Holographic Entanglement Entropy on Generic Time Slices
aa r X i v : . [ h e p - t h ] M a y YITP-17-21IPMU17-0038
Holographic Entanglement Entropy on Generic Time Slices
Yuya Kusuki a , Tadashi Takayanagi a,b and Koji Umemoto aa Center for Gravitational Physics, Yukawa Institute for Theoretical Physics (YITP),Kyoto University, Kyoto 606-8502, Japan b Kavli Institute for the Physics and Mathematics of the Universe,University of Tokyo, Kashiwa, Chiba 277-8582, Japan
Abstract
We study the holographic entanglement entropy and mutual information forLorentz boosted subsystems. In holographic CFTs at zero and finite temperature,we find that the mutual information gets divergent in a universal way when theend points of two subsystems are light-like separated. In Lifshitz and hyperscalingviolating geometries dual to non-relativistic theories, we show that the holographicentanglement entropy is not well-defined for Lorentz boosted subsystems in general.This strongly suggests that in non-relativistic theories, we cannot make a real spacefactorization of the Hilbert space on a generic time slice except the constant timeslice, as opposed to relativistic field theories. t X (cid:1111) (cid:583) A (cid:583) B A B (cid:1) (cid:1) (cid:1) (cid:2) (cid:2) (cid:3)(cid:1)(cid:4)(cid:2)(cid:4)(cid:3) (cid:1) (cid:3) (cid:2) (cid:4)(cid:3) (cid:1) (cid:3) (cid:5) (cid:4) (cid:2) (cid:4)(cid:3) (cid:2) Figure 1: The left picture explains entanglement entropy and mutual information can bedefined on any space-like time slices in relativistic field theories. A constant time slice isdescribed as Σ , while generic one (deformed one) by Σ . We can define the entanglemententropy S A , S A , S B and S B and their mutual informations. The right picture describesa special setup with a Lorentz boosted interval and an unboosted interval. The dottedcurve Σ represents the time slice in this setup. Entanglement entropy provides us quite a lot of information on quantum states in quan-tum many-body systems and quantum field theories [1, 2, 3]. To define the entanglemententropy, we first decompose the total Hilbert space H tot into two subsystems H A and H B such that H tot = H A ⊗ H B . Then we trace out the subsystem B and define thereduced density matric ρ A for the subsystem A . The von-Neumann entropy of ρ A is theentanglement entropy.In quantum field theories, a quantum state or equally a wave functional is defined ona time slice. Therefore, in order to define entanglement entropy in quantum field theorieswe need to specify the time slice which defines the quantum state. If we divide the timeslice into the region A and B , then we have the decomposition of the Hilbert space as H tot = H A ⊗ H B . In most of all examples studied so far, we choose the simplest time slice(canonical time slice) defined such that the canonical time t takes a fixed value. However,in Lorentz invariant field theories, we can choose any time slice as long as it is space-like todefine a quantum state. Therefore we can consider entanglement entropy on such generictime slices as in the left picture of Fig.1. For example, this plays a very important rolein the entropic proof of c-theorem [4] and F-theorem [5].The entanglement entropy and mutual information on generic time slices provide uswith more general information of quantum field theories e.g. properties of ground statesand their reduced density matrices. For instance, in two dimensional conformal field the-ories (2d CFTs), the mutual information for two intervals on a generic time slice [5, 6, 7]1ives us the information of four point functions for generic values of cross ratios. To studythese generalized quantities is the main purpose of this paper. If we consider 2d CFTs, forexample, then the simplest entanglement entropy S A is the one for which A is an interval.If we assume this interval is on a generic time slice, we can always relate it to an intervalon a constant time slice by a Lorentz boost and thus the entanglement entropy S A isessentially reduced to the result for a constant time slice. To get a non-trivial result weconsider a mutual information I ( A, B ) = S A + S B − S A ∪ B , where A is a boosted interval,while B is not. This is depicted in the right picture of Fig.1. The mutual informationfor such generalized setups have been studied in two dimensional field theories: refer to[6] for free field theories and to [7] for orbifold theories (based on the computations in[8]). One of the main aim of this paper is to study this quantity by using the holographicentanglement entropy [9, 10, 11]. In two dimensional CFTs, the special feature of holo-graphic entanglement entropy and mutual information have been well understood fromfield theoretic computations [12, 13, 14, 15]. Our direct calculation using the holographicentanglement entropy allows us to obtain results in higher dimensional CFTs.On the other hand, in non-relativistic field theories, it is a highly non-trivial questionwhether we can specify a quantum state by choosing a generic time slice. Thereforethe second aim of this paper is to study this question by calculating the holographicentanglement entropy for gravity duals of non-relativistic scale invariant field theories (orso salled Lifshitz-like fixed points) [16] and its modification called hyper scaling violatinggeometries [17, 18, 19]. As we will see later, if we take a generic time slice, we can definethe entanglement entropy S A only when the size of the subsystem A is sufficiently large.This shows that the real space factorization of Hilbert space is not always possible on thegeneric time slices in non-relativistic field theories.This paper is organized as follows. In section two, we analyze the mutual informationfor boosted subsystems in AdS / CFT at zero and finite temperature. In section three, westudy the mutual information for boosted subsystems in higher dimensional AdS/CFTsetups, including a finite temperature case. In section four, we study the holographicentanglement entropy for gravity duals of non-relativistic scale invariant theories whenwe boost the subsystem. We also analyze the same problem for hyper scaling violatinggeometries. In section five, we summarize our conclusions. / CFT Case
Consider a two dimensional holographic CFT on R , , whose coordinate is defined by( x, t ). We define the subsystem A and B by two intervals whose end points are P A,B and Q A,B . In particular we choose the points in R , as follows (refer to the right picture ofFig.1): P A = (0 , , Q A = ( x, t ) ,P B = ( b, , Q B = ( b + r, . (2.1)2e are interested in the limit where the interval A (= [ P A , Q A ]) and the interval [ Q A , P B ]are both null. Therefore we parameterize x + = t + x = 2 t + ǫ , x − = x − t = ǫ , b = x + t + ǫ , (2.2)and consider the limit ǫ → ǫ → I ( A, B ) between A and B is defined as I ( A, B ) = S A + S B − S A ∪ B , (2.3)where the UV cut offs cancel out.The CFT on R , is dual to the gravity on Poincare AdS ds = R (cid:18) dz − dt + dx z (cid:19) . (2.4)The holographic EE [9] in AdS / CFT reads S A = L A G N , (2.5)where L A is the geodesic length which connects the two end points of A at the AdSboundary. The central charge c of the 2d CFT is related to the AdS radius [20] c = 3 R G N . (2.6) By applying the AdS / CFT , the holographic entanglement entropy S A is given by S A = c (cid:20) x − t δ (cid:21) , (2.7)where δ is the UV cut off or lattice constant. If we set t = 0, this is reduced to thewell-known formula in [21, 22]. Note that S A is well-defined only | x | > | t | i.e. P A and Q A are space-like separated.In the same way, we can calculate the mutual information I ( A, B ) = S A + S B − S A ∪ B ,assuming I ( A, B ) ≥ I ( A, B ) = c " r √ x + x − ( b + r ) p ( b − x + )( b − x − ) ≃ c (cid:20) r √ ǫ (2 t + r ) √ ǫ (cid:21) . (2.8)If the above expression gets negative we should interpret it as I ( A, B ) = 0, which is dueto the phase transition phenomena in holographic CFTs [12, 13, 14] (refer to Fig.2).3igure 2: The sketch of the transition in the holographic computation of S A ∪ B . When theinvariant length of subsystem A gets smaller, the extremal surface for the computation ofholographic entanglement entropy changes into the disconnected ones (the right picture)and the mutual information becomes vanishing. A similar phase transition occurs whenthe invariant length between Q A and P B changes.This is comparable to the result in two dimensional rational CFTs (RCFTs) [7] I ( A, B ) = c tr (2 t + r ) ǫ − log d tot , (2.9)where d tot is the total quantum dimension. In the holographic CFT, we expect d tot = ∞ and this leads to the different result.The holographic result (2.8) shows that when Q A and P B get close to the light likeseparation, the mutual information gets divergent as I ( A, B ) ∼ − ( c/
6) log ǫ . The samebehavior can also be seen in the RCFT result (2.9). On the other hand, the invariantlength of subsystem A gets smaller (i.e. ǫ gets smaller), the mutual information goes tozero. When ǫ and ǫ are the same order, the holographic result (2.8) behaves differentlythan that for the RCFT (2.9). Let us perform the same analysis for the BTZ black hole dual to a finite temperatureCFT: ds = R z (cid:18) − f ( z ) dt + dz f ( z ) + dx (cid:19) , (2.10) f ( z ) = 1 − z z H , (2.11)where R is the radius of AdS spacetime and z H is a positive parameter related to theinverse temperature via β = 2 πz H . The space coordinate x is assumed to be non-compact.4igure 3: We plotted S A (2.12) as a function of x and t , which are the space and time-likewidth of the interval A for the BTZ black hole d = 2. We subtracted the holographicentanglement entropy for a disconnected geodesic from the one for connected one: ∆ S A = S A − S (dis) A to remove the UV divergence. Note that the interval A has to be space-likeand therefore we need to require x > t . We set the parameters z H = R = G N = 1.First, consider a general situation and set the end points as P A = (0 ,
0) and Q A = ( x, t ).The calculation for HEE can be accomplished by noting that the BTZ black hole isobtained from a quotient of pure AdS as in [10]: S A = c (cid:20) β π δ sinh (cid:18) πβ ( x + t ) (cid:19) sinh (cid:18) πβ ( x − t ) (cid:19)(cid:21) . (2.12)Again S A is well-defined if P A and Q A are space-like separated. This is plotted in Fig.3.Now it is straightforward to compute the mutual information between arbitrary sub-regions. In particular we focus on the previous choice (2.1) and then we obtain I ( A, B ) = c sinh (cid:16) πrβ (cid:17) sinh (cid:16) π ( b + r ) β (cid:17) + c sinh (cid:16) πβ ( x + t ) (cid:17) sinh (cid:16) πβ ( x − t ) (cid:17) sinh (cid:16) πβ ( x − b + t ) (cid:17) sinh (cid:16) πβ ( x − b − t ) (cid:17) . (2.13)As a check, when we take the zero temperature limit β → ∞ , it reproduces (2.8). Inparticular, if we use the same limit ǫ → ǫ → I ( A, B ) ≃ c sinh (cid:16) πrβ (cid:17) sinh (cid:16) πtβ (cid:17) sinh (cid:16) π ( b + r ) β (cid:17) sinh (cid:16) πbβ (cid:17) + c (cid:20) ǫ ǫ (cid:21) . (2.14)We focus on the divergence come from the light-like limit between Q A and P B . Themutual information is plotted in Fig.4. 5 (cid:1) (cid:1) (cid:1)(cid:1) (cid:2)(cid:3) (cid:1) (cid:2) (cid:4) (cid:1) (cid:3) (cid:5) (cid:2) (cid:5) (cid:3)(cid:4) Figure 4: In the left graph, we describe the setup of the two intervals parametrized by ǫ and θ , defined as ( x, t ) = ( r cos θ, r sin θ ) and b = x + t + ǫ . The dashed line denotes alight-like surface. In the right graph, we plotted the mutual information I ( A, B ) (2.13)as a function of ( ǫ , θ ) fixing r = 0 .
5. The horizontal coordinate and depth coordinate are ǫ cos θ and ǫ sin θ . Here we study the holographic entanglement entropy and mutual information for boostedsubsystems in higher dimensional AdS/CFT setups (i.e. AdS d +1 / CFT d with d ≥ d +1 / CFT d First, we consider a pure AdS d +1 whose metric is given by ds = R − dt + dz + dx + P d − i =1 dy i z , (3.1)where R is the radius of anti-de Sitter space. The subsystem A is specified by the boostedstrip defined by − ∆ x/ ≤ x ≤ ∆ x/ , − ∆ t/ ≤ t ≤ ∆ t/ , x = ∆ x ∆ t · t, − L ≤ y , y , . . . , y d − ≤ L , (3.2)where we take the limit L → ∞ . The extremal surface γ A is specified by the functions x = x ( z ) and t = t ( z ). The HEE is computed by extremizing the functional: S A = L d − R d − G N Z z ∗ δ dzz d − p x ′ ) − ( t ′ ) , (3.3)where δ is the UV cut off and L is the length of the infinite space length. The extremalsurface extends for the region δ ≤ z ≤ z ∗ , where z = z ∗ is the turning point.6ven though we can find the extremal surface directly, we can easily obtain the finalanswer by boosting the standard result of holographic entanglement entropy [9] for thecanonical time slice ∆ t = 0. If we define the width of the strip by the invariant length l ≡ p (∆ x ) − (∆ t ) , then the holographic entanglement entropy is given by the knownformula S A = L d − R d − G N " d − (cid:18) δ (cid:19) d − − k d (cid:18) l (cid:19) d − , (3.4)where we defined k d = π ( d − / d − d − Γ (cid:16) d d − (cid:17) Γ (cid:16) d − (cid:17) d − . (3.5)Then the mutual information I ( A, B ) for the setup (2.1) is I ( A, B ) = L d − R d − k d G N (cid:18) r + b (cid:19) d − − (cid:18) √ x − t (cid:19) d − + p ( x − b ) − t ! d − − (cid:18) r (cid:19) d − ≃ L d − R d − k d G N " − (cid:18) √ tǫ (cid:19) d − + (cid:18) √ tǫ (cid:19) d − − (cid:18) r (cid:19) d − + (cid:18) r + b (cid:19) d − . (3.6)This shows that when Q A and P B get close to the light-like separation, the mutual in-formation gets divergent as ∼ ( ǫ ) − ( d − / . On the other hand, as the invariant length ofsubsystem A gets smaller, the mutual information goes to zero. We can repeat the computation of I ( A, B ) for the d + 1 dimensional AdS black brane (inPoincare coordinate). This metric is given by ds = − R f ( z ) z dt + R dz z f ( z ) + R z dx + R z d − X i =1 dy i ,f ( z ) = 1 − (cid:18) zz H (cid:19) d , (3.7)where R is the radius of anti-de Sitter space and z H is the location of the horizon, relatedto the inverse temperature by β = πd z H . The extremal surface γ A is specified by thefunctions x = x ( z ) and t = t ( z ). We choose the subsystem A to be defined by (3.2). Theholographic entanglement entropy is computed by extremizing the functional: S A = L d − R d − G N Z z ∗ δ dzz d − s f ( z ) + ( x ′ ) − f ( z )( t ′ ) , (3.8)7here δ is the UV cut off. The constant z ∗ describes the turning point where t ′ and x ′ get divergent. The equations of motion for t and x read f ( z ) t ′ z d − q f ( z ) − f ( z )( t ′ ) + ( x ′ ) = 1 q ,x ′ z d − q f ( z ) − f ( z )( t ′ ) + ( x ′ ) = 1 p , (3.9)where p and q are positive integration constants. These equations can be solved as t ′ = 1 qf ( z ) r q + f ( z ) (cid:16) z d − − p (cid:17) ,x ′ = 1 p r q + f ( z ) (cid:16) z d − − p (cid:17) . (3.10)The turning point z ∗ condition allows us to eliminate the parameter q by the relation1 q = f ( z ∗ ) (cid:18) p − z d − ∗ (cid:19) . (3.11)The sizes of interval ∆ t and ∆ x are rewritten as follows:∆ t q Z z ∗ dzf ( z ) p h ( z ) , ∆ x p Z z ∗ dz p h ( z ) , (3.12)where we introduced the function h ( z ): h ( z ) = f ( z ) (cid:18) z d − − p (cid:19) − f ( z ∗ ) (cid:18) z d − ∗ − p (cid:19) . (3.13)Now S A is found as S A = R d − L d − G N Z z ∗ δ dzz d − p h ( z ) . (3.14)In order to z ∗ to be the turning point, z ∗ should be the smallest solution to h ( z ) = 0.This condition requires 1 p ≤ z − d +2 ∗ (cid:18) d − d − (cid:18) d − d (cid:19) z d ∗ (cid:19) . (3.15)Note that when the inequality is saturated, z = z ∗ is a double root. The behaviors of entanglement entropy S A and the mutual information I ( A, B ) areplotted in Fig.5. At this special value of z ∗ , ∆ x and ∆ t go to infinity due to the double zero. When z is very close to z ∗ , S A as a function of ∆ x and ∆ t , which are the spaceand time-like width of the strip A for the AdS black brane d = 3. We again subtractedthe disconnected entropy from the connected one ∆ S A = S A − S (dis) A to remove the UVdivergence. Note that the strip A has to be space-like and therefore we need to require∆ x > ∆ t . We set the parameters z H = R = G N = 1. In the right graph, we plotted themutual information I ( A, B ) with the choice of subsystems (2.1) for the AdS black braneas a function of ( ǫ , θ ) defined by ( x, t ) = ( r cos θ, r sin θ ) and b = x + t + ǫ fixing r = 0 . We are interested in S A when the subsystem A is boosted such that it is almost null. This corresponds to the following limit: p, q, z ∗ → , with q ( z ∗ ) d = finite , p ( z ∗ ) d = finite . (3.16)In particular we have q p = 1 f ( z ∗ )(1 − p z − d − ∗ ) ≃ p z − d − ∗ , (3.17)where p z − d − ∗ = O ( z ∗ ) is the next leading contribution in the limit.In this light-like limit, we can approximate f ( z ) ≃ we can find (∆ x ) / (∆ t ) ≃ q ( f ( z ∗ )) p = 1 − d − d − z d ∗ ≤
1. This means that there exist extremal surfaceswith ∆ t > ∆ x . However such an extremal surface should be discarded when we compute the holographicentanglement entropy as we require the existence of a space-like surface [23] whose boundary is the unionof the subsystem A and its extremal surface γ A . Also such a time-like configuration contradicts with theoriginal covariant computation [10]. If we take a double scaling limit by combining the zero temperature z H → ∞ limit and the infiniteboost, then we get the AdS plane wave geometry [24]. The holographic entanglement entropy wasanalyzed in [25]. Our light-like limit discussed in here is different from this. However we can confirm thesame conclusion of the singularity structure also in this double scaling limit. t ≃ q Z z ∗ z d − dz p − ( z/z ∗ ) d − = √ π Γ (cid:16) d d − (cid:17) Γ (cid:16) d − (cid:17) ( z ∗ ) d q , ∆ x ≃ p Z z ∗ z d − dz p − ( z/z ∗ ) d − = √ π Γ (cid:16) d d − (cid:17) Γ (cid:16) d − (cid:17) ( z ∗ ) d p , (3.18)where note that ∆ t and ∆ x are finite in the limit (3.16). We find the light-like property∆ t ≃ ∆ x from (3.17).Thus the invariant length of the subsystem A becomes l = √ ∆ x − ∆ t ≃ √ π Γ (cid:16) d d − (cid:17) Γ (cid:16) d − (cid:17) z ∗ . (3.19)Finally the holographic entanglement entropy is estimated as S A ≃ R d − L d − G N Z z ∗ δ dzz d − p − ( z/z ∗ ) d − = R d − L d − G N δ − ( d − d − − √ πd − (cid:16) d d − (cid:17)(cid:16) d − (cid:17) z − ( d − ∗ . (3.20)This reproduces the result in pure AdS (3.4) as expected.The mutual information for the choice (2.1) can be done in the similar way. It isnow obvious that in the light-like limit ǫ → ǫ → ∼ ( ǫ ) − ( d − / and ∼ ( ǫ ) − ( d − / ) is exactly the same as that inpure AdS (3.6), while the finite contributions change.In the same way, we expect that this singular behavior (3.6) is common to all asymp-totically AdS geometries as the only near boundary region is involved in the computationof extremal surface in the light-like limit. In Fig.6, we also numerically confirmed thisfact. The d + 1 dimensional gravity background [16], ds = R − z − ν dt + z − dz + z − dx + z − d − X i =1 dy i ! , (4.1)10 .2 0.4 0.6 0.8 1.0 1.2 ϵ Δ S A AdS BBpure AdS Figure 6: We plotted the holographic entanglement entropy S A for an interval (∆ x, ∆ t ) =( t + ǫ, t ) as a function of ǫ at t = 0 .
5. The upper curve describes the AdS black braneand the lower one does the pure AdS . We subtracted the area law divergence ∝ Lδ fromthe both entropies.is dual to a fixed point (called Lifshitz-like fixed point) with a non-relativistic scale in-variance ( t, x, ~y ) → ( λ ν t, λx, λ~y ) . (4.2)The parameter ν is called the dynamical exponent and the gravity dual is sensible onlywhen ν ≥ ν = 1 corresponds to the pure AdS d +1 . We would like to study the holographic entanglement entropy in this theory. We choosethe strip shape subsystem A which extends in ~y direction as in (3.2). Its holographicentanglement entropy is computed as the area of extremal surface. The profile of thisextremal surface is described by t = t ( z ) , x = x ( z ) . (4.3)Note that when t ( z ) = 0 (the constant time slice), the computation is the same as thatfor the Poincare AdS d +1 . Below we would like to focus on more general solutions.The holographic entanglement entropy S A is computed by extremizing the functional: S A = L d − R d − G N Z z ∗ δ dzz d − q x ′ ) − z − ν ) ( t ′ ) , (4.4)where δ is the UV cut off. The constant z ∗ describes the turning point where t ′ and x ′ get divergent.The equations of motion for t and x read z − ν ) t ′ z d − p x ′ ) − z − ν ) ( t ′ ) = 1 q ,x ′ z d − p x ′ ) − z − ν ) ( x ′ ) = 1 p , (4.5)11here p and q are integration constants.The turning point z ∗ satisfies1 + q z − ν ) − d − ∗ − q p z − ν ) ∗ = 0 . (4.6)Thus we can eliminate q by 1 q = z − ν ) ∗ (cid:18) p − z d − ∗ (cid:19) . (4.7)The sizes of interval ∆ t and ∆ x are calculated as∆ t q Z z ∗ dzz − ν q ˜ h ( z ) , ∆ x p Z z ∗ z − ν dz q ˜ h ( z ) , (4.8)where we introduced the function ˜ h ( z ):˜ h ( z ) = z − ν ) (cid:18) z d − − p (cid:19) − z − ν ) ∗ (cid:18) z d − ∗ − p (cid:19) . (4.9)Thus S A is re-expressed as S A = L d − R d − G N Z z ∗ δ dz z − d − − ν ) q ˜ h ( z ) . (4.10)As in (3.15), we also have the following condition because z ∗ should be the smallestsolution to ˜ h ( z ) = 0: 1 p ≤ d + ν − z d − ∗ ( ν − . (4.11)And since the left side of (4.7) should be positive, we can get the other bound for p . Hencewe finally get the following inequality: r ν − d + ν − ≤ pz d − ∗ ≤ . (4.12)It is obvious that there is a (non-relativistic) scale symmetry(∆ t, ∆ x, z ∗ , p ) → ( λ ν ∆ t, λ ∆ x, λz ∗ , λ d − p ) . (4.13)Therefore we focus on the ratio ∆ t (∆ x ) ν = f ν,d ( p, z ∗ ) . (4.14)12 .5 0.6 p f ( p,z * ) d = d = d = Figure 7: The plot of f ν,d ( p, z ∗ ) as a function of p at ν = 2 and z ∗ = 1 for d = 2 (blue), d = 3 (green) and d = 4 (red). Note that the domain of f ν,d ( p, z ∗ ) perfectly matches(4.12).Actually owing to the scale symmetry, this ratio only depends on the combination: z − d ∗ p .Thus we can set z ∗ = 1 without loss of generality. We plotted this function in Fig.7 when ν = 2 in various dimensions. As is clear from this plot, there is an upper bound for thisratio. ∆ t (∆ x ) ν ≤ f maxν,d . (4.15)For example, we find f maxν =2 ,d =2 = 0 . , f maxν =2 ,d =3 = 0 . , f maxν =2 ,d =4 = 0 . . (4.16)Note also that this bound (4.15) is reduced to the space-like condition ∆ t ∆ x < ν = 1 (i.e. the pure AdS space). d = ν = 2 In the special case: d = ν = 2, we can analytically find the geodesic [27]. First we cansolve z ∗ as z ∗ = q p − s(cid:18) q p (cid:19) − q . (4.17)Finally we find ∆ t q p log s q + 2 p q − p − q , ∆ x q p log s q + 2 p q − p . (4.18)The holographic entanglement entropy is found as S A = R G N log p qδ p q − p ! . (4.19)13 .05 0.10 0.15 0.20 Δ t - - Δ S A Figure 8: The plot of holographic entanglement entropy S A as a function of ∆ t at ∆ x = 1.We subtracted the entanglement entropy for ∆ t = 0 from S A . We set R = 4 G N in (4.21).There are two branches which correspond to two extremal surfaces. We need to choosethe lower blanch for the computation of S A , while the upper one is unphysical.If we define η = p q , then we find∆ t (∆ x ) = log (cid:16) e − η q η − η (cid:17) h log (cid:16)q η − η (cid:17)i , (4.20)which has the upper bound mentioned before. Finally we obtain S A = R G N log (∆ x ) η δ p − η h log (cid:16)q η − η (cid:17)i . (4.21)The special value η = 0 corresponds to ∆ t = 0 and in this case we reproduced theknown result S A = R G N log ∆ xδ [28], which actually takes the identical form as the one in arelativistic CFT. To see the general behavior, we plotted S A in Fig.8 as a function of ∆ t when we fix ∆ x = 1. It is monotonically decreasing as ∆ t gets larger, though it is alwayspositive. Now we would like to come back to our original question: can we take generic time slicesin non-relativistic scale invariant theories as in relativistic field theories ? Our analysisshows that the holographic entanglement entropy is defined only when the condition (4.15)is satisfied. Consider a deformation of the constant time slice (canonical time slice) t = 0to a generic time slice described by t = t ( x ). If we can define a quantum state on this14ime slice, we expect that the total Hilbert space H tot can be decomposed into the directproduct of H x which corresponds to each lattice point x such that H tot = ⊗ x H x , (4.22)assuming an appropriate lattice regularization. Then we should be able to define theentanglement entropy for any intervals on this non-canonical time slice. However, bychoosing the length of the interval A enough small, we can easily find an interval A onthis non-canonical time slice which does not satisfy the bound (4.15). This shows that theentanglement entropy is not well-defined and shows an inconsistency of the real spacedecomposition of the Hilbert space (4.22). This also suggests that a quantum state in thenon-relativistic scale invariant theory may be not well described or may be at least highlynon-local on a time slice unless it is a constant time slice t =const. It might be useful torecall that extremal surfaces for holographic entanglement entropy exist only for specialsubsystems in the NS5-brane gravity backgrounds [9], where the dual theory is expectedto be non-local.Let us comment on the mutual information for Lorentz boosted subsystems. If theseparation of Q A and P B in (2.1) is almost light-like, the bound (4.15) is satisfied only if b − x ≃ t is large enough. In that case, there is no singular behavior in the light-like limitas we can also see from the plot Fig.8. This is in contrast with the relativistic results inprevious sections. Finally we would like to study a more general class of gravity duals, called the hyperscaling violating geometry. [17, 18, 19]. This describes a non-relativistic theory witha scale symmetry violation and its gravity dual is given by the following class of d + 1dimensional metric: ds = R z − a dz − z − b dt + z − dx + z − d − X i =1 dy i ! , (4.23)where a and b are given by a = 1 − θd − − θ , b = 1 + ( d − ν − d − − θ . (4.24)The parameter ν is the dynamical exponent as before and the new one θ is the hyperscaling violation exponent. Obviously the metric (4.23) does not have any scale symmetryfor θ = 0 as opposed to the pervious Lifshitz case (i.e. θ = 0). One might worry that in such cases, the holographic entanglement entropy formula itself can breakdown. Even though we do not have a solid argument which denies this possibility, it is natural to believewe can still apply the extremal area formula, remembering that it can be derived from the bulk-boundaryprinciple without using special properties of AdS spaces [29, 30]. Also note that the Bekenstein-Hawkingformula of black hole entropy can be applied to various spacetimes, not only AdS but also flat spaces. It is intriguing to note that another interesting and exotic aspect of holographic entanglemententropy, related to entanglement wedge, in the Lifshitz background has been found in [27].
15e can repeat the previous analysis of holographic entanglement entropy S A for thesame choice of strip subsystem A . Introducing positive integration constants p and q asbefore, we find ∆ t Z z ∗ dz z − a +1 q z − b +2 − q p z − b +4 + q z − d − − b +4 , (4.25)∆ x qp Z z ∗ dz z − a +1 − b +2 q z − b +2 − q p z − b +4 + q z − d − − b +4 , (4.26) S A = L d − R d − G N Z z ∗ δ dz qz − d − b − a +5 q z − b +2 − q p z − b +4 + q z − d − − b +4 . (4.27)We can eliminate q dependence by q = (cid:18) z − b ∗ q p − z d − ∗ (cid:19) − .Actually, even though the full gravity dual does not have any scale symmetry, we canfind the following scale symmetry for the integral equations (4.25) and (4.26):(∆ t, ∆ x, z ∗ , p ) → ( λ ν ∆ t, λ ∆ x, λ − θd − z ∗ , λ d − − θ p ) . (4.28)Thus again the ratio ∆ t (∆ x ) ν is universal. Combining with numerical studies, we eventuallyfind that this ratio has an upper bound as in (4.14). Therefore the conclusion of theexistence of extremal surfaces for the hyperscaling violation geometry is the same as thatfor the Lifshitz geometry. In this paper we studied computations of holographic entanglement entropy and mutualinformation when we take a non-canonical time slice (refer to the right picture in Fig.1).In the first half of this paper we studied them in CFTs at zero and finite temperature byusing the AdS/CFT. We find that the mutual information I ( A, B ) gets divergent as theseparation of one end point of A and that of B becomes light-like. From the viewpoint ofthe deformed time slice which includes both A and B (refer to the left picture in Fig.1),this light-like limit corresponds to the limit where the region A and B get causally touchedwith each other. Moreover, we confirmed that the same behavior in the light-like limitoccurs in finite temperature holographic CFTs by studying extremal surfaces in AdS blackbranes.In the latter half of this paper, we performed a similar analysis for gravity duals of non-relativistic scale invariant theories (Lifshitz geometries). Such a theory is characterizedby the dynamical exponent ν . In this case, the holographic entanglement entropy fora single interval or strip shows non-trivial behavior under a Lorentz boosting. Notethat for a pure AdS, the boosting a single subsystem is trivial as the theory is Lorentzinvariant. By studying the extremal surfaces in Lifshitz geometries, we find that the16xtremal surface exists only if the time-like width ∆ t and space-like width ∆ x of thesubsystem satisfy ∆ t (∆ x ) ν ≤ f max . Here f max is a certain O (1) constant, which depends onthe dynamical exponent ν and the dimension d . The presence of this bound for the choicesof subsystems in holographic entanglement entropy tells us that a real space decompositionof Hilbert space is not possible on generic time slices except the canonical ones. This mayalso suggest that we cannot define a quantum state in a standard way by taking anynon-canonical time slice. Moreover we found that the same result is obtained for moregeneral geometries so called the hyperscaling violation. It would be an interesting futureproblem to analyze the non-canonical time slice from the non-relativistic field theoryviewpoint. At the same time, another interesting question is computations of boostedmutual information for more generic shapes of subsystems such as round balls [31, 32],where quantum corrections in gravity will play an important role [12, 33, 34] in additionto the classical contributions [35].Finally it is worth mentioning possible applications of our analysis. Our results ofnon-relativistic scale invariant theories show that if an observer is Lorentz boosted, thenthe observer will probe some sort of non-locality, which prevents us from a real spacedecomposition of the Hilbert space. This suggest that if we do an experiment of a con-densed matter system at a non-relativistic critical point from a boosted observer, we mayencounter interesting non-local effects. This deserves a future study. Acknowledgements
We are grateful to Pawel Caputa, Naotaka Kubo, Tomonori Ugajin and Kento Watanabefor useful discussions. TT is supported by the Simons Foundation through the “It fromQubit” collaboration, JSPS Grant-in-Aid for Scientific Research (A) No.16H02182 andWorld Premier International Research Center Initiative (WPI Initiative) from the JapanMinistry of Education, Culture, Sports, Science and Technology (MEXT).
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