YITP Workshop on Quantum Information Physics@YITP Dynamics of Entanglement Entropy From Einstein Equation Tokiro Numasawa Kyoto University, Yukawa Institute for Theoretical Physics based on arXiv:1304.7100 (PR D 88(2013)026012) collaborate with Masahiro Nozaki(YITP), Andrea Prudenziati, Tadashi Takayanagi(YITP) 14 年 8 月 6 日水曜日
Main Results AdS(gravity) side: Einstein equation CFT side: constraint equation for entanglement entropy 14 年 8 月 6 日水曜日
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. 14 年 8 月 6 日水曜日
(2) Gravity Motivation In the AdS/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. 14 年 8 月 6 日水曜日
(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? 14 年 8 月 6 日水曜日
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 : A small B ∆ E A = T ent ∆ S A [Bhattacharya-Nozaki-Ugajin-Takayanagi 12] Energy in A “Temperature”:depend only on the geometry of A is the difference between EE for excited states and ground states. ∆ S A For example, if the subsystem is a round ball, then 2 π l T ent = d + 1 14 年 8 月 6 日水曜日
Holographic Entanglement Entropy In the AdS/CFT correspondence , EE in a CFT d corresponds to the minimal surface in the bulk: B � S A = Area ( γ A ) A � γ A 4 G N γ A :minimal surface z The minimal surface shares the boundary with the subsystem . A 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 . 14 年 8 月 6 日水曜日
How to calculate Holographic EE for excited states B � S A = Area ( γ A ) A � γ A 4 G N γ A :minimal surface Excited states bulk metric is changed from the AdS metric G αβ = G (0) αβ + ε G (1) Induced metric : αβ + O ( ε 2 ) γ A = γ (0) A + εγ (1) Minimal surface : A + O ( ε 2 ) γ (0) Beause is a minimal surface, In the first order of , ε A Z 1 p G (0) G (1) αβ G αβ (0) ∆ S A = 8 G N γ (0) A We choose the subsystem A to be a round ball . 14 年 8 月 6 日水曜日
Case of AdS 3 /CFT 2 First, we consider the case the bulk theory is pure Einstein gravity: Z ⇣ 1 R + 6 ⌘ S = L 2 16 π G N We expand perturbatively the metric around the AdS solution in the GF coordinate: ds 2 = L 2 dz 2 + g µ ν ( z, x ) dx µ dx ν , g µ ν = η µ ν + ε h µ ν z 2 Then, we get the EOM in the first order of ε ( ∂ 2 t − ∂ 2 x ) H ( t, x ) = 0 where h tt = h xx = z 2 H ( t, x ) , ∂ t h tx = z 2 ∂ x H ( t, x ) , ∂ x h tx = z 2 ∂ t H ( t.x ) 14 年 8 月 6 日水曜日
Using , we can write the variation of EE as H ( t, x ) z# Ll 2 γ A Z t, ξ + l ⇣ ⌘ d ϕ cos 3 ϕ H ∆ S A ( ξ , l, t ) = 2 sin ϕ ϕ 32 G N A � l ξ If we use the wave equation for derived from the Einstein H ( t, x ) equation , we can get the following equations: ( ∂ 2 t − ∂ 2 ξ ) ∆ S A ( ξ , l, t ) = 0 l − 1 ξ − 2 h i ∂ 2 4 ∂ 2 ∆ S A ( ξ , l, t ) = 0 l 2 This is the counterpart of perturbative Einstein eq. 14 年 8 月 6 日水曜日
Derivation of first law from Einstein eq We consider the small subsystem limit (don’t assume the l → 0 translational invariance). In this limit , HEE is written as follows: Ll 2 ∆ S A ( ξ , l, t ) ' H ( t, ξ ) 24 G N On the other hand, from the formula of Holographic energy momentum tensor we can find the following relations: L T CFT = H ( t, ξ ) tt 8 π G N Ll Z dl T CFT ' l · T CFT ∆ E A = = H ( t, ξ ) tt tt 8 π G N From these relations , we can get the first-law like relation: T ent = 3 ∆ E A = T ent ∆ S A , π l 14 年 8 月 6 日水曜日
Case of AdS 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: h ∂ 2 l 2 − ∂ 2 ∂ x 2 − ∂ 2 ∂ ∂ l 2 − 1 ∂ l − 3 i ∆ S A = 0 ∂ y 2 l [Bhattacharya-Takayanagi 13] This equation contains no time derivatives. The time evolution of EE is determined by the IR boundary condition. boundary IR If we take the limit of , we can find the first-law like relation. l → 0 14 年 8 月 6 日水曜日
The meaning of the equation Roughly speaking , the differential equation is hyperbolic PDE: ( @ 2 l − @ 2 x ) ∆ S A ( t, ~ x, l ) ≈ 0 ~ ∆ S A ( t, ~ x, l ) ≈ f ( l − | ~ x | ) + g ( l + | ~ x | ) Consider the case of local excitation. ∆ S A ≈ � ( l − | ~ x | ) ∆ S A = 0 B B ∆ S A 6 = 0 A A l ~ x The differential equation put a constraint that is non-trivial ∆ S A only when the intersect with the excited region ! ∂ A 14 年 8 月 6 日水曜日
Case of Einstein-Scalar theory We consider a gravity with matter(scalar field). 1 ( R − 2 Λ ) + 1 Z Z � ( ∂φ ) 2 + m 2 φ 2 � S = 16 π G N 4 In this case , the differential equations for entanglement entropy is modified as follows: ・ Case of AdS 3 /CFT 2 O :operator dual to the bulk scalar ( ∂ 2 t � ∂ 2 ξ ) ∆ S A ( ξ , l, t ) = h O i h O i l � 1 4 ∂ ξ � 2 h i ∂ 2 ∆ S A ( ξ , l, t ) = h O i h O i l 2 dual ! R µ ν − 1 2 Rg µ ν + Λ g µ ν = T µ ν ・ Case of AdS 4 /CFT 3 h ∂ 2 l 2 � ∂ 2 ∂ l 2 � ∂ ∂ ∂ l � 3 i ∂ x 2 � ∆ S A = h O i h O i ∂ y 2 First-law like relation also holds. 14 年 8 月 6 日水曜日
・ ・ 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. Already done by Raamsdonk et.al . 14 年 8 月 6 日水曜日
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 A = ∆ S A [Blanco-Casini-Hung-Myers 13] where A � B � R 2 − | ~ x 0 | 2 x − ~ Z T CFT ∆ H A = 2 ⇡ ( t 0 , ~ x ) tt 2 R A generator of isometry of the causal development of the round ball A is the radius of subsystem . A R This is the integrated version of our results. In the small size limit , we can reproduce the first-law like relation . ∆ E A = T ent ∆ S A 14 年 8 月 6 日水曜日
Einstein eq from first-law like relation From the gravitational view point, means there is a ∆ H A = ∆ S A relation between the two functionals of linearized metric: Z Z f E ( h µ ν ) = f S ( h µ ν ) ˜ A A This is a nonlocal constraint, but the Einstein eq is a local constraint. This achieved by the following way. t z ˜ A ~ x Σ A z = 0 14 年 8 月 6 日水曜日
We denote Einstein eq by . G ab = 0 We can find a -form which satisfy the following properties: ( d − 1) χ Z Z χ = ∆ H A , χ = ∆ S A ˜ A A A ˜ Σ A d χ = − 2 f ( x ) G tt vol Σ ___ component of Einstein eq tt ~ x From the first-law like relation, Z Z Z 0 = ∆ S A − ∆ E A = z χ = χ − χ ˜ A A ∂ Σ Then , from the stokes’ theorem, Z Z d χ = 0 χ = ∂ Σ Σ Since we can choose arbitrary ball, this equality folds for any . Σ A Then we can conclude that the integrand becomes : 0 d χ = 0 G tt = 0 component of Einstein eq tt Other components can be shown the same way. 14 年 8 月 6 日水曜日
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