Entanglement entropy in holographic moving mirror and Page curve

We calculate the time evolution of entanglement entropy in two dimensional conformal field theory with a moving mirror. For a setup modeling Hawking radiation, we obtain a linear growth of entanglement entropy and show that this can be interpreted as the production of entangled pairs. For a setup, which mimics black hole formation and evaporation, we find that the evolution follows the ideal Page curve. We perform these computations by constructing the gravity dual of the moving mirror model via holography. We also argue that our holographic setup provides a concrete model to derive the Page curve for black hole radiation in the strong coupling regime of gravity.


INTRODUCTION
Moving mirrors have been known for a while as a class of instructive models that mimic Hawking radiation [1] based on quantum field theory [2,3] where unitarity is manifest. On the other hand, in the case of black hole evaporation, it has been a significant problem to understand whether unitarity is maintained in the gravitational theory. One manifestation of unitary black hole evaporation is the Page curve for the entropy of Hawking radiation [4]. Based on the fine grained entropy formula [5][6][7][8], this has been derived semiclassically for field theories coupled to gravity [9][10][11] and confirmed by direct gravity replica computations [12,13]. See e.g.  for further progress along this direction. For recent related works refer to [63][64][65].
In this article, we first present concrete calculations of entanglement entropy in moving mirror setups and show that this leads to an ideal Page curve. This itself provides a novel nonequilibrium setup, where quantum entanglement evolves rapidly. Moreover, we present a close connection between moving mirror models and black hole radiation via a particular version of the anti de-Sitter (AdS)/conformal field theory (CFT) correspondence [66], namely, in the case when the CFT is defined on a manifold with a boundary [67,68]. For earlier studies of entanglement entropy in moving mirror models refer to [69][70][71][72][73].

MOVING MIRROR FROM CONFORMAL MAPS
A moving mirror setup in two dimensions is specified by the trajectory of a mirror profile x = Z(t). We consider a CFT which lives on the right region, i.e. x ≥ Z(t). A conformal transformation (here, we set u = t − x and v = t + x) [2,3] IG. 1. The moving mirror setup (left) and its conformal transformation into a static mirror (right). The Mirror trajectory is depicted by the thick curve. The shaded region shown in the right panel corresponds to an inside horizon region, which is missing in the left picture. maps this into a simple setup with a static mirrorũ−ṽ = 0, as depicted in Fig. 1. Here, we choose the function p(u) such that the mirror trajectory is given by v = p(u), i.e.
For example, we can calculate the energy stress tensor from the conformal anomaly via the map (1), such that where the components T uv and T vv are vanishing.
As an example of a CFT, consider a massless free scalar φ. We impose the Dirichlet boundary condition φ(t, Z(t)) = 0 along the mirror trajectory. A complete set of positive frequency solutions to the equations of motion ∂ u ∂ v φ = 0, which satisfy the latter boundary condition, reads Then, φ can be expanded in terms of these modes [2] as where a in ω and a in † ω are the annihilation and creation operators, respectively. The in-coming vacuum |0 in is defined by the state annihilated by a in ω and the out-going arXiv:2011.12005v2 [hep-th] 2 Dec 2020 vacuum |0 out is given by a Bogoliubov transformation of |0 in . The expectation value of the energy stress tensor, i.e. 0 in |T uu |0 in , reproduces (3) for c = 1.

MOVING MIRROR WITH HORIZON
For a typical example which models Hawking radiation from a black hole, we choose where the parameter β plays the role of an inverse temperature. Its profile is depicted in the left picture of Fig. 1. In the early time limit t → −∞, we have Z(t) 0, while in the late time limit, the mirror trajectory gets almost lightlike, Z(t) −t−βe −2t/β . As depicted in Fig. 1, the regionũ ≥ 0 in the extended coordinates is missing for the original coordinates. This region is analogous to the inside horizon region in the black hole formation process. The energy stress tensor (3) reads which vanishes at early time u → −∞, and becomes a constant thermal flux, T uu c 48πβ 2 , at late time u → ∞. Let us calculate the entanglement entropy S A for a semi infinite subsystem A given by [x 0 , ∞] at time t. We can calculate S A from the one point function of the twist operator [74][75][76] on the upper half planex > 0 by using the conformal map (1) via the replica method. We find Here, is the UV cutoff (lattice spacing) of the CFT and S bdy is the boundary entropy [68,74,77]. Note that this formula holds for any CFT.
If we fix the end point of the subsystem A, i.e. x 0 , we can approximate (8) at late time t → ∞, and find The first term linear in t arises from entangled pair production due to the moving mirror, while the second log t term comes from the standard vacuum entanglement as the length of the complement of A grows linearly.
To study the first contribution in more detail, we allow changing the value of x 0 time dependently as In the late time limit u → ∞, we obtain We choose ξ 0 to be positive, but sufficiently small. If the left end point of A, given by (u, v) = (2t−ξ 0 , ξ 0 ), satisfies v + p(u) > 0, we get the linear growth (see the left panel in Fig. 2), In this way, we may conclude that the entangled pair production occurs along the spacelike curve v + p(u) = 0, and the propagation of the entangled pairs gives the linear growth of the entanglement entropy (12). This is sketched in the right panel of Fig. 2. We can also confirm this from the free scalar example (5), where the spacial distribution of the pair production looks like where we have defined p i := p(u i ) for brevity. The first two terms are divergent at v 1 + p 2 = 0 and v 2 + p 1 = 0. This shows that the entangled pairs are produced along the curve v + p(u) = 0, and they propagate in opposite directions at the speed of light.

ADS/BCFT AND ENTANGLEMENT ENTROPY
To compute S A for generic subsystems, we need to specify the target CFT. For holographic CFTs, we can calculate S A via the gravity dual of a CFT defined on a manifold M with a boundary ∂M , i.e. boundary CFT (BCFT) [78], known as AdS/BCFT [68]. In this description, the dual geometry is given by extending the boundary ∂M into the bulk AdS, which leads to a codimension one surface Q, called the end of the world brane. This surface Q obeys the Neumann boundary condition where h ab is the induced metric and K ab is the extrinsic curvature. The parameter T is the tension of the brane Q and depends on the boundary condition of the CFT at ∂M . The condition (14) implies the presence of boundary conformal invariance. Refer to [79] for an equivalent formulation using Chern-Simons gravity, and to [80] for comparisons with CFT calculations. We can find a gravity dual by applying the following coordinate transformation, which is a special case of [81,82] and is a bulk extension of the map (1), Using (3), this leads to the metric In Poincare AdS 3 (16), by solving the boundary condition (14), the profile of Q is given by X = −λη, where we have defined λ = T √ 1−T 2 , and introduced new coordinates U = T − X and V = T + X. The metric on Q is given by that of Poincare AdS 2 (see the left panel in Fig. 3), Thus, the gravity dual in terms of the (U, V, η) coordinates is given by a part of Poincare AdS 3 defined by X + λη > 0. Note that the surface Q at the boundary z = 0 coincides with the mirror trajectory v = p(u) via the map (15). The gravity dual in terms of the coordinates (u, v, z), which is given by the metric (17) and is sketched in the right panel of Fig. 3, only covers the region U < 0 as U = p(u) is always negative for any u. This is the bulk extension of the mentioned inner horizon region shown in the right panel in Fig. 1. In the coordinates (u, v, z), the metric of the brane Q reads This covers only the part T < −λη of (18). In this way, the gravity dual of the moving mirror has a horizon, analogous to a single sided AdS black hole. In general, the holographic entanglement entropy [5,6] in AdS/BCFT can be computed [68] as where A(Γ A ) is the length of Γ A which satisfies ∂Γ A = ∂A ∪ ∂I s , where I s (i.e. island) is a region on the surface Q. The three dimensional Newton constant is denoted by G N . The minimum in (20) is taken over all possible choices of I s and Γ A . When A is an interval [x 0 , x 1 ] at time t, there appear to be two candidates for Γ A . One is a connected geodesic between x = x 0 and x = x 1 . The other one is a union of two disconnected ones, each of which departs from x = x 0 (or x = x 1 ) and ends on Q, respectively. In our setup, they are explicitly given by S A = Min[S con A , S dis A ], where the disconnected and connected geodesic contributions S dis A and S con A read The boundary entropy is a function of the tension and is given by S bdy = c 6 log (1 + T )/(1 − T ). When A is semi infinite, i.e. x 1 → ∞, we always have S A = S dis A , and this reproduces (8). When A is a finite interval, S dis A is initially favored and this gives the linear growth as in (12). At later time, S con A is favored and this leads to a saturation as depicted in the right panel of Fig. 4. We have also plotted S A for the massless free Dirac fermion case, which is shown in the left panel of Fig. 4. Refer to Appendix A for detailed computations of S A in the free Dirac fermion and holographic CFT case.

PAGE CURVE FROM MOVING MIRROR
A typical moving mirror model which mimics an evaporating black hole is found by setting (22) whose mirror trajectory x = Z(t) and energy flux T uu are depicted in Fig. 5. When β is small, we can approximate the trajectory as Z(t) 0 for t < 0, Z(t) −t for 0 < t < u 0 /2, and Z(t) = −u 0 /2 for t > u 0 /2. The energy flux is nonvanishing, T uu c 48πβ 2 , namely, only for the period 0 < u < u 0 . We can again calculate the holographic entanglement entropy S A = Min[S con A , S dis A ], using (21) as plotted in Fig. 6. In particular, when A is a semi infinite line, S A takes the form of the Page curve. For 0 < t < u 0 /4, S A grows linearly dS A dt c 6β , and for u 0 /4 < t < u 0 /2, it decreases linearly, i.e. dS A dt − c 6β . Note that as is clear from Fig. 6, the disconnected result S dis gives the dominant contribution (refer to [70] for an earlier calculation of the connected result S con ).
The initial linear growth of S A can be understood as in the previous example (12) by considering entangled pair production along the curve v + p(u) = 0. Moreover, the linear decay of S A is explained by reflections of the left moving partner, as shown in the left picture of Fig. 5. When A is a finite interval, we have two Page peaks. The first peak occurs when only the originally right moving particles are crossing A. The second peak appears when only the reflected particles are crossing A.

BRANE WORLD GRAVITY AND ISLAND
The gravity dual of our moving mirror setup can be interpreted in an alternative way by regarding the surface Q as an end of the world brane in the brane world setup [83][84][85]. This situation is depicted in the left panel of Fig. 7. According to this interpretation, the CFT defined in the region x ≥ Z(t) will be coupled to a two dimensional gravity theory on Q. By estimating the effective Newton constant on Q via Kaluza-Klein reduction [44,[83][84][85], which we denote by G The gravitational entropy of AdS 2 , i.e. the brane Q, will thus be equal to the boundary entropy S bdy . We can regard S A in (8) as the entanglement entropy of the subregion A in a system consisting of a CFT on x ≥ Z(t) and a gravitational theory on Q, glued along the moving mirror. Then, we can interpret the first and second term in (8) as the bulk entropy contribution S bulk A∪Is and the area term Area(∂Is) 4G N , respectively, in the island formula [9][10][11][12][13]. Note that here we use the standard formula for computing holographic entanglement entropy without invoking the quantum extremal surface prescription. The density matrix under consideration is pure and radiation is manifestly unitary.
In our moving mirror model, the entropy (8) shows linear growth in the region defined by the equation v + p(u) > 0. This part in the entanglement entropy would arise from the island. This region is not covered by the coordinate patch (19), as sketched in the left panel of Fig. 7. An interesting feature in our setup is the entanglement between the gravitational theory on Q and the CFT by the amount (23). Also it is important to note that the presence of energy flux from the boundary is different from standard BCFTs (Cardy states [86]) which have no energy flux condition T (w) −T (w) = 0 at the boundary.
The radiation in the CFT looks similar to the setups [9][10][11][12][13], where the Page curve was derived. However, unlike these, in our BCFT model, we find that there is no radiation present in the gravitational system on Q. That is, the flux does not come from the gravitational system, but is created on the boundary. Indeed, the holographic energy stress tensor [87] on Q is proportional to h ab as it follows from (14). Hence, it can just be regarded as a negative cosmological constant. The absence of radiation from Q is obviously consistent with the fact that the mirror is completely reflective. We can regard our analysis as a derivation of the Page curve in the strong coupling regime of gravity, while [9][10][11][12][13] focus on the weak coupling regime. Indeed, by changing the profile of the brane Q, we can deform our setup of the moving mirror such that it incorporates radiation resulting from the gravitational sector on Q, see right panel of Fig. 7. If we modify the surface Q to make it close to the standard AdS boundary located at z = , the matter energy stress tensor on Q will be approximated by (7). This provides a special and concrete example of the setup considered in [11], see also [15-17, 37, 42, 44, 48] for related works. The holographic dual of this modified setup is a two dimensional CFT coupled to two dimensional gravity. One advantage of our procedure is that our calculation based on the BCFT analysis is much easier than the one based on the conformal welding problem [10,13]. An interesting future direction will be relating the two realizations in an explicit way [88].

CONCLUSION
In this article, we have presented a gravity dual of two dimensional CFT with a moving mirror, which mimics black hole formation and evaporation. We have explicitly calculated the time evolution of entanglement entropy in the presence of the mirror. We have found that it follows the ideal Page curve. This can be explained by the creation of entangled particles, their propagation, and reflection from the mirror. We have also discussed that modifying the profile of the end of the world brane in the gravity dual results in a model for two dimensional black hole radiation. In order to understand unitary evolution for realistic black hole evaporation, we will have to incorporate the singularity. We expect that the presence of spacelike boundaries in the CFT and its gravity dual [44,48] will be relevant [88].