Contents

YITP-13-14

IPMU13-0045

Holographic Local Quenches and Entanglement Density

Masahiro Nozaki , Tokiro Numasawa , 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 propose a free falling particle in an AdS space as a holographic model of local quench. Local quenches are triggered by local excitations in a given quantum system. We calculate the time-evolution of holographic entanglement entropy. We confirm a logarithmic time-evolution, which is known to be typical in two dimensional local quenches. To study the structure of quantum entanglement in general quantum systems, we introduce a new quantity which we call entanglement density and apply this analysis to quantum quenches. We show that this quantity is directly related to the energy density in a small size limit. Moreover, we find a simple relationship between the amount of quantum information possessed by a massive object and its total energy based on the AdS/CFT.

1 Introduction

Quantum entanglement provides us with a powerful method of investigating various quantum states and classifying their quantum structures. Moreover, quantum entanglement is a useful tool when we would like to study excited quantum systems which are far from thermal equilibrium, for example, those under thermalization processes. Even though in such systems we cannot define the thermodynamical entropy and temperature etc, we can always define the entanglement entropy (refer to e.g. the reviews [2, 3, 4, 5, 6]).

Quantum quenches provide us with ideal setups to study thermalizations of quantum systems, which can be realized even in real experiments such as cold atoms. They are unitary evolutions of pure states triggered by sudden change of parameters such as mass gaps or coupling constants [7, 8, 9]. By using the AdS/CFT [10, 11, 12], we can relate this problem to the dynamics of gravitational theories [13, 14, 15, 16].

One typical class of quantum quenches is called global quenches and they occur from homogeneous changes of parameters [7]. Holographic duals of global quenches are dual to black hole formations as have been discussed in [13, 15]. A useful quantity to understand how thermalizations occur is the entanglement entropy [7]. The holographic entanglement entropy (HEE) [17, 18, 6] has been calculated for various quantum quenches [14, 19, 20]. Refer also to [21] for a computation of HEE for a stationary system which is described by an excited pure state dual to an AdS plane wave. In general, non-local probes such as the entanglement entropy are useful to measure the thermalization time. On the other hand, local quantities such as an expectation value of energy momentum tensor shows an immediate thermalization and is not suitable to see if a given system is completely thermalized [19]. In this sense, the entanglement entropy can serve as a non-equilibrium substitute of thermodynamical entropy.

Another class of quantum quenches is the local quench and this is triggered by a shift of parameters within a localized region or simply at a point. One of the aims of the present paper is to provide a simple construction of holographic dual for local quench and is to calculate the holographic entanglement entropy. A local quench shows how localized excitations in a given quantum system propagate to other spatial regions. Local quenches have been studied in two dimensional CFTs [8, 9]. However, local quenches in higher dimensions have not been understood well. This partially motivates us to study the local quenches in AdS/CFT, which often allows us higher dimensional calculations.

We will argue that a simple holographic description of a system just after the local quench is a free falling particle-like object in an AdS space. It is pulled into the horizon of AdS space due to the gravitational force and this is the reason why we observe the non-trivial time-dependence of entanglement entropy. Therefore this problem is deeply connected to a fundamental question: what is the non-gravitational (or CFT) counterpart of gravitational force via holography ? We will suggest an intuitive answer to this question in the end.

The time evolution of quantum entanglement under local quenches is more complicated than the global quenches because it is inhomogeneous. To understand its structure clearly we introduce a new quantity which we call entanglement density. It is defined by taking the derivatives of the entanglement entropy with respect to the positions of two boundary points of the subsystem . This quantity counts the number of entangled pairs at a given position. The strong subadditivity guarantees that this quantity is always positive. As we will see this analysis reveals the detailed structure of quantum entanglement under local quenches as well as global quenches.

One more motivation to study local quenches is to estimate the amount of quantum information possessed by a massive object or radiations. We will employ the entanglement entropy for local quenches in order to measure the amount of information included in a localized excited lump. We will evaluate this quantity by using the holographic entanglement entropy (HEE) and obtain the simple conclusion that it is given by the total energy of the object times its size up to a numerical factor.

The paper is organized as follows: In section two, we explain our holographic setup of local quench using a free falling particle in AdS. We calculate the holographic energy stress tensors in this model. In section three, we compute the holographic entanglement entropy for assuming that the back-reaction due to the falling particle is very small. In section four, we perform an exact analysis of holographic entanglement entropy for . In section 5, we introduce a new quantity which we call entanglement density and we investigate the evolution of quantum entanglement structures under local quenches by using this. In section 6, we study the relation between the amount of information of an object and its total energy using the results in the previous sections. We also interpret our results of local quenches using the idea of the entanglement renormalization and discuss the holographic interpretation of gravitational force. In section 7 we summarize our conclusions and discuss future problems. In appendix A, we show an explicit perturbative calculation of back-reactions due to the falling particle.

2 Holographic Local Quenches as Falling Particles

Local quenches in quantum systems are triggered by a sudden local change of the Hamiltonian at a specific time. One typical class of examples will be joining two separated semi-infinite systems at each endpoint as studied in [8, 9] (see the upper picture in Fig.1). When this quench process happens, an interaction between two endpoints is instantaneously introduced. From the viewpoint of the new Hamiltonian, an locally excited state is generated just after this local quench. Therefore we can generally characterize a local quench by local excitations. These excitations will propagate to other regions under the time-evolution.

Now we would like to construct gravity duals for local quenches via the AdS/CFT. Even though we have not found a simple gravity dual of the original model i.e. joining two CFTs, it is not difficult to find a holographic model for local excitations. It is given by a falling massive particle in a Poincare AdS space (see Fig.2). At , the particle is situated near the AdS boundary and its back-reaction to the metric is highly localized near the particle. Under its time evolution it falls into the AdS horizon and its back-reaction spreads out. In the dual CFT, at , the excitations are concentrated in a small localized region (see the lower picture in Fig.1), whose radius is defined to be . Therefore we can regard this state at as the one just after the local quench. Later the excitations expand at the speed of light as we will see e.g. from its holographic energy stress tensor later. In this way, we can regard this setup as a gravity dual of local quench.

We will employ the beautiful construction of back-reacted solutions found by Horowitz and Itzhaki in [22]. The basic idea is to start from a black hole in a global AdS space and map it into a Poincare AdS by the coordinate transformation. This leads to a falling black hole solution. Though we are thinking of a massive particle with a finite size or equally a star, instead of a black hole, the asymptotic solution which is outside of the star is the same as that for a black hole as usual.

Figure 1: Setups of local quenches. The upper picture describes a process of jointing two systems which are defined on semi-infinite lines. The lower one describes localized excitations on an infinitely extended system. We define the parameter which measures the size of excited region at the beginning of the local quench.
Figure 2: A falling massive particle in AdS and the calculation of holographic entanglement entropy for two different choices of the subsystem . It is clear from this picture that the back reaction due to the falling particle gets significant when in the left picture and in the right one because the particle is on top of .

2.1 A Falling Massive Particle in AdS

Consider a dimensional AdS space (AdS) in the Poincare coordinate

(2.1)

The radius of AdS is defined to be and the coordinate of AdS is represented by .

In this AdS space, we introduce a massive object (mass ) with a very small size which is larger than the Schwartzschild radius. Its motion in the AdS space is described by the trajectory . In general, the action of a particle with mass in a spacetime defined by the metric is given by

(2.2)

We assume that the particle is situated at and we gauge fix by setting . Then the trajectory is specified by the function . In the pure AdS background (2.1), the action looks like

(2.3)

The solution to the equation of motion derived from (2.3) is given by

(2.4)

where and are integration constants. Below we will set by using the time translation invariance. When the particle moves from the horizon to the boundary. It reaches at . Later (), it again falls into the horizon as depicted in Fig.2. Thus the energy of the particle in the AdS space is calculated as

(2.5)

2.2 Einstein Equation

The gravity action coupled to the massive particle reads

(2.6)

where the cosmological constant is given by and is the Newton constant.

The equation of motion becomes

(2.7)

where the bulk energy-stress tensor is given by

(2.8)

We will show a direct perturbative calculations of this back-reaction in appendix A. However, below we will take a different step in order to analytically construct the back-reacted solutions. See the paper [23] for analytical calculations of back-reactions to a scalar field in an AdS space. Refer also to [24] for a more extensive analysis and a relation to expanding qluon plasmas, where the back-reacted solutions are called conformal solitons (see [25] for spacetime structures of conformal solitons).

2.3 Coordinate Transformation from Global AdS to Poincare AdS

Now consider the global AdS space defined by the metric

(2.9)

We can show this is (locally) equivalent to the Poincare AdS space (2.1) via the following coordinate transformation:

(2.10)

Here, the coordinate of is described by such that . Also we defined . The arbitrary constant is introduced for the later purpose, which corresponds to the boost transformation of symmetry. If we set , (2.10) is reduced to the standard one which can be found in e.g. [12].

2.4 Back-reacted Metric for a Falling Massive Particle

In the global coordinate, we can consider a static particle situated at . Following the idea in [22], we would like to map it into the Poincare AdS. After the coordinate transformation (2.10), its trajectory is mapped into

(2.11)

Thus this corresponds to the previous trajectory (2.4) with the identification

(2.12)

The back-reacted geometry outside of the massive object is obtained from the AdS black hole solution [26]:

(2.13)

Note that in the AdS case (), the solution (2.13) for is not a black hole solution but a solution with a deficit angle.111We do not have to worry about the singularity because we replace the region near it with a star solution. The mass parameter in (2.13) is related to the mass of the particle via

(2.14)

Therefore, we can find the back-reacted metric by performing the coordinate transformation (2.10) to the metric (2.13). This can be done in a straightforward manner by noting

(2.15)

2.5 Holographic Energy Stress Tensor

One way to understand the time evolution of the CFT state dual to the falling particle in AdS, is to calculate the holographic energy stress tensor. For this purpose, it is useful to employ the Fefferman-Graham gauge of the coordinates given by the expression

(2.16)

where . We are considering the case where the boundary metric coincides with the flat Minkowski metric as in the Poincare AdS. Then near the AdS boundary , behaves like

(2.17)

In this setup, the holographic energy stress tensor [27] is calculated from the formula:

(2.18)

Note that the metric we find from the coordinate transformation (2.10) of (2.13) is not in the form of the Fefferman-Graham gauge (2.16). Thus we need to perform a coordinate transformation further to achieve this gauge so that we can employ (2.18). It is also useful to define the light-cone coordinate and , where for and for . Finally we obtain the following energy stress tensor for :

(2.19)
(2.20)
(2.21)

The angular coordinates for and for are those of the polar coordinates of and , respectively. For , the above expression agrees with the one in [22].

Note that we can easily confirm the traceless condition

(2.22)

and the conservation law

(2.23)

In particular, the energy density in each dimension is given by

(2.24)

By using these expressions we can confirm

(2.25)

for any , by using the relation (2.14). Therefore the total energy agrees with that of the particle (2.5) as expected.

To understand how the excitations propagate, it is useful to look at the time evolution of . This is plotted in Fig.3. We can observe a peak on the light-cone as can be understood as the light-like propagation (or shock waves as in [22]) of the initial excitations at .

The parameter parameterizes the width of this peak and therefore it measures the size of the localized excitations. Notice that for , the heights of the two peaks are equal and stay constant due to the total energy conservation. We sketched the essence of this behavior in Fig.4 for . In the zero width limit we find gets delta-functionally localized:

(2.26)
Figure 3: The profiles of the energy density for (left), (middle) and (right) as a function of and within the range . We set .
Figure 4: A sketch of time evolution of energy density and entangled pairs for . The understanding of detailed structure of quantum entanglement and entangled pairs is the main subject of this paper as will be studied in later sections.

3 Perturbative Analysis of Holographic Entanglement Entropy under Local Quenches

The entanglement entropy is defined as the von-Neumann entropy when we trace out the subsystem , which is the complement of . The subsystem is an arbitrary chosen space-like region on a time slice. Therefore, in a time-dependent background, depends on the time even if we fix the shape of the region . This is expected to be an important quantity which characterizes various non-equilibrium processes in quantum many-body systems such as quantum quenches [7, 9, 8].

In holographic setups, we can calculate by using the holographic entanglement entropy (HEE). In dimensional static gravity backgrounds, we can calculate by the formula [17]

(3.1)

where is the dimensional minimal area surface on a time slice which ends on the boundary of i.e. . For spherically symmetric subsystems, we can manifestly prove (3.1) in pure AdS spaces via the bulk to boundary relation [11] as shown in [28]. In time-dependent backgrounds, there is no natural time-slice in the bulk AdS and we need to employ the covariant version of HEE [18]. This is given by redefining to be an extremal surface in the Lorentzian spacetime.

Below we would like to compute the holographic entanglement entropy in the gravity dual of the local quench obtained from the coordinate transformation of the back hole solution (2.13) as explained in the previous section. In this section we perform a perturbative calculation assuming that the back-reaction from the particle is very small, keeping only the leading term proportional to . This allows us a relatively simple calculation which is applicable to any dimension .

3.1 Perturbative Calculations of HEE under Local Quenches

Consider the first order perturbation of the metric

(3.2)

where represents the metric of pure AdS (2.1); is the leading perturbation due to the back-reaction, which is of order in our case. We obtained from the direct calculation explained in the previous section, though we will not write its complicated expression explicitly.

If we know the extremal surface in the pure AdS, then the perturbed area of an extremal surface can be found as

(3.3)

Here is the induced metric on the surface :

(3.4)

where is the coordinate of the codimension two surface . The embedding function is that of in the pure AdS. Notice the useful fact that we do not need to know the precise shape of in the perturbed metric to calculate (3.3).

3.2 Explicit Calculations of HEE under Local Quenches

Consider the case (AdS) first. We take the subsystem to be a round disk with the radius , defined by . The corresponding extremal surface (or equally minimal surface) is given by the half sphere [17]

(3.5)

We can take and then we find that the area density is given by

(3.6)

where .

The area perturbation is found by integrating (3.6) over and . By plugging this into (3.1) we obtain the increased amount of HEE, denoted by , compared with for the ground state dual to the pure AdS:

(3.7)

where we used the relation (2.12). Note that the area law divergence [29] is canceled in because is defined from by subtracting the entanglement entropy for the ground state. For example, at we find

(3.8)

The time-evolution of (3.7) is plotted in Fig.5. Notice that respects the time-reversal symmetry. When , the entanglement entropy initially grows with the time for and reaches the maximum at as in the left graph in Fig.5. Later, it decreases as

(3.9)

When , the entanglement entropy always decreases for . At late time we have the behavior (3.9). Notice also that the width of the peak around is estimated as when . These results can be intuitively understood because at the particle in AdS passes through the minimal surface (3.5). Refer to the Fig.2 again. In the dual CFT, they can be naturally understood if we remember the excitations propagate at the speed of light as we will discuss later.

Notice also an interesting property that the height of the peak at stays constant, given by , and thus does not decrease under the time-evolution. This can also be seen from the right graph in Fig.5. This is rather different from the behavior of energy stress tensor studied in the previous section (see Fig.3).

Figure 5: The plots of at for . The left and middle one describe as a function of . We choose (left) and (middle), respectively. The right 3d graph expresses as a function of and for . The horizontal coordinate corresponds to . We took the range and . We set .

It is also intriguing to shift the center of the subsystem relative to the trajectory of the falling particle in the plane. We take the minimal surface in the pure AdS to be

(3.10)

where is the distance between the center of and the falling particle.

We plotted the results of in Fig.6 for different values of . In general one may notice that the peak is broadened so that it is spread over . In the gravity dual, this is easily explained from the propagation of gravitational waves from the falling particle. This also qualitatively agrees with the results of local quenches in two dimensional CFTs [9].

Figure 6: The plots of for as a function of time . The left, middle and right graph correspond to and , respectively. We set .

It is straightforward to generalize the above results to other dimensions. In and , the expression of (3.7) for is replaced with222In the subsystem is chosen to be an interval .

(3.11)

We can confirm that the behaviors of in and are very similar to that in and thus we will not show them the explicitly.

At late time , they are approximated by

(3.12)

This shows that decays like at late time. Notice that the perturbative calculation is always justified at late time as the back-reaction to for a finite clearly gets suppressed.

3.3 Small Subsystem Limit: An Analogue of the first law of thermodynamics

In the small limit of the subsystem size , we can trust the perturbative results of (3.7) and (3.11). This is because the surface is situated near the AdS boundary and therefore the deviation of the metric from the pure AdS is very small [30]. In asymptotically AdS backgrounds which are static and translation invariant, a relation which looks like the first law of thermodynamics has been found in the small size limit of [30]:

(3.13)

where is the energy in the subsystem given by . The effective temperature is defined by , where the constant only depends on the shape of the subsystem and is independent from the details of the CFT we consider. When is a dimensional ball with the radius as we choose in this paper, we have

(3.14)

Thus it is intriguing to see if this relation (3.13) holds in our time-dependent and inhomogeneous setup of local quenches. We can calculate the energy density from (2.24), and from (3.7) and (3.11) in the limit as follows:

(3.15)

By using these expressions we can explicitly confirm the relation (3.13) for any .

3.4 Large Subsystem Limit

Before we go on, we would like to write down the result in the large subsystem limit . In this limit, we find from (3.7) and (3.11):

(3.16)

Note that these are time-independent.

4 Exact Holographic Entanglement Entropy for d Local Quenches

We can actually find exact extremal surfaces (i.e. geodesics) in the AdS case (). We take the subsystem to be an interval at a constant time in the dual CFT. In the holographic calculation, first we obtain a geodesic in the metric (2.13), which is asymptotically global AdS. Then we map it into an asymptotically Poincare AdS metric by the coordinate transformation (2.10).

Initially, we will assume and thus the geometry (2.13) has the deficit angle at . If we were allowed to take to be instead of , the geometry gets smooth. Later we will come back to the case i.e. the BTZ black hole.

In the near AdS boundary limit, the map between the point in (2.13) and the point in the Poincare AdS is given by333To fix the signs of and , we need to go back to the original coordinate transformation (2.10).

(4.1)

Notice that is interpreted as the UV cut off (or lattice spacing) in the dual CFT. In the above expressions, we chose the range of and to be .

We can specify the geodesic in (2.13) by

(4.2)

Its length reads

(4.3)

The minimal length condition is summarized as

(4.4)

where and are integration constants.

4.1 Symmetric Intervals

Consider the case where the subsystem is given by an interval at time . The excitations of the local quench occur at because the falling particle is situated at . In the global AdS, this is mapped into an interval at a constant time slice , where we can assume without losing generality (see Fig.7). Since we can assume is constant on , the extremal surface condition (4.4) gets simplified to

(4.5)

where is an integration constant and is the minimum value of on , i.e. the turning point.

4.1.1 Case 1:

Let us first assume . The curve with the minimum length is given by the geodesic which takes the angle range (see the left picture of Fig.7). The point corresponds to the turning point . Thus we find444In this paper, the function takes values between and .

(4.6)

It is useful to rewrite this into

(4.7)

Finally, the HEE is given by

(4.8)

In the case , on the other hand, the curve is given by the geodesic which takes the angular range (see the right picture of Fig.7). Thus its turning point is at . This guarantees the basic property of entanglement entropy written as for pure states, where is the complement of . Thus for , we find the correct HEE from (4.8) by replacing with in (4.7).

Figure 7: The sketch of in the asymptotically global AdS background (2.13). The left and right picture corresponds to and , respectively.

4.1.2 Case 2:

For , the geometry (2.13) becomes the BTZ black hole. In this case, the basic result can be obtained from the previous one by the analytic continuation : the relation (4.7) is replaced with

(4.9)

For , the holographic entanglement entropy can be found from the same expression (4.8), which is denoted by . For , is given by as in the case.

In this calculation we implicitly assume that we are considering a star solution and that outside of the star is described by the BTZ black hole solution (2.13). Thus this is dual to a pure state in the CFT and indeed our construction satisfies .

4.1.3 Final Results

The final result is plotted in Fig.8. For a small the exact result nicely agrees with the one from the perturbation theory (3.11). Even for a large , the agreement is very good except the peak at . Indeed, we can analytically show that in both the limit and , approaches to

(4.10)

where we employed the well-known relation [31]. This reproduces the well-known result of the entanglement entropy for ground states in CFTs [32, 33, 3]. Remember that corresponds to the UV cut off (lattice spacing) in the dual CFTs.

Figure 8: The exact plots of as a function of time . The left, middle and right graph correspond to , respectively. The blue and red curve correspond to the result from the exact analysis and perturbation theory, respectively. They coincide almost completely for . The dotted black one for corresponds to the thermal local quench given by (4.11). We set and .

4.1.4 Thermal Local Quenches

In the previous analysis for , we obtained by replacing a black hole with a star in order to have a gravity dual of a pure state, which is usually assumed in the study of quantum quenches. If we deal with a falling black hole instead of a massive star, we will obtain as follows

(4.11)

where denotes the smaller one among and . The term corresponds to a half of black hole entropy and arises because wraps a half of horizon. The topological condition of of HEE requires that the subsystem should be homologous to . Therefore in the presence of the black hole horizon, we cannot simply change the geodesic from the one passing through to the one passing through . If we want to deform in this way, we need also to wrap on the horizon as its disconnected part. We plotted for this thermal local quench in the third graph of Fig.8.

4.2 General Formulation

Now let us extend the previous analysis to more general setups where the subsystem is given by an arbitrary interval at the time . The surface is defined by the geodesic curve whose two end points are given by in the Poincare AdS3 and equally by in the global AdS. Note that their relations are given by: