Nonlinear entanglement growth in inhomogeneous spacetimes

Entanglement has become central for the characterization of quantum matter both in and out of equilibrium. In a dynamical context entanglement exhibits universal linear temporal growth in generic systems, which stems from the underlying linear light cones as they occur in planar geometries. Inhomogeneous spacetimes can lead, however, to strongly bent trajectories. While such bent trajectories crucially impact correlation spreading and therefore the light-cone structure, it has remained elusive how this influences the entanglement dynamics. In this work we investigate the real-time evolution of the entanglement entropy in one-dimensional quantum systems after quenches which change the underlying spacetime background of the Hamiltonian. Concretely, we focus on the Rindler space describing the spacetime in close vicinity to a black hole. As a main result we find that entanglement grows sublinearly in a generic fashion both for interacting and noninteracting quantum matter. We further observe that the asymptotic relaxation becomes exponential, as opposed to algebraic for planar Minkowski spacetimes, and that in the vicinity of the black hole the relaxation time for large subsystems becomes independent of the subsystem size. We study entanglement dynamics both for the case of noninteracting fermions, allowing for exact numerical solutions, and for random unitary circuits representing a paradigmatic class of ergodic systems.


I. INTRODUCTION
Quantum entanglement [1] is a key feature of quantum systems and has become a cornerstone of many modern branches of physics, such as quantum information [2] (in particular, quantum cryptography [3][4][5], quantum teleportation [6,7], quantum computing [8][9][10][11][12]) and quantum many-body physics [13,14]. In the latter, the importance of entanglement is reflected by its universal properties, both under equilibrium and non-equilibrium conditions. Ground states of gapped one-dimensional quantum systems obey an area law [15][16][17]. In nonequilibrium the entanglement entropy of generic systems shows generally a linear growth in time, which is today explained through a semiclassical picture of ballistic quasiparticle propagation [18][19][20][21][22][23][24][25][26][27][28][29][30][31]. Although the applicability of the quasiparticle picture has been confirmed for a range of integrable models, including inhomogeneous initial states [32,33], this picture cannot be directly applied when the post-quench Hamiltonian is not translationally invariant. A prominent example is the logarithmic spreading of entanglement in a many-body localized phase [34][35][36][37][38] in disordered models. These models are, however, constructed on flat homogeneous spacetime backgrounds and therefore their local properties are the same everywhere. It has remained, however, an open question how entanglement grows in spatially inhomogeneous systems due to inhomogeneity of the spacetime itself.
In this work we investigate the entanglement entropy growth generated by a quench of the underlying spacetime metric of the Hamiltonian. As a prototypical example of an inhomogeneous spacetime we choose a Rindler space, which can be viewed an asymptotic spacetime near a black hole horizon [39][40][41]. We find that in the case of a global inhomogeneous quench the initial entangle-ment entropy growth is sublinear and attains a constant size-independent value in a long-time limit. This is the most noticeable difference in comparison to the translationally invariant case where the entanglement entropy growth is linear. We contend that this behavior is universal applying for both integrable and ergodic systems. We find that some important aspects of the physical picture can be captured qualitatively via semiclassical arguments based on the maximal speed of correlation propagation. The correlations spread within bent light-cones defined by null geodesic of the (1+1) Rindler metric which suggests that our results capture the universal properties of the continuum theory.
This papers is organized as follows. In Sec. II we discuss the basic concepts which are central to this study and introduce the model. In Sec. III we calculate the dynamics of correlations after a Rindler quench and draw semiclassical arguments that are helpful to understand some physical aspects of the entanglement evolution. In Sec. IV we present numerical results for finite subsystems and, by achieving a data collapse, we identify sizeindependent universal behavior of entanglement. While in Sec II -IV we focus on the case of free fermions on the 1D lattice, Sec. V is devoted to random unitary circuits [42][43][44][45] that represent minimally structured ergodic models. We estimate the coarse grained entanglement entropy, which is equivalent to the minimal membrane description [43], and show the remarkable qualitative agreement of results for integrable and ergodic systems. In Sec. VI we conclude.

A. Entanglement entropy
Quantum entanglement has developed into a central concept in quantum many-body physics. One of the key consequence for a state |Ψ to be entangled is that it cannot be written as a simple product of states belonging to different subsystems. Quantification of entanglement is possible through various entanglement measures [46]. In this work we want to study the entanglement entropy defined as whereρ A = Tr B |Ψ Ψ| is a reduced density matrix of a subsystem A traced over the rest of the system, a subsystem B. Entanglement between two spatial intervals can be generated dynamically through a quantum quench when a system is prepared in an initial state which is not an eigenstate of the post-quench Hamiltonian. Calabrese and Cardy [18] have shown by path integral methods of (1+1) quantum field theory that, up to a time t * ∝ L/2, the entanglement entropy growths linearly with a rate independent of the subsystem size L and attains a constant value ∝ L afterwards, yielding a volume law, which is due to the finite subsystem size.
The linear spreading of entanglement entropy turns out to be very general and holds in various non-interacting [19][20][21][22][23][24][25] and short-range interacting [26][27][28][29][30] models, also for inhomogeneous initial states [32,33]. The linear growth of entanglement is due to the maximal speed of information. In the (1+1) Minkowski spacetime the rate of any information propagation is bounded by the speed of light c and all particle trajectories lie within a light-cone defined by null geodesic. In the lattice systems, the role of the speed of light c plays the Lieb-Robinson velocity [47], being the emergent maximal velocity of correlation spreading.

B. Rindler metric
Although the entanglement dynamics in spatially invariant quantum systems seems to be completely understood, the entanglement dynamics in inhomogeneous spacetimes is a completely open question. In this paper, we study the entanglement evolution in systems on a Rindler background described by a (1+1) metric [39][40][41] which serves us as a nontrivial example of an inhomogeneous metric. In the literature the Rindler metric appears primarily in two related contexts. First of all, the Rindler metric is an asymptotic Schwarzschild metric [48] in the vicinity of a black hole horizon at x = 0. Secondly, the Rindler metric describes a flat Minkowski spacetime in a hyperbolically accelerated reference frame, i.e., it characterizes the motion of uniformly accelerated observers [49]. Comparing Minkowski and Rindler metrics, one could interpret c(x) = x as a spatially varying speed of light. Indeed, the geodesics equations come down to a simple dx/dt = ±x with a straightforward solution with x 0 the initial position. When one switches to a proper τ = x 0 t/c of an observer at rest at x = x 0 , then The above result means that the light-cones in the Rindler spacetime are distorted and the strongest bending takes place in a vicinity of a horizon at x = 0. Conversely, the bending of the light cone is negligible for τ /x 0 1, i.e., at very early times or far away from the horizon.
Because the information in the Rindler spacetime does not propagate linearly, it is a central open question how the entanglement is propagating in such a setup. In the following we argue the entanglement entropy in generic quantum systems in a Rindler spacetime first grows sublinearly and asymptotically at long-times attains a constant size-independent value in a thermodynamic limit.

C. Setup
In the following we will consider the Hamiltonian description of free fermions in inhomogeneous dimensional spacetimes. As a first step we investigate (1+1) dimensional metric, but the generalization to higher dimensions is possible, and we plan to purse this path in the next future. Let us consider a system of spinless fermions on a one dimensional (1D) lattice of length N with open boundary conditions. We assume that an initial state is a spatially homogeneous product state where |0 is a particle vacuum state, i.e., it is annihilated by any annihilation operatorĉ n |0 = 0. Since the initial state is a product state, its entanglement entropy, Eq. (1) vanishes with respect to any bipartition. Suppose that at time t = 0 we perform a quench such that the evolution of a system t > 0 is described by a new Hamiltonian where c † n , c n are standard fermionic creation and annihilation operators on a lattice. It has been shown in Ref. [50] (see also Refs [51][52][53][54][55]) that the massless Dirac fermions propagation on static spacetimes can be described by a lattice free fermion Hamiltonian with a spatially varying tunneling amplitude proportional to the determinant of the metric. In this context, the Hamiltonian Eq. (7) may be viewed as a spinless version of the discrete relativistic Hamiltonian on a curved (1+1) dimensional spacetime. Here, we choose t n = c = 1 or t n = n. The first choice of tunneling amplitudes amounts to the Minkowski metric, Eq. (2), and the corresponding Hamiltonian is obviously a free fermionic Hamiltonian. From now on, we will refer to this choice of tunneling amplitudes as the homogeneous quench. The second choice of tunneling amplitudes (t n = n) entails the Rindler metric, Eq. (3), yielding the Rindler Hamiltonian We note that this Hamiltonian, in a context of the Bisognano-Wichmann theorem, can be also interpreted as a modular or entanglement Hamiltonian. The study of entanglement Hamitlonian for lattce models has recently attracted substantial attention [56][57][58][59][60][61][62][63][64].
In this work, we investigate the entanglement evolution generated by the Rindler quench. In Sec. IV we present the exact numerical diagonalization results for the entanglement entropy and in Sec. V we consider an analogous quench protocol in a unitary circuit setup. Before going to the main results, in Sec. II E for illustrative and comparison we review the semi-classical pair quasiparticle picture for the homogeneous quench, which is well understood in the existing literature [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33], and in Sec. III we give its geometric interpretation in terms of propagation of correlations. On the contrary, because of the lack of the translational invariance in the Rindler Hamiltonian, Eq. (8), the (quasi-)momenta are no longer good quantum numbers and the quasiparticle picture of counterpropagating pairs with opposite momenta does not hold. Nevertheless, since the speed of correlation spreading is still bounded in Sec. III we present heuristic arguments that allow us to extract important features of the entanglement dynamics, such as sublinear growth and a long-time asymptotically constant behavior. Most of all, we show that this approach allows us to identify relevant scaling parameters of the model.

D. Entanglement entropy for free fermionic systems: the Peschel formula
For the free fermionic models the entanglement entropy, between a subsystem A and its complement B, can be efficiently calculated via correlation functions [65,66], as long as an initial state can be described by a Slater determinant. In this case, the density matrix can be written as an exponential of free fermionic operatorŝ where H = [h ij ] i,j∈A matrix is diagonalized by the same transformation as a single particle correlation matrix C = [ ĉ † iĉ j ] i,j∈A . It has been found that the two matrices are related [65] Using the thermal form of a reduced density matrix, Eq. (9), one can readily obtain an expression for the entanglement entropy following a quench at t = 0 where λ n (t)'s are eigenvalues of equal time correlation matrix restricted to a subsystem A. According to Eq. (11) the maximal value of the entanglement entropy S max = L log 2 is obtained if all λ n = 1/2, which corresponds to a trivial reduced density matrix, Eq.

E. Quasiparticle geometric picture
In the quasiparticle pair picture [18], an initial state |Ψ 0 , being excited from the point of view of a quenched HamiltonianĤ, serves as a source of quasiparticle pair excitations. Each pair emitted from the same point in space is entangled and contribute to the total entanglement between A and B, if at a time t a pair is shared between the two regions. Accordingly, the entanglement entropy in the quasiparticle pair picture reads where f (p) is a quasiparticle production rate and v(p) = dE(p)/dp is a quasiparticle velocity. If there exists a maximal quasiparticle velocity v max , than the entanglement entropy grows always linearly and saturates as a consequence of finite subsystem size at times t t * = The exact numerical lattice calculations of the correlation function C n,n (t) = ĉ † n (t)ĉ n , see Eq. (14), followed by a Rindler quench at t = 0. We plot C n,n (t) as a function of n and t for different values of n . We observe a non-linear spreading of correlations which form distorted light-cone structure. The light cone edged (red dashed lines) are described by a continuum theory, see Eq. (4). Panel C: Because of the light-cone bending, the number of lattice sites that are in a causal relation with a site m depend on a spatial direction. We assume that the entanglement entropy is equal to the number of distinct pairs that can be correlated across the boundary between A and B, i.e., it is proportional L/2. The applicability of Eq. (13) goes beyond a simple qualitative understanding. In fact, in the thermodynamic limit t, L → ∞, t/L = const. the predictions of the quasiparticle picture exactly reproduce the behavior of entanglement entropy in 1D translationally invariant integrable models, based on the knowledge of the steady state and its excitations. [28]. In the Appendix A, using the quasiparticle formula, Eq. (13), we give an analytic form of the entanglement entropy for the staggered initial state and the homogeneous quench, Eq. (7). There, we show that the analytical formula is nonanalytical at t * = L/2, where the second derivative of entanglement entropy is discontinuous. Also in the Appendix A we illustrate that even for relatively small L's the quasiparticle picture quite accurately reproduces the numerical data, although the nonanalyticity can only be observed in the infinite subsystem limit.

III. CORRELATION SPREADING
The quasiparticle picture has a simple geometric interpretation. The fastest quasiparticles propagate at the maximum velocity v max = max k |dE/dk| = 1, the Lieb-Robinson velocity [47], which is the maximal velocity of correlation spreading where |Ψ 0 is an initial state. In homogeneous systems, the correlations spread within linear light cones, which agrees with a quasiparticle pair picture. Geometrically, the contribution to the entanglement entropy S(t) coming from each boundary between A and B is proportional to the length of an interval, d(t), covered by a light-cone placed at this boundary. Assuming that the entanglement entropy is proportional to the number of degrees of freedom which can become correlated at a tim t < t * = L/2, the total entanglement entropy is then given by; The quasiparticle picture of counterpropagating pairs with opposite quasimomenta cannot be applied directly if the post-quench Hamiltonian does not possess translational invariance. However, similarly to the homogeneous quench, we can relate the entanglement entropy growth with the spreading of correlations. On Fig. 1(B) we plot the correlation function, Eq. (14) for a Rindler quench and for different choices of lattice positions n . The exact numerical calculations show that the correlations spread nonlinearly within bent light-cones that are described by Eq. (4) with a very good agreement. On Fig. 1(C) we present a cartoon picture of a bent light cone originating at a lattice site m. Through a simple calculation a distance, d − (t) and d + (t), covered by a left and right part of the light cone accordingly is simply Now, we aim to apply semiclassical arguments to describe the entanglement dynamics. While this captures qualitative aspects of the entanglement dynamics, quantitative differences remain as we discuss in detail in the following. The direct application of application of a homogeneous result would lead to a superlinear growth of the entanglement entropy. Let us heuristically assume the entanglement entropy is proportional to the number of distinct pairs that can be correlated through each boundary between A and B. This number is equal to min(d + , d − ) = d − , see Fig. 1(C), and the total entanglement entropy is wherem = (m + L/2) is a position of the middle of A, and where τ = tm is a proper time of an observer placed atm. By switching to the proper time τ , any stationary observer perceives the same local value of the speed of light c(m)/m = 1, which allows us to directly compare results with different observers' positions and different spacetimes.
The heuristic formula, Eq. (17), gives as an important insight into the behavior of entanglement entropy. First of all, it depends only on one parameterm, and the dimensionless rescaled quantity S(τm)/m is parameterfree. In Sec. IV we find numerically the same scaling for free lattice fermions. Furthermore, the formula predicts universal sublinear growth growth of entanglement. Let us look at two opposing limits: 1. if τ /m 1, then S(τ ) ∝ 4τ , so that locally and for early times a linear growth is recovered, Let us stress that the second limit is not the finite-size effect, but rather stems from general properties of the Rindler spacetime where the horizon plays a role of an effective spacetime boundary.
In the following sections we show that the entanglement generated by the Rindler quench has both of the above properties. Nevertheless, we also find that the heuristic formula, Eq. (17), is too naive and does not recover quantitatively the numerical data. In fact, Eq. (17) considerably underestimate the entanglement entropy. One possible explanation is that the entanglement entropy requires important corrections from multipartite entanglement, which is not captured in the above picture.

IV. NUMERICAL RESULTS
In this section we present numerical data for the entanglement entropy dynamics for the Rindler quench and stress the differences with the homogeneous case. We calculate the entanglement entropy by exact diagonalization of equal-time correlation matrices, see Sec. II D. In Sec. IV A we show that the entanglement growth features universal behavior by achieving a data collapse, while in Sec. IV B we discuss finite-size effects in more details. Surprisingly, the finite-size analysis allows us to identify a universal long-time asymptotic behavior.

A. Universal sublinear growth
In the case of the homogeneous quench, the entanglement entropy depends only on the subsystem size L . Form comparison we include data for both the Rindler (solid lines) and homogeneous quench (dashed lines). τ denotes a proper time of an observer, see discussion in the main text. Apart from the finite subsystem size saturation to the Smax = L log 2, we find that all curves for different subsizes L's and fixed mean positionsm overlap, which allows us to identify a generic functional behavior. The numerical data confirm thatm is the only relevant parameter and that the entanglement entropy grows sublinearly. Furthermore, we observe peculiar finite-size features of the Rindler quench: although the entanglement entropy grows slower then in the homogeneous case, the entanglement entropy saturates much faster. and the generic behavior can be numerically extracted by overlaping data for different subsystem sizes L. For the Rindler quench, due to the inhomogeneous metric, we find that the entanglement entropy depends also on the mean position of the subsystem A, see the semiclassical discussion in Sec. III. Therefore, in order to extract the universal behavior from the numerics, we need to fix a mean positionm first and then perform the finite data overlap. On Fig. 2 the entanglement entropy is plotted versus a proper time τ = tm of observer located atm. For the sake of comparison, we include data for both the Rindler (solid lines) and homogeneous (dashed lines) Comparison of the entanglement entropy curves in the case of the Rindler quench, obtained from the finitesize data collapse for a fixedm. All curves are rescaled accordingly to sm(τ ) = Sm(mτ )/m. The inset shows χm = [ dτ (sm=500(τ ) − sm (τ )) 2 ] 1/2 , which is the measure of distance between two functions. In the limit 1/m → 0 we observe a converge of data.

quenches.
At early times and small subsystem sizes L the data for Rindler and homogeneous quenches are indistinguishable (this is expected as initial light cone bending from the point of view of local observers is small , see Eq. (5)). The inhomogeneity of the spacetime becomes important at times scales τ ≈m, where the entanglement entropy for the Rindler quench exhibits sublinear growth behavior. For both quenches, we observe the collapse of the data and a generic entanglement function Sm(τ ) can be identified as an envelope of a family of curves, Sm ,L (τ ), with fixedm and different L. In other words, Sm(τ ) can be defined a limit Sm(τ ) = lim L→2m Sm ,L (τ ).
According to the semiclassical formula, Eq. (17), the sm(τ ) curves should overlap. Indeed, although small deviations are found for smallm's, in the limit 1/m → 0 we observe a convergence of the data, i.e., s(τ ) defined as has a thermodynamic limit, see Fig. 3. This suggests a universal sublinear behavior of the entanglement entropy growth. On the contrary to the homogeneous case, the entropy production rate (the first derivative of entanglement entropy) decreases over time and, as we argue in the next section, the entropy is asymptotically constant in a long-time limit. We definite a crossover time τ * of the entanglement entropy growth when the finite subsystem size saturation effects start to take place. Initially, for small ratios L/m 1, the critial time is τ * ≈ L/2, which agrees with the result from the homogeneous quench. The critical time grows drastically when the ratio ratio L/m increases and approaches its maximally attainable value equal to 2 − 2/m. τ * /L becomes infinite in the limitm → ∞ such that L/m → 2.

B. Asymptotic long-time dynamics
In this section we analyze finite subsystem size effects in entanglement entropy evolution, which allows us to identify universal asymptotic long-time dynamics. On Fig. 2 there are visible two striking saturation differences between the Rindler and homogeneous quenches. First of all, a crossover time τ * -a time when the evolution starts to thermalize due to a finite size of a subsystemincreases together the growth of a subsystem size L. Secondly, the finite size thermalization is much faster.
Let us first quantify the first observation. We recall thatm denotes a middle position of a subsystem A of length L, and therefore the L/m ∈ [1/m, 2-2/m]. The lower limit corresponds to a situation when A of a unit length is far away from the origin m = 1, while the upper limit corresponds to A located maximally close to the origin. Previously, we have identified a generic function Sm(τ ) which is an envelope of a family of curves Sm ,L (τ ) with fixed L. We define a crossover time τ * (L) as |Sm ,L (τ ) − Sm(τ )| < and on Fig. 4 we plot τ * /L as function of L/m form = 200 and = 0.03. While the choice for the threshold is, of course, arbitrary, we find that the final result for τ * (L) doesn't depend crucially on it as long as is sufficiently small. As expected, we observe that τ * ≈ L/2 when the subsystem A is small comparing to its distance to the origin, which recovers the known homogeneous result. This is perfectly understandable, since for small L/m the effective speed of light c(m) ∝ m is locally constant and does not change significantly on the extent of a subsystem, i.e., In physical terms, the thermodynamic limitm → ∞ while L/m → 2 means that one edge of the subsystem A is placed basically at the horizon, where the local light velocity vanishes asymptotically. As a consequence, it would require an infinite time for the signal to propagate throughout the subsystem and the crossover time τ * has to diverge accordingly. Note that this behavior was already predicted on a semiclassical level, Sec. III. Finally, in Fig. 5 we investigate the entanglement entropy saturation rate, i.e., how fast the subsystem thermalizes. We plot the distance |S(τ ) − S max | to the maximal value S max = L log 2 in time. It is known that in a case of global quench the entanglement entropy saturates as L 2 /t [18] (see also Appendix A). On the other hand, in a case of a Rindler quench, we can see that after times τ ≈m a subsystem thermalizes exponentially fast, which we find very intriguing: although the entanglement entropy grows sublinearly, it saturates much faster than in a homogeneous case. Comparison of the entanglement entropies for a Rindler quench for a free fermionic chain obtained from exact numerical data (blue) and from from the random unitary circuits model (red).

V. RANDOM UNITARY CIRCUITS
In the previous sections we have investigated the entanglement entropy evolution after the Rindler quench in a simple system of free fermions on a lattice. In particular, we have found that a simple heuristic semiclassical arguments (Sec. III) give us important insight into the universal behavior of entanglement dynamics, although it fails to quantitatively reproduce the numerical data (Sec. IV). The semiclassical arguments cannot account for multiparticle contribution to the entanglement, which is most probably the reason of quantitatively differences. On the other hand, such mutliparticle contribution can be captured in random unitary circuit setups, which provide minimally structured toy models of generic nonintegrable systems. For this reason, in this part we estimate the entanglement entropy for a specific system of random unitary circuits, where the coarse grained entanglement dynamics is equivalent to the minimal membrane description [43].
Let us denote S(x, t) for the entanglement entropy for an arbitrary quantum state, where x is a position of a bipartite cut. If, like in the previous section, we would rather consider a bipartition where a subsystem A is a finite segment A = [m 1 , m 2 ], then the entanglement entropy of such a bipartition is S A (t) = S(m 1 , t)+S(m 2 , t). After [43] we write down the equation for the leading order coarse-grained dynamics of the local rate of the entanglement entropy where Γ(s) is a production rate where α, γ are free parameters. Let us note that in general the above formula should be treated as Tyler expansion of the entanglement line tension, where the higher order terms do not contribute to the coarse grained dynamics as long as ∂ x S(x, t) is sufficiently small. Our goal is to describe the entanglement entropy dynamics in a Rindler quench scenario. To make it work, we simply notice that in Rindler Universe, Eq. (3), due to spatially varying speed of light c(x) = x, the local dynamics is the slower the closer are to x = 0, and so γ = βx should be proportional to x, yielding Now, by utilizing an ansatz for separation of variables it is straightforward to solve the corresponding differential equation which has an elementary solution Finally, then the entanglement entropy between A and its complement reads wherem = (m 1 + m 2 )/2 and τ =mt is a proper time of an observer located atm. As we have obtained from the semiclassical analysis, where S(τ ) =m(1 − e −τ /m ), see Sec. III, the entanglement entropy growth predicted by the random unitary circuit, Eq. (29), is linear at early times and attains a constant values at long times scales. On the contrary, we know that the entanglement entropy of random unitary circuit models could in principle capture multipartite contribution to the entanglement. Therefore, although the two formulas obey the same scaling s(τ ) = S(τm)/m and are qualitatively similar, we can expect quantitative differences both in the short and long time behavior. In particular, at short time scales the first non vanishing correction to the linear growth in S(τ ) is quadratic while in S A (τ ) the first correction is cubic. At long time scales it is straightforward to find that S A (τ ) ∝ 1−2e −2 √ αβτ /m . Notice that the entanglement entropy S A (τ ), Eq. (29), depends on two free parameters which correspond to different choices of the entropy production rate. Since α, β are two unknown model specific parameters, S A (τ ) should also recover the numerical results obtained for non-interacting fermions. Let us therefore fit s A (τ ) = S A (τm)/m to the numerical data from the Sec. IV. We plot the results in Fig. 6 and find that the curve remarkably agrees with the numerical data with fitting coefficients α ≈ 2.4, β ≈ 0.45. This suggests that our result captures universal properties of entanglement entropy dynamics after the Rindler quench.

VI. CONCLUSIONS
The main goal of this work was to study entanglement growth in inhomogeneous spacetimes, where correlations do not propagate within straight light cones. We have taken as an example the (1+1) Rindler metric, which is known to have a spacetime horizon that strongly distorts the light cones in its vicinity. We have shown that the entanglement initially grows sublinearly and in a longtime limit is asymptotically constant. This behavior can be qualitatively understood via semiclassical arguments that base on the knowledge of correlation spreading. Correlations spread within distorted light-cones that are described by null geodesics of the Rindler spacetime. This suggests that we have captured basic properties of the continuum theory and that the similar reasoning can be applied to other spacetimes.
We have found indications that our observations are universal. For the paradigmatic example of an ergodic system, we have studied the entanglement growth also for a random unitary circuit model. Choosing a specific random unitary circuit setup, we have derived a leading coarse-grained dynamics of the entanglement entropy that has the same characteristics as in a case of free fermions on the lattice. This suggests that our results are applicable to a generic quantum system. This opens the way towards studying entanglement production in more general inhomogeneous spacetimes. and v(p) = dE(p)/dp = cos(p) is a simple lattice dispersion relation for free particles. For a global homogeneous quench that we consider the main part of the article we obtain S(t) = S lin (t) , 0 ≤ t ≤ L/2 S sat (t) , t > L/2 , where S lin (t) = 4 ln 2 π t, S sat (t) = 2L ln 2 π arccos L 2t + 2t − √ 4t 2 − L 2 L .
(A4) From general considerations, Calabrese and Cardy [18] argue that at late times t L/2 the entanglement entropy S(t) saturates as ∝ L 2 /t. In our specific case, we can calculate the asymptotic behaviour exactly. The Taylor expansion of S sat (t) at small x = L/2t yields It is convenient to rescale s(t) ≡ S(tL)/L, which does not depend on L, such that the thermodynamic limit t, L → ∞, but t/L=const. is straightforward. The function s(t) belongs to the C 1 differentiability class, i.e., its derivative is not differentiable at t = t * . We plot s(t) together with the numerical data on Fig. 7. We see that even for relatively small L's the numerical data reproduces the analytic curve almost perfectly, where the divergence can be observed in a vicinity of t * . Yet, the finite-size analysis confirms non analytical behaviour of s(t) in the thermodynamic limit, see insets of Fig. 7. In the main text of this article, we have shown that the saturation effects of entanglement entropy after a Rindler quench is far from being universal. On Fig. 8, as before, we plot the first derivative of the entanglement entropy for different subsystem sizes L and fixedm = 200 , see Fig. 7 for a comparison. The derivative gets smoother with increasing L and the crossover point becomes less pronounced.
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