YITP-13-29

IPMU-13-0083

Dynamics of Entanglement Entropy from Einstein Equation

Masahiro Nozaki , Tokiro Numasawa , Andrea Prudenziati , and Tadashi Takayanagi

Yukawa Institute for Theoretical Physics, Kyoto University,

Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan

Kavli Institute for the Physics and Mathematics of the Universe,

University of Tokyo, Kashiwa, Chiba 277-8582, Japan

We study the dynamics of entanglement entropy for weakly excited states in conformal field theories by using the AdS/CFT. This is aimed at a first step to find a counterpart of Einstein equation in the CFT language. In particular, we point out that the entanglement entropy satisfies differential equations which directly correspond to the Einstein equation in several setups of AdS/CFT. We also define a quantity called entanglement density in higher dimensional field theories and study its dynamical property for weakly excited states in conformal field theories.

## 1 Introduction

The AdS/CFT correspondence [2, 3] is a remarkable fundamental relation which connects gravitational systems and quantum field theories as an equivalence. In spite of its recent successful developments, we are still far from the complete understanding of the basic mechanism of AdS/CFT correspondence (or gauge/gravity duality). The aim of the present paper is to report a modest progress in this direction. In particular we would like to study what the Einstein equation in the gravity side corresponds to in the quantum field theory side.

Since the AdS/CFT correspondence relates gauge invariant quantities between both sides, the Einstein equation itself, which is written in term of the spacetime metric, is not directly interpreted in the dual quantum field theory. Therefore we need to find a counterpart of Einstein equation for gauge invariant quantities. We argue that the holographic entanglement entropy (HEE) is one of the best quantities for this purpose.

The entanglement entropy in quantum field theories and more generally quantum many-body systems has been intensively studied recently (see e.g. the review articles [4, 5, 6, 7]). In AdS/CFT we can holographically calculate the entanglement entropy in a gravity dual as an area of minimal surface as conjectured in [8, 9]. This holographic entanglement entropy (HEE) calculation was proved in [10] quite recently by using the bulk to boundary relation [3]. See [11] for a proof when the subsystem is a round ball. Also refer to [12, 13, 14] for strong supports towards a proof within AdSCFT. This holographic calculation of entanglement entropy was also developed in time-dependent backgrounds [15], which has been applied to the quantum quenches [16, 17, 18], the de-Sitter space [21] and energy flow [22]. In [17], a falling particle in AdS was considered as the holographic dual of local quenches [20] and its HEE has been computed. In [18], an analytical framework for holographic counterpart of global quantum quenches and their entanglement entropies in CFTs [19] has been discovered. In this paper we will study how a small perturbation of HEE evolves dynamically by solving the Einstein equation in AdS spaces. A final goal will be to rewrite the perturbative Einstein equation in terms of the HEE and we will do this explicitly in several examples of AdSCFT. At the same time, our results describe the behavior of entanglement entropy for weakly excited states.

Note that the complete information of HEE for arbitrary subsystems is essentially the same as that of the spacetime metric in the gravity dual (see e.g. [24]). For this correspondence we do not need to know the details of the Lagrangian of matter fields coupled to the Einstein gravity. On the other hand, if we want to reproduce the spacetime metric only from the information of holographic energy stress tensor [23] we need to employ the precise form of Einstein equation and thus this requires the details of matter fields.

When the size of subsystem is small, we can find a simple relation between its entanglement entropy and the total energy inside it for excited states. This is called the first law-like relation and has been first obtained in [26] when excited states are static and translationally invariant. Our analysis in this paper provides a proof of this first law relation for spherical subsystems in the presence of time-dependent excitations.

We will also study the dynamics of entanglement density introduced in [17] for two dimensional field theories. We will extend this quantity in higher dimensions. Then we consider evolutions of entanglement density imposed by the Einstein equation.

This paper is organized as follows: In section two we will present a general strategy of conducting our perturbative analysis of HEE and review the first law-like relation. In section three, four and five, we present our analysis of HEE in AdSCFT, AdSCFT and AdSCFT, respectively. In section six, we study the higher dimensional entanglement density and its perturbations. In section seven, we summarize our conclusions.

## 2 Perturbative Calculation of Holographic Entanglement Entropy

### 2.1 General Strategy

We consider a perturbation of the pure AdS metric in the Fefferman-Graham (FG) gauge as follows:

(2.1) |

The coordinate () describes the dimensional Lorentzian space where the dimensional conformal field theory (CFT) lives in. The parameter describes the radius of the AdS space.

We consider the perturbative expansion of the form:

(2.2) |

assuming that is very small. We are interested in only the linear order of . The dynamics of this perturbation is determined by the Einstein equation as usual

(2.3) |

where is the energy stress tensor for the matter coupled to the Einstein gravity ( are indices of coordinates of the dimensional space).

The holographic entanglement entropy (HEE) in a dimensional AdS gravity is given in terms of the area of dimensional extremal surface in a given spacetime so that the boundary of coincides with that of the subsystem [8, 15]. In this paper we are interested in the first order correction of HEE under a metric perturbation. This can be conveniently computed as follows. First we start with the extremal surface whose shape is already known in the pure AdS and calculate its area. Because the pure AdS spacetime is static, is a minimal area surface on a canonical time slice. Next we evaluate the area of the same surface in the perturbed metric. The differen between these two gives precisely the first order correction of HEE . Thus we do not need to know how the shape of the extremal surface is modified under the metric perturbation. This simply stems from the fact that satisfies the extremal surface condition in the pure AdS space.

Therefore we can calculate the shift of HEE due to the metric perturbation (2.2) from the formula

(2.4) |

where is a coordinate of the dimensional extremal surface as employed in [17]. and represents the induced metric on with respect to the pure AdS and its first order perturbation. In this paper we consider examples where the subsystem is given by a round ball for which we know the analytical expression of the surface in the pure AdS.

### 2.2 First Law-like Relation

A useful property which is enjoyed by the perturbed HEE is the first law-like relation [26]. It takes the following form:

(2.5) |

When we choose to be a round ball with radius , the entangling temperature takes the universal value

(2.6) |

Also denotes the total energy in the region and is written as

(2.7) |

where is the energy density in CFT. If we perform the expansion of defined in (2.2) as

(2.8) |

in the near AdS boundary limit , the holography energy stress tensor can be obtained as follows [23]

(2.9) |

This relation (2.5) was originally confirmed in static, isotropic and translationally invariant perturbations of the metric. Therefore it is intriguing to see if it holds without this assumptions. This is another motivation of this paper.

## 3 Analysis of Perturbed HEE in AdSCft

To study the HEE in AdSCFT, we set in (2.1). We choose the subsystem A for the entanglement entropy to be an interval . Then the corresponding minimal surface is parameterized as

(3.1) |

Then the shift of HEE (2.4) reads in terms of the perturbation of the metric :

(3.2) |

where we employed the FG gauge in the final expression.

### 3.1 HEE in AdS Pure Gravity

As the simplest example in AdSCFT, we would like to calculate in the pure Einstein gravity for AdS. The equations of motion for the metric perturbation reads

(3.3) |

Note that this system is topological in that there are no propagating degrees of freedom as usual in three dimensional pure gravity.

By requiring that is order so that only normalizable modes are excited, these equations can be solved as follows:

(3.4) |

where the function satisfies

(3.5) |

Finally, by using this solution, (3.2) is rewritten as follows:

(3.6) |

where we employed the relation by setting in (2.9).

By taking the Fourier transformation

(3.7) |

we obtain

(3.8) |

In this way, we find that is related to the metric via this non-local transformation. Moreover, it is straightforward to see that it satisfies

(3.9) |

In summary, we find from (3.5) and (3.9) that satisfies the following “equations of motion for entanglement entropy”:

(3.10) | |||

(3.11) |

The first equation (3.10) shows that the quantum entanglement propagates at the speed of light in the direction. The second one (3.11) describes an evolution in the width (or radial) direction, which is analogous to the wave equation in an AdS spacetime. We believe that the presence of two constraint equations for as in (3.10) and (3.11) is peculiar to the AdSCFT duality and this is due to the fact that the gravity does not have propagating degrees in three dimension.

### 3.2 HEE from Einstein-Scalar Theory

Since the pure AdS does not have any propagating degrees of freedom, it is more interesting to consider a Einstein-matter theory on AdS. Though we will not write down all components of Einstein equation, we would like to note that the component reads

(3.12) |

where is the energy momentum tensor in the dimensional gravity. This should be distinguished from the holographic energy tensor for the dimensional dual CFT. By integrating this equation, the shift of holographic entanglement entropy (3.2) is expressed as follows:

(3.13) |

In particular, we consider the Einstein-scalar theory which is defined by a free scalar field (mass ), which is minimally coupled to the Einstein gravity. The Einstein equation is given by (2.3) with the energy stress tensor is given by

(3.14) |

where we normalize the scalar field appropriately.

We consider a perturbation of the scalar and AdS metric in the FG gauge as follows:

(3.15) |

where is an infinitesimally small parameter of the perturbation.

The equation of motion of this perturbation is given by

(3.16) |

After the Fourier transformation, the normalizable solution for this equation is given by

(3.17) |

where denotes the expectation value of an operator dual to the scalar field (up to a normalization factor) [3, 25].

The entanglement entropy is decomposed into two parts:

(3.18) |

The is the same as that in the pure AdS case and is given by (3.6). The other term comes from the contribution from the matter field and is expressed as

(3.19) |

is given by

(3.20) |

where we defined .

We can also act a differential operator to get rid of the contribution and obtain the following constraint equations which are satisfied by :

(3.22) |

(3.23) |

where we did not write explicitly and as they can be easily obtained from (3.19).

In summary, we have obtained a counterpart of the perturbative Einstein equation in terms of entanglement entropy. Therefore we can regard (3.22) and (3.23) as the perturbative equations of motion for the entanglement entropy. Once we specify the expectation value of energy density and the scalar operator as functions of and , then the differential equations (3.22) and (3.23) determine the time evolution of .

They share a similar structure with that of the Einstein equation because the left hand side comes from the gravitational physics ( entanglement) and the right hand side corresponds to the matter fields ( operator expectation values). Note also that they are manifestly gauge invariant (as far as we fix the AdS boundary coordinate), as opposed to the Einstein equation where the metric changes under the coordinate transformation.

One may notice that we only used the Einstein equation involving the space-like component . In order to take into account the one for the time-like component we need to consider the entanglement entropy for boosted subsystems. We will leave the details of these for a future problem.

### 3.3 Proof of First Law-like Relation

When is very small, we can neglect the contributions from the matter fields . Therefore in this limit, we find

(3.24) |

We can show this relation (3.24) confirms the time-dependent version of the first law-like relation (2.5) of the entanglement entropy for AdSCFT. Indeed, since by definition we find we can easily reproduce (2.5) from
(3.24)^{1}^{1}1note that due to our conventions for the case we should replace in (2.6) ..

## 4 Analysis of Perturbed HEE in AdSCft

Now we move on to higher dimensional cases and here we especially consider a AdSCFT example. Since the pure Einstein gravity is already dynamical in four or higher dimension, we will concentrate on the pure gravity just for simplicity.

### 4.1 Einstein Equation

We consider a metric perturbation of the AdS space. The metric is again given by (2.1) and (2.2) with . We require that is order in the limit so that we can keep only normalizable deformations.

By performing the Fourier transformation

(4.1) |

the perturbative Einstein equation leads to

(4.2) |

for all components of perturbations . This is easily solved as

(4.3) |

Note that the explicit form of the Bessel function reads

(4.4) |

We can show the following behaviour near the AdS boundary :

(4.5) |

Since we are interested in non-singular and normalizable solutions, we restrict to the range .

### 4.2 Calculations of HEE

We consider the shift of HEE by choosing the subsystem to be a disk with a radius . In the pure AdS, the HEE is computed as the area of the minimal surface given by the half of sphere parameterized by

(4.8) |

with the range and .

The shifted amount of the HEE, denoted as due to the linear perturbation of the metric is found by using (2.4):

We take the Fourier transformation of with respect to , which is denoted by :

(4.10) |

Then we find

(4.11) |

where we defined

(4.12) |

We can perform integral by using the formula of Bessel functions

(4.13) |

as follows:

(4.14) |

Here the angle was introduced such that i.e. and ; we employed (4.6) to get the final equation.

Thus we can express in term of the (holographic) energy stress tensor as follows

(4.15) |

where is defined by

(4.16) |

### 4.3 First law-like Relation and Translationally Invariant Limit

By taking the limit in (4.2), we obtain

(4.17) |

where we employed the relation (4.7). On the other hand the total energy in reads

(4.18) |

by using (2.9). Therefore we can confirm the first law-like relation (2.5) with . In other words the energy density and are related to each other via

(4.19) |

Moreover, it is also intriguing to note that we can obtain the same result (4.19) when we take the translationally invariant limit even if we keep and finite. This fact can be shown by explicitly evaluating as follows:

(4.20) |

### 4.4 Metric Shift in CFTs

In the main part of this paper we have considered only normalizable perturbations which
correspond to the dynamical change of state in a given CFT. However, we would like to
study non-normalizable modes only in this subsection. Indeed, when we consider the time-dependent background corresponding to quantum quenches induced by a sudden change of the metric, we need to take into account the non-normalizable modes^{2}^{2}2If we consider quantum quenches induced by excitations of matter fields
such as scalar fields, we will have milder metric backreactions such as the Vaidya
metric. Refer to e.g.[27] for recent developments of holographic
quantum quenches..

For this, as a solution to the Einstein equation, we assume

(4.21) |

where the Hankel function is defined by

(4.22) |

Near the AdS boundary , it approaches

(4.23) |

Then the HEE behaves like

(4.24) |

where is the UV cut off and is related to the lattice spacing via .

The divergent part of the HEE is simply obtained as

(4.25) |

where is the UV cut off (lattice spacing). This agrees with the expectation from the area law.

The more non-trivial contribution is the finite term, evaluated as follows:

(4.26) |

## 5 Analysis of Perturbed HEE in AdSCft

Here we analyze the perturbed HEE in the AdSCFT setup. Since the calculations are parallel with the previous section, our presentation will be brief.

### 5.1 Solutions to Perturbative Einstein Equation

The Einstein equations for the Fourier transformed metric perturbations

are equivalent to

(5.1) |

together with