m + 1) k + n + 3 j (cid:18) mk + n + 3 j + 1 (cid:19)(cid:41) . (4.8)By using eq. (2.12) we obtain S finitehigh = R G (5) N (cid:18) Ll (cid:19) (cid:110)(cid:112) ξ ( π ˆ T l ) + S ( π ˆ T l ) (cid:111) , (4.9)where we defined S ≡ S + S + S , for convenience. We note that the finite leading tem-perature dependent term (first term in eq. (4.9)) scales with the volume of the rectangularstrip, L l , while the sub-leading term is area dependent. Hence the first term describes thethermal entropy while the second term corresponds to the entanglement entropy betweenthe strip region and its complement, and within this thermal limit the largest contributioncomes from the near horizon part of the minimal surface.– 12 – igure 2 . A naive sketch of the case where two disjoint strips A and B are separated by thedistance x with the choices for minimal surfaces. The union of brown curves represents the choiceof minimal surface for A ∪ B when the separation distance is small enough. We mentioned in the section 3.2 that the area of an extremal surface has a divergent naturein general and it needs to be regulated. This fact immediately implies the dependency ofthe holographic entanglement entropy to the choice of a cut-off hypersurface near theboundary. To avoid a regulator-dependent measure of entanglement, one could borrowanother quantity from quantum information theory called the mutual information which isa well-defined entanglement measure in the context of QFT [23]. For given disjoint regions
A, B ⊂ C , the mutual information is defined by I ( A : B ) = S ( A ) + S ( B ) − S ( A ∪ B ) , (5.1)where S ( A ∪ B ) denotes the entanglement entropy of the composite region ρ AB . First wenote that this measure is positive-semidefinite, since by using the subadditivity inequalityof the von Neumann entropy which states that S ( A ) + S ( B ) ≥ S ( A ∪ B ), one can eas-ily show that I ( A : B ) ≥ ρ AB = ρ A ⊗ ρ B . It was also shown that mutual informa-tion incorporates the total amount of correlations between two subsystems or equivalentlytwo separate spacetime regions A and B [24]. More importantly, mutual information isregulator-independent since the UV divergences of S ( A ) and S ( B ) are canceled by thosein S ( A ∪ B ) .In our set up, we let the two disjoint systems both be infinite rectangular strips ofsize l which are separated by the distance x on the boundary (Fig.2). For the minimalsurface γ A ∪ B , satisfying the condition ∂γ A ∪ B = ∂ ( A ∪ B ) , we have two choices: whenthe separation distance is large enough, one can deduce that the A ( γ A ∪ B ) > A ( γ A ∪ γ B ) Simply, I ( A : B ) quantifies the amount of common information between A and B . – 13 –ence it follows that one would have S ( A ∪ B ) = S ( A ) + S ( B ) which then results in thevanishing mutual information [25]. On the other hand when x is small enough, A ( γ A ∪ B )would be equal to A ( γ x ) plus the area of the minimal surface corresponding to the entireunion of the regions A, B and x . Therefore one can assume that there would be a criticalseparation distance larger than which the mutual information vanishes and the two regions A and B become disentangled. This has been shown in [25]. For the non-vanishing mutualinformation we have I ( A : B ) = 2 S ( l ) − S ( x ) − S (2 l + x ) . (5.2)We will use this relation to discuss the behavior of mutual information in different thermallimits. Since the mutual information is a linear combination of entanglement entropies, one couldsimilarly investigate its behavior with respect to the thermal limits which we discussed insection 4. In addition to those cases, we are able to compare the location of horizon to thenewly introduced separation distance as well, which would be specified by the dimensionlessratio x/z h . In the field theory, it would mean that the parameter ˆ T x introduces an extratemperature limit. Therefore we identify ( l/z h (cid:28) ∧ ( x/z h (cid:28)
1) or ( ˆ
T l (cid:28) ∧ ( ˆ T x (cid:28) x/z h (cid:28) ∧ ( l/z h (cid:29)
1) or ( ˆ
T x (cid:28) ∧ ( ˆ T l (cid:29) x/z h (cid:29) ∧ ( l/z h (cid:29) T x (cid:29) ∧ ( ˆ T l (cid:29)
1) characterizes the high temperature regime where ˆ T is defined ineq. (2.12). By using eqs. (4.6) and (5.2), the mutual information in the low temperature limit where z h (cid:29) l, x is given by I low = R G (5) N (cid:40) c (cid:34) (cid:18) Ll (cid:19) − (cid:18) L l + x (cid:19) − (cid:18) Lx (cid:19) (cid:35) − (cid:18) l + xz h (cid:19) (cid:18) Lz h (cid:19) f ( ξ ) (cid:41) , (5.3)By eq. (2.12) we obtain I low = R G (5) N (cid:40) c (cid:34) (cid:18) Ll (cid:19) − (cid:18) L l + x (cid:19) − (cid:18) Lx (cid:19) (cid:35) − (cid:18) l + xl (cid:19) (cid:18) Ll (cid:19) f ( ξ ) ( π ˆ T l ) (cid:41) , (5.4)where the first terms in brackets matches the result we expect for T = 0 case [36] and thefinite temperature-dependent term obeys the area-law behavior which has been proved tobe true generally in [24]. In the intermediate temperature limit where x (cid:28) z h (cid:28) l , the mutual information isobtained by using eqs. (4.7), (4.6) and (5.2), and it is given by I int = R G (5) N (cid:40) − c (cid:18) Lx (cid:19) + (cid:18) Lz h (cid:19) ( S − ξ − (cid:18) xz h (cid:19) (cid:18) Lz h (cid:19) (cid:112) ξ − (cid:18) xz h (cid:19) (cid:18) Lz h (cid:19) f ( ξ ) (cid:41) , (5.5)– 14 –here S ≡ S + S + S . As one can see, the mutual information in this limit does notdepend on the characteristic length of the system. By using eq. (2.12) we obtain I int = R L G (5) N ( π ˆ T ) (cid:40) − c ( π ˆ T x ) + ( S − ξ − ( π ˆ T x ) (cid:112) ξ − f ( ξ ) ( π ˆ T x ) (cid:41) . (5.6)One could also go further and investigate the case where two strips touch each other i.e.when x ∼ x → x → I int = R G (5) N (cid:40) − c (cid:18) Lx (cid:19) + (cid:18) S − ξ (cid:19) ( π ˆ T L ) (cid:41) , (5.7)by keeping in mind that in all of the above expressions, c is a numerical coefficient and f ( ξ )depends only on ξ where both are defined in eq. (4.5). We note that the leading term in thelast expression obeys an area-law divergence with respect to the separation distance x , andthe finite sub-leading term scales with the area of strip, L , times temperature squared.This area law behavior corresponds to the case where the volume-law thermodynamicentropy contribution to the entanglement is absent and eq. (5.7) is a measure of purequantum entanglement. This unique behavior has been also observed for the differentbackgrounds in [36, 37]. As we discussed earlier in this section, for x/z h (cid:29) T x (cid:29) A ∪ B for large separation distances becomes the disjoint union of the two strips minimalsurfaces, hence the mutual information identically vanishes. In this section we study the critical phenomena of the underlying field theory using theinformation-theoretic measure we introduced in the previous section. Mutual information,a scheme-independent quantity, is considered to serve as an order parameter in the stronglycoupled plasma in our setup and we investigate whether the static critical exponents of thetheory could be read off from its behavior near or at the critical point . We first begin byrecalling the notation we introduced in subsection 2.2 for the critical point which was char-acterized by the dimensionless quantity ξ = 2 (1 − √ − λ ) /λ where λ ≡ ( µ/T ) / ( π/ √ ξ → λ →
1, we observe that the mutual information, whichdepends on the parameters of the theory, remains finite and its leading behavior at thecritical point, omitting the first constant term in brackets, is proportional to √ − λ as I low ∼ − R G (5) N (cid:18) l + xl (cid:19) (cid:18) Ll (cid:19) ( π ˆ T l ) (cid:18) (3 b + 23 b ) − b + 23 b ) (cid:112) − λ (cid:19) , (6.1) The role of entanglement entropy as a probe of phase transitions in field theories with holographic dualis pointed out previously in [39–41]. – 15 –here we have defined b ≡ (cid:0) − w + 3 w + 2 ( w − a a ) (cid:1) a and b ≡ (cid:0) ( w − a a − − w (cid:1) a , (6.2)such that f ( ξ ) = b (1 + ξ ) + b ( ξ /
6) . It is easy to see that this result, i.e. beingproportional to √ − λ , also features in the intermediate regime. Therefore such behavioris independent of the thermal limits and regardless of whether we take the limit where theseparation distance x goes to zero or not, it is true for all the results we have obtained sofar for the mutual information in subsection 5.1. So we can conclude I low ∼ I int ∝ (cid:18) µT − µ c T c (cid:19) / . (6.3)By comparing eq. (6.3) to the expected power-law behavior at the critical point (cid:18) µT − µ c T c (cid:19) /δ , (6.4)analogous to the power-law behavior of the critical isotherm evaluated at the critical tem-perature, one may conclude that δ = 2 . Hence by considering the mutual information asan order parameter, we were able to obtain one of the independent critical exponents ofthe underlying theory.In order to obtain the other remaining independent exponent—by following the ther-modynamic analogy and the same discussions in the beginning of this section—we can usethe slope of mutual information near the critical point for this purpose. We note thatalthough the mutual information is finite there, we see that its derivative with respect to λ will tend to infinity as we approach the critical point. For the slope of mutual informationin any thermal limit one could write dI/dλ = ( dI/dξ ) ( dξ/dλ ) where dξdλ = 4 (cid:16) − √ − λ (cid:17) λ √ − λ . (6.5)Therefore at the critical point, one could easily see that dξ/dλ behaves as (1 − λ ) − / hence it diverges. The only remaining fact that needs to be checked is whether dI/dξ isfinite or it tends to zero at the critical point. By using eqs. (5.3) and (5.5) we obtain dI low dξ = − R L G (5) N ( l + x ) z h (cid:16) b + ξ b (cid:17) , (6.6)and dI int dξ = R L G (5) N (cid:20) z h (cid:18) d S dξ − (cid:19) − x z h √ ξ − x z h a (cid:16) b + ξ b (cid:17)(cid:21) , (6.7)as well as dI int dξ (cid:12)(cid:12)(cid:12) x → = R L G (5) N (cid:20) z h (cid:18) d S dξ − (cid:19) (cid:21) , (6.8) This result is similar to the critical exponent calculated for this theory using the thermodynamicquantity, charge density. – 16 –here b and b are defined in eq. (6.2). We can see that in all cases, dI/dξ remains finiteat the critical point ξ = 2. Therefore we reach the conclusion that the mutual informationdiverges near the critical point with the power-law behavior given by (cid:18) µT − µ c T c (cid:19) − / ≡ (cid:18) µT − µ c T c (cid:19) − γ , (6.9)where γ = 1 / .Finally, by using the following known scaling relations for the static critical exponents α + β (1 + δ ) = 2 , α + 2 β + γ = 2 , (6.10)we obtain β = 1 / α = 1 / .Remarkably, these exponents are identical to those calculated previously for this modelwithin the thermodynamic framework [30, 33]. The dynamic critical exponent of this modelhas been also obtained via different quantities in [42–44]. It is interesting to note that thesame identical values for these four static critical exponents have been also obtained forcompletely different gravitational backgrounds such as Born-Infeld AdS black holes andtopological charged black holes in Horava-Lifshitz gravity [45–47]. Summary
In this work we have argued that information-theoretic measures like mutualinformation could also be used in order to study the critical phenomena of the stronglycoupled field theories in the large-N limit. We based our claim on the result of our analyticcalculations for the entanglement entropy and mutual information for the strongly coupledplasma at finite temperature and chemical potential with a critical point using the holo-graphic methods. It is known, as we have also observed here, that despite the volume-lawbehavior of entanglement entropy in the high temperature limit, mutual information scaleswith the area of the system, therefore it has the upper hand in capturing the full quantumentanglement structure of the field theories. Based on this observation, we analyzed thecritical behavior of the underlying plasma using our analytical results for the mutual in-formation in various thermal limits and we found out that although it was constant at thecritical point with the exponent δ − = 1 / γ = 1 / α, β, γ, δ ) = (cid:18) , , , (cid:19) , (6.11) By assuming the correspondence between entanglement entropy and its thermodynamic counterpartand using the same arguments we made in the beginning of this section, we could calculate the slope ofentanglement entropy in eqs. (4.6) and (4.9) in order to obtain the exponent α instead, which is analogousto the exponent of specific heat capacity at constant chemical potential, C µ , evaluated near the criticalpoint. In doing so, we obtain α = 1 / We could use different names and notations for these critical exponents as these labels are associatedwith the behavior of quantities in the vicinity of the critical point, approached along the first-order lineexcept for the critical isotherm, while there is no such first-order transition in this model and the phasediagram is one-dimensional. However, to avoid any confusion we would rather use these notations instead. – 17 –hich is in exact agreement with the prior thermodynamics results in the literature. Sinceentanglement entropy (hence mutual information) has more advantages than the thermo-dynamic entropy , and it captures the critical phenomena as well, our result suggests thatit would be a proper candidate for further investigations regarding the various physicalproperties of the strongly coupled systems, specially in the ongoing research program ofunderstanding the rich phase structure of hot QCD at finite density. Acknowledgment
H. E. would like to thank high energy, cosmology and astroparticle physics group at ICTPwhere the main parts of the calculations of this paper was done and K. Papadodimas fortheir warm hospitality. H. E. would also like to thank M. Ali-Akbari for fruitful discussions.
A Mathematical Relations
In this appendix we present some useful relation which we used in our work. • Newton’s binomial and trinomial expansion
Newton’s generalized binomialexpansion for | y | < | x | is given by( x + y ) r = ∞ (cid:88) k =0 (cid:18) rk (cid:19) x r − k y k , ( x + y ) − r = ∞ (cid:88) k =0 ( − k (cid:18) r + k − k (cid:19) x − r − k y k . (A.1)Similarly the generalized trinomial expansion for | y + z | < | x | is given by( x + y + z ) r = ∞ (cid:88) k =0 k (cid:88) j =0 (cid:18) rk (cid:19) (cid:18) kj (cid:19) x r − k y k − j z j , ( x + y + z ) − r = ∞ (cid:88) k =0 k (cid:88) j =0 ( − k (cid:18) r + k − k (cid:19) (cid:18) kj (cid:19) x − r − k y k − j z j , (A.2)where x, y, r ∈ R and r >
0. Note that for any real numbers p and q we have (cid:18) pq (cid:19) = Γ( p + 1)Γ( q + 1) Γ( p − q + 1) . (A.3) • Asymptote of Polylogarithm
By analytic continuation, polylogarithm function,Li s ( z ) , can be extended to | z | ≥ Re ( s ) > | z | > s ( z ) ∼ − [ln ( z )] s Γ ( s + 1) . (A.4) Although we should point out that the exact equivalence of entanglement entropy with Bekenstein-Hawking entropy is not clear enough as discussed in [48]. – 18 –
Numerical Constants
Here is the list of all numerical constants defined throughout the paper: a ≡ ∞ (cid:88) j =0 Γ (cid:0) j + (cid:1) √ π Γ ( j + 1) (2 + 3 j ) = 3 √ π Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) ,a ≡ ∞ (cid:88) j =0 Γ (cid:0) j + (cid:1) √ π Γ ( j + 1) (4 + 3 j ) = √ π Γ (cid:0) (cid:1) (cid:0) (cid:1) ,a ≡ ∞ (cid:88) j =0 Γ (cid:0) j + (cid:1) (4 − j ) √ π Γ ( j + 1) (2 + 3 j ) (4 + 3 j )= 3 √ π (cid:20) Γ (cid:18) (cid:19) Γ (cid:18) (cid:19) −
35 Γ (cid:18) (cid:19) Γ (cid:18) (cid:19)(cid:21) − F (cid:18) , ,
73 ; 83 ,
103 ; 1 (cid:19) , (B.1)and w ≡ √ π ∞ (cid:88) j =1 Γ (cid:0) j + (cid:1) Γ ( j + 1) (3 j −
1) = 12 / F (cid:18) ,
23 ; 53 ; − (cid:19) ,w ≡ √ π ∞ (cid:88) j =1 j Γ (cid:0) j + (cid:1) Γ ( j + 1) (3 j −
1) (3 j + 1) = 116 F (cid:18) , ,
32 ; 53 ,
73 ; 1 (cid:19) ,w ≡ √ π ∞ (cid:88) j =1 Γ (cid:0) j + (cid:1) Γ ( j + 1) (3 j −
1) (3 j + 1)= 316 F (cid:18) , ,
32 ; 53 ,
73 ; 1 (cid:19) − √ F (cid:18) ,
53 ; 73 ; − (cid:19) . (B.2)– 19 – Minimal Surface Area in the High Temperature Limit
The regularized area of eq. (3.14) in the high temperature limit is given by A finitehigh = L R lz c (cid:32) ξ (cid:18) z c z h (cid:19) (cid:33) + L R z c (cid:40) ξ (cid:18) z c z h (cid:19) − (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) + 1 + ξ ξ (cid:18) z c z h (cid:19) (cid:32) ξ (cid:18) z c z h (cid:19) (cid:33) − − ξ (cid:18) z h z c (cid:19) (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) ξ (cid:18) z c z h (cid:19) + (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) − − − ξ ξ (cid:18) z h z c (cid:19) (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) (cid:34) −
16 + 16 (cid:114) ξ (cid:16) z c z h (cid:17) + ξ (cid:18) z c z h (cid:19) × (cid:32) − ξ (cid:18) z c z h (cid:19) + 5 ξ (cid:18) z c z h (cid:19) (cid:33)(cid:35)(cid:41) + L R z c (cid:40) ∞ (cid:88) k =2 k (cid:88) n =0 ∞ (cid:88) m =0 Γ (cid:0) k + (cid:1) Γ (cid:0) m + (cid:1) Γ ( k + n + 2) π Γ ( n + 1) Γ ( k − n + 1) Γ ( k + n + m + 3) × ( − k + n ξ k − n + m (1 + ξ ) n (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) − m − (cid:18) z c z h (cid:19) k + n + m ) × (cid:40) m + 1 k + n − (cid:20) m + 1 k + n (cid:18) mk + n + 1 (cid:19)(cid:21) + (cid:18) ξ (cid:16) z c z h (cid:17) (cid:19) ( m + 1) k + n (cid:18) mk + n + 1 (cid:19)(cid:41)(cid:41) + L R z c (cid:40) ∞ (cid:88) k =2 k (cid:88) n =0 ∞ (cid:88) m =0 ∞ (cid:88) j =1 Γ (cid:0) k + (cid:1) Γ (cid:0) j + m + (cid:1) Γ ( k + n + 3 j + 2) π Γ ( j + 1) Γ ( n + 1) Γ ( k − n + 1) Γ ( k + n + m + 3 j + 3) × ( − k + n ξ k − n + m (1 + ξ ) n (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) − m − (cid:18) z c z h (cid:19) k + n + m ) × (cid:40) m + 1 k + n + 3 j − (cid:20) m + 1 k + n + 3 j (cid:18) mk + n + 3 j + 1 (cid:19)(cid:21) + (cid:18) ξ (cid:16) z c z h (cid:17) (cid:19) ( m + 1) k + n + 3 j (cid:18) mk + n + 3 j + 1 (cid:19)(cid:41)(cid:41) . (C.1)– 20 – Sub-leading Corrections in the Near Horizon Limit
In this appendix we will investigate the convergence of characteristic length and behaviorof area for z c → z h . We note that the large terms of the series in eq. (3.9) for thecharacteristic length scale l grow as − m ξ m k − / (1 + ξ ) k j − / (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) − m (cid:18) z c z h (cid:19) k + m ) , (D.1)which diverges for z c = z h . We can overcome this situation by isolating the divergent termof eq. (D.1) from eq. (3.9) so that l converges. Hence the regularized l becomes l z c ∞ (cid:88) k =1 ∞ (cid:88) m =1 ∞ (cid:88) j =1 (cid:40) k (cid:88) n =1 (cid:40) G knmj F knmj (cid:18) z c z h (cid:19) k + m + n ) (cid:41) − − m ξ m (1 + ξ ) k π k / j / (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) − m (cid:18) z c z h (cid:19) k + m ) (cid:41) + z c z c ξ π (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) − (cid:18) z c z h (cid:19) ζ (cid:18) (cid:19) Li (cid:32) (1 + ξ ) (cid:18) z c z h (cid:19) (cid:33) . (D.2)where we made use of the following relations for eq. (D.1) in the process of regularizationLi s ( z ) = ∞ (cid:88) k =1 z k k s ,ζ ( p ) = ∞ (cid:88) j =1 j p , (D.3)and the fact that remaining summation over m in eq. (D.1) can be performed. Note thatin eq. (D.3), ζ ( p ) is the Riemann zeta function and Li s ( z ) is the polylogarithm functionof order s . As a mathematical curiosity, one might consider the appearance of Riemannzeta function and polylogarithm with rational order (or even with the integer order) aninteresting phenomena due to their direct link to number theory.Now since the minimal surface remains at a finite distance from horizon [38], we couldsafely assume z c = z h (1 − ε ) , where ε < ε at leading order, we obtain ε = 12 ln (1 + ξ ) − π / (3 + 2 ξ ) (cid:112) ln (1 + ξ )4 ζ (cid:0) (cid:1) ξ (cid:20) σ + 12 − (cid:18) lz h (cid:19)(cid:21) , (D.4) By approximating the series in the limit where all the free indices are set to infinity. The case for s ∈ N in both ζ ( s ) and Li s ( z ) is the subject of wide interest in the number theory literature.See for example [50]. – 21 –here we defined σ ≡ ∞ (cid:88) k =1 k (cid:88) n =1 ∞ (cid:88) j =1 m =1 ( − k + n Γ (cid:0) k + (cid:1) Γ (cid:0) j + m + (cid:1) (1 + ξ ) n − m Γ(3 j + k + n + 2) ξ k + m − n π Γ( j + 1)Γ( n + 1)Γ( k − n + 1)Γ(3 j + k + m + n + 3) − ∞ (cid:88) k =1 ∞ (cid:88) m =1 ∞ (cid:88) j =1 − m − ξ m (1 + ξ ) k − m π k / j / . (D.5)Finally, we are ready to calculate the sub-leading corrections to the minimal surface areain the near horizon limit. Similarly, we observe that the large terms of the series in eq.(C.1) for the area behave as3 − m ξ m (1 + m ) (1 + ξ ) k k − / j − / (cid:34) ξ (cid:18) z c z h (cid:19) (cid:35) − m (cid:18) z c z h (cid:19) k + m ) . (D.6)Hence by following the same regularization procedure as we did for l by isolating this piecefrom eq. (C.1) and performing its sum, together with the assumption z c = z h (1 − ε ) , weexpand the resulted expression at the first order in ε and by the help of eq. (D.4) we obtain A finitehigh = R (cid:18) Lz h (cid:19) (cid:18) lz h (cid:19) (cid:112) ξ + R (cid:18) Lz h (cid:19) ( S + S + S ) − R (cid:18) Lz h (cid:19) (cid:40) ξ ) / ζ (cid:0) (cid:1) ζ (cid:0) (cid:1) ξ (cid:20) σ + 12 − (cid:18) lz h (cid:19)(cid:21) + (1 + ξ ) / ζ (cid:0) (cid:1) π (3 + 2 ξ ) + σ (cid:41) , (D.7)where σ ≡ ∞ (cid:88) k =2 ∞ (cid:88) j =1 ∞ (cid:88) m =0 − m (1 + m ) (3 + 2 ξ ) ξ m (1 + ξ ) k − m π √ ξ k / j / . (D.8)Note that the expression in the second line of eq. (D.7) is the desired sub-leading contri-bution to the area of the minimal surface in the high temperature limit. References [1] M. Rangamani and T. Takayanagi, “Holographic Entanglement Entropy,” Lect. Notes Phys. , pp.1 (2017) [hep-th/1609.01287].[2] C. Holzhey, F. Larsen and F. Wilczek, “Geometric and renormalized entropy in conformalfield theory,” Nucl. Phys. B , 443 (1994) [hep-th/9403108].[3] P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum field theory,” J. Stat.Mech. , P06002 (2004) [hep-th/0405152].[4] H. Casini and M. Huerta, “Entanglement entropy in free quantum field theory,” J. Phys. A , 504007 (2009) [hep-th/0905.2562].[5] T. Nishioka, “Entanglement entropy: holography and renormalization group,” Rev. Mod.Phys. , no. 3, 035007 (2018) [hep-th/1801.10352].[6] L. Bombelli, R. K. Koul, J. Lee and R. D. Sorkin, “A Quantum Source of Entropy for BlackHoles,” Phys. Rev. D , 373 (1986). – 22 –
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