Dynamics of Entanglement Entropy From Einstein Equation

Transcription

1 YITP Workshop on Quantum Information Dynamics of Entanglement Entropy From Einstein Equation Tokiro Numasawa Kyoto University, Yukawa Institute for Theoretical Physics based on arxiv: (PR D 88(2013)026012) collaborate with Masahiro Nozaki(YITP), ndrea Prudenziati, Tadashi Takayanagi(YITP)

2 Main Results ds(gravity) side: Einstein equation CFT side: constraint equation for entanglement entropy

3 Motivations (1)Field theoretical motivation In a quantum field theory, excited states properties are not well studied, so we study the excited states properties in CFTs ( critical point theory of quantum many body systems.) We consider the weakly excited states. To study universal properties, we need to study the physical observables that can be defined in any theory. We study entanglement entropy for excited states.

4 (2) Gravity Motivation In the ds/cft context, Entanglement entropy Minimal surface So entanglement entropy is directly related to the bulk metric. On the other hand, excited states deformation of metric The deformed metric also satisfy the Einstein equation. From these, we can expect that there should be a counterpart of the bulk Einstein equation which constrains the behavior of entanglement entropy.

5 (3)Thermodynamics Motivation It is known that first-law like relation holds for entanglement entropy in CFTsif the excitation is small, static,and translational invariant. Is this true for the time dependent excitation?

6 First law for entanglement entropy First law like relation holds for entanglement entropy in conformal field theory if the excitation is sufficiently small and translational invariant : small B E = T ent S Energy in [Bhattacharya-Nozaki-Ugajin-Takayanagi 12] Temperature :depend only on the geometry of S is the difference between EE for excited states and ground states. For example, if the subsystem is a round ball, then T ent = 2 l d +1

7 Holographic Entanglement Entropy In the ds/cft correspondence, EE in a CFTd corresponds to the minimal surface in the bulk: S = rea( ) 4G N :minimal surface B z The minimal surface shares the boundary with the subsystem. Entanglement entropy is a nonlocal observable, and this is reflected to the fact that the minimal surface extends to the bulk. So naively, we think that we can detect the bulk using the minimal surface or entanglement entropy in the boundary CFT viewpoint.

8 How to calculate Holographic EE for excited states S = rea( ) 4G N :minimal surface B Excited states Induced metric : Minimal surface : bulk metric is changed from the ds metric G = G (0) + "G(1) + O("2 ) = (0) + " (1) + O("2 ) (0) Beause is a minimal surface, In the first order of ", S = 1 8G N (0) p G (0) G (1) (0) G We choose the subsystem to be a round ball.

9 Case of ds 3 /CFT 2 First, we consider the case the bulk theory is pure Einstein gravity: S = 1 16 G N R + 6 L 2 We expand perturbatively the metric around the ds solution in the GF coordinate: ds 2 = L 2 dz2 + g µ (z,x)dx µ dx Then, we get the EOM in the first order of " (@ 2 2 x)h(t, x) =0 z 2, g µ = µ + "h µ where h tt = h xx = z 2 H(t, t h tx = z x H(t, x h tx = z t H(t.x)

10 Using H(t, x) S (,l,t)= Ll2 32G N, we can write the variation of EE as d' cos 3 'H t, + l 2 sin ' l If we use the wave equation for H(t, x) derived from the Einstein equation, we can get the following equations: z# γ ξ ϕ (@ t 2 ) S (,l,t)=0 l i l 2 S (,l,t)=0 This is the counterpart of perturbative Einstein eq.

11 Derivation of first law from Einstein eq We consider the small subsystem limit translational invariance). l! 0 (don t assume the In this limit, HEE is written as follows: S (,l,t) ' Ll2 24G N H(t, ) On the other hand, from the formula of Holographic energy momentum tensor we can find the following relations: T CFT tt = L H(t, ) 8 G N E = dl Ttt CFT ' l T CFT tt = Ll 8 G N H(t, ) From these relations, we can get the first-law like relation: E = T ent S, T ent = 3 l

12 Case of ds 4 /CFT 3 We consider the case the bulk theory is a pure Einstein gravity. The equation for EE that is the counterpart of Einstein eq becomes as follow: i l 2 This equation contains no time derivatives. S =0 The time evolution of EE is determined by the IR boundary condition. [Bhattacharya-Takayanagi 13] boundary IR If we take the limit of l! 0, we can find the first-law like relation.

13 The meaning of the equation Roughly speaking, the differential equation is hyperbolic PDE: 2 2 ~x ) S (t, ~x, l) 0 S (t, ~x, l) f(l ~x )+g(l + ~x ) Consider the case of local excitation. S (l ~x ) S 6=0 l ~x B S =0 B The differential equation put a constraint that S is non-trivial only when intersect with the excited region!

14 Case of Einstein-Scalar theory We consider a gravity with matter(scalar field). 1 S = (R 2 )+ 1 (@ ) 2 + m G N 4 In this case, the differential equations for entanglement entropy is modified as follows: Case of ds3/cft2 (@ t 2 ) S (,l,t)=hoihoi l i l 2 S (,l,t)=hoihoi Case of ds4/cft3 @l 3 l 2 S = hoihoi O :operator dual to the bulk scalar dual! R µ 1 2 Rg µ + g µ = T µ First-law like relation also holds.

15 Conclusion We derive the equations for entanglement entropy dual to the bulk Einstein equation. We calculate the variation of entanglement entropy explicitly and confirm that the first-law like relation is satisfied if we take the limit subsystem is sufficiently small. Future problem We assume that the theory is invariant under the conformal transformation. If a theory doesn t have conformal invariance, are there relations? We linearize Einstein equations. What is the nonlinear version? The inverse of our results, or derivation of Einstein eqs from constraints for EE. lready done by Raamsdonk et.al.

16 Gravitation from Entanglement thermodynamics [Lashkari-McDerott-Raamsdonk 13] [Faulkner-Guica-Hartman-Myers-Raamsdonk 13] If the subsystem is a round ball, the first-law like relation holds also when the subsystem is not small: t H = S [Blanco-Casini-Hung-Myers 13] where H =2 R 2 ~x ~x 0 2 Ttt CFT (t 0, ~x) 2R B generator of isometry of the causal development of the round ball R is the radius of subsystem. This is the integrated version of our results. In the small size limit, we can reproduce the first-law like relation E = T ent S.

17 Einstein eq from first-law like relation From the gravitational view point, H = S means there is a relation between the two functionals of linearized metric: f E (h µ )= This is a nonlocal constraint, but the Einstein eq is a local constraint. This achieved by the following way. à f S (h µ ) t z ~x à z =0

18 We denote Einstein eq by G ab =0. We can find a (d 1) -form which satisfy the following properties: d = 2f(x)G tt vol From the first-law like relation, 0= S E = Then, from the stokes = H, = tt à d =0 à Since we can choose arbitrary ball, this equality folds for any. Then we can conclude that the integrand becomes : d =0 G tt =0 = S component of Einstein eq tt 0 component of Einstein eq à ~x z Other components can be shown the same way.