Inhomogeneous quenches as state preparation in two-dimensional conformal field theories

The non-equilibrium process where the system does not evolve to the featureless state is one of the new central objects in the non-equilibrium phenomena. In this paper, starting from the short-range entangled state in the two-dimensional conformal field theories ($2$d CFTs), the boundary state with a regularization, we evolve the system with the inhomogeneous Hamiltonians called M\"obius/SSD ones. Regardless of the details of CFTs considered in this paper, during the M\"obius evolution, the entanglement entropy exhibits the periodic motion called quantum revival. During SSD time evolution, except for some subsystems, in the large time regime, entanglement entropy and mutual information are approximated by those for the vacuum state. We argue the time regime for the subsystem to cool down to vacuum one is $t_1 \gg \mathcal{O}(L\sqrt{l_A})$, where $t_1$, $L$, and $l_A$ are time, system, and subsystem sizes. This finding suggests the inhomogeneous quench induced by the SSD Hamiltonian may be used as the preparation for the approximately-vacuum state. We propose the gravity dual of the systems considered in this paper, furthermore, and generalize it. In addition to them, we discuss the relation between the inhomogenous quenches and continuous multi-scale entanglement renormalization ansatz (cMERA).


D Vertex Operator Four-point Function in Free Dirac Fermion Boundary
State 35 1 Introduction A major trend in twenty-first-century theoretical physics is the extensive utilization of quantum information-theoretic ideas and techniques in a wide range of seemingly disparate subfields of theoretical physics.For example, in the study of holography [1], an understanding of the quantum mechanical properties possessed by the system is expected to lead to a deeper grasp of the subject [2,3].One such quantum information-theoretic concept is that of quantum error correction which originated from the field of quantum computing but has since been used for bulk reconstruction in holography [4,5] as well as the study of measurementinduced phase transition [6,7] by the condensed matter community.Another topic from the field of quantum computation that might be of relevance to condensed matter physics is the preparation of quantum states [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27].In particular, obtaining low entropy states will allow the simulation of exotic phases of matter ranging from antiferromagnetic spin liquids to hightemperature superconductors [28,29].One of the desired outcomes of state preparation is the preparation of these states faster than an adiabatic evolution of the system.In holography, preparing quantum states corresponds to the production of asymptotically AdS spacetimes.Since two-dimensional conformal field theories (2d CFTs) possess the infinite-dimensional Virasoro symmetry, they may allow an analytic treatment of vacuum state preparation.In this paper, we consider the inhomogeneous quenches induced by the so-called Möbius and sine-squared deformed (SSD) Hamiltonians in 2d CFTs [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] and explore the utility of these quenches in vacuum state preparation.The densities of these inhomogeneous Hamiltonians are modulated by envelope functions that vary in space.In addition, these inhomogeneous Hamiltonians can be thought of as the Hamiltonians of systems that exist on a curved spacetime described by a metric whose time component is determined by the envelope function.Originally, the modulation of energy density was used to remove the effect of the boundaries of spin systems with finite size [47][48][49].Subsequently, these inhomogeneous deformations were generalized to 2d CFTs [50,[50][51][52][53][54][55][56].These inhomogenous Hamiltonians were used in the implementation of 2d Floquet CFTs [35,36,40,41,57,58].In addition, they have also been used to explore a variety of non-equilibrium phenomena and quantum information-theoretic aspects of 2d CFTs [59][60][61][62][63][64] and non CFTs [65].
In [32], the authors found that the time evolution induced by the SSD Hamiltonian can be used to approximately prepare the vacuum state.In the setup considered in [32], the system begins in the thermal state and subsequently undergoes time evolution with the SSD Hamiltonian which has a spatial point where the envelope function vanishes.If the subsystem does not include this point, the entanglement entropy associated with this subsystem evolves to that of the vacuum state.Regardless of the spatial location and the size of subsystems, the mutual information evolves into the mutual information for the vacuum state.In [66], this setup is generalized to a Floquet time evolution, and the authors consider the cooling effect of the inhomogeneous Hamiltonian.
In previous studies, the SSD time evolution was found to deform the entanglement structure of the long-range entangled state, where the entanglement entropy is proportional to the subsystem size, so that the resulting reduced density matrix is approximately given by that of the vacuum state.In this paper, we will explore whether this vacuum entanglement structure emerges during time evolution with the SSD Hamiltonian, starting from a short-range entangled state.The short-range entangled state considered in this paper is the regularized boundary state [67,68].Since short-range entangled states are easily prepared in the laboratory, this study may pave the way for creating low entropy states using quantum quenches.

Summary
In this paper, starting from a short-range entangled state which we take to be the regulated boundary state, we evolve the system with Möbius/SSD Hamiltonians in 2d CFTs.The major difference from previous studies [8,9] is that we aim to explore whether vacuum states occur in CFTs with differing ability to scramble information which may determine the speed of information processing [69].Our findings are as follows: Entanglement entropy: During Möbius time evolution, the time dependence of entanglement entropy exhibits a periodic behavior in time called quantum revival.During the SSD time evolution, when the edges of subsystems are not located at the origin, regardless of the spatial location of the subsystems, at sufficiently late times, the entanglement entropies are approximated by that of the vacuum state.If the subsystem size, l A , is much smaller than the system size, L, the time regime for the entanglement entropy for the single intervals to cool down to the vacuum one is t 1 ≫ t * ≈ O(L L A /ϵ), where t 1 and ϵ are time and the parameter determining the short-range entanglement of the initial state.We argue that t * characterizes the time for the subsystem to cool down to the vacuum one for the SSD time evolution.In addition, when the edges of subsystems are located at the origin, for sufficiently large times, the entanglement entropy logarithmically grows with time in the holographic CFTs while it approaches the saturation value with a power law in the free fermion CFTs.Mutual information: During the SSD time evolution, in the large time regime, mutual information eventually saturates to that of the vacuum one for both the free fermion CFT and the holographic CFTs while it goes to zero in the quasiparticle picture.This indicates that non-local correlations emerge under time evolution by these inhomogeneous Hamiltonians.The time dependence of entanglement entropy and mutual information suggests that the SSD time evolution prepares the state with the entanglement structure and non-local correlations possessed by the vacuum state.Gravity dual: We propose the gravity dual of the system considered and also generalize it.In particular, the end of the world brane approaches to the asymptotic boundary and eventually collides with the cutoff surface.We argue that this is a time scale over which holographic calculations become unreliable.Furthermore, we discuss an interpretation of the Möbius/SSD time evolution as a type of tensor network called the continuous multi-scale entanglement renormalization ansatz (cMERA).

Organization of this paper
In Section 2, we will describe the details of the inhomogeneous quench, how to compute entanglement entropy in the twist operator formalism, and the evolution of local operators induced by the Möbius/SSD Hamiltonians.In Section 3, we will present the time evolution of entanglement entropy in the 2d Dirac fermion CFT and in the quasiparticle picture that captures the dynamics of entanglement in the Dirac fermion CFT, as well as the entanglement entropy for the single interval in the 2d holographic CFTs which possess a gravitational dual.In Section 4, we will report the time evolution of mutual information in 2d CFTs considered in this paper.In Section 5, we will present the gravity dual of the systems considered and give an interpretation of the quenches as a cMERA tensor network.In Section 6, we discuss the relation between the inhomogeneous quenches and renormalization group, and comment on future directions.

Inhomogeneous quenches from the boundary state
Here, we will describe the inhomogeneous quenches considered in this paper.Suppose that we prepare the system in the boundary state with a proper regularization, e −ϵH |Ψ 0 ⟩, [67] and then evolve it with inhomogeneous Hamiltonian H Inho : where N 2 is the normalization constant that guarantees that ⟨Ψ(t)|Ψ(t)⟩ = 1.The operator H is the homogeneous Hamiltonian defined as H = L 0 dxh(x), where h(x) is Hamiltonian density and L is the system size.One of the inhomogeneous Hamiltonians considered in this paper is the Möbius Hamiltonian defined as [36,57,58,70,71], where θ is a real parameter.The system considered in this paper is on the spatial circle with the circumference L. For ∞ > θ > 0, H Inho = H Möbius is called Möbius Hamiltonian, and in the SSD limit where θ → ∞, it reduces to so-called sine-squared deformed Hamiltonian (SSD Hamiltonian) defined as On the other hand, when θ = 0, H Möbius reduces to the homogeneous Hamiltonian.In this paper, by using the time dependence of entanglement entropy and mutual information during the evolution induced by the Möbius/SSD Hamiltonian in the two-dimensional conformal field theories (2d CFTs), we will explore the entanglement structure of the state.The entanglement entropy and mutual information are defined as the von Neumann entropy for the reduced density matrix and a linear combination of the entanglement entropies respectively.Denote the density matrix of the state as ρ(t) = |Ψ(t)⟩ ⟨Ψ(t)|.Divide the Hilbert space into A and its complement A, and define the reduced density matrix associated to A as ρ A (t) = tr A ρ(t).The entanglement entropy for A is defined as the von Neumann entropy of Let us now turn to the detailed definition of the mutual information.Divide the Hilbert space into A ∪ B and A ∪ B, define the entanglement entropies associated with A, B, and A ∪ B, and define the mutual information as the linear combination of these entanglement entropies, where S A , S B , and S A∪B are the entanglement entropies associated with A, B and A ∪ B.

The parameter region considered in this paper
The parameter regime where we explore the inhomogeneous non-equilibrium process in this paper is where l V denotes the size of the subsystem V and t 1 is the time associated with the Möbius/SSD Hamiltonian.

Entanglement entropy in the twist operator formalism
To employ the Euclidean path-integral formalism suitable to the analytical computation of the entanglement entropy in 2d CFTs, we define the Euclidean density operator as where ϵ is a regularization parameter, and τ 1 is a real Euclidean time.The normalization parameter N guarantees that trρ E = 1, and it satisfies Divide the Hilbert space into V and V, the complement space to V, and then define a reduced Euclidean density matrix associated with V as ρ E,V = tr V ρ E .Subsequently, define Euclidean entanglement entropy of V as the von Neumann entropy of ρ E,V : where we call S (n) E,V the n-th Rényi entanglement entropy.In the path-integral formalism, n-th Rényi entanglement entropy is given by tr V ρ n E,V which is the partition function on a n-sheeted geometry where each of sheets is a finite cylinder.
Let us assume that V is a single interval.In the twist operator formalism, tr V ρ n E,V is given by the two-point function of the twist and anti-twist operators.As a consequence, the n-th Rényi entanglement entropy is given by where v 1 and v 2 denote the edges of V, respectively.Here, T n and T n are the primary operators called the twist and anti-twist operators.Their conformal dimensions are Here, we assume that L > v 1 > v 2 > 0. As in [32], the evolution of the primary operator O(w, w) induced by Möbius/SSD Hamiltonian is given by dw New,α=0,1 dw hn dw New,α=0,1 dw hn O(w New,α=0,1 , w New,α=0,1 ), (2.10) where (w New,α=0,1 , w New,α=0,1 ) is the location of the operator during the Euclidean time evolution induced by the Möbius (α = 1) and SSD (α = 0) Hamiltonians, respectively.The details of (w New,α , w New,α ) are reported in Appendix A.1.The Euclidean Rényi entanglement entropy is given by the "free energy" of two point function on the finite cylinder where the length along Euclidean time is 2ϵ, and the circumference of the spatial circle is L, (2.11) In the von Neumann limit where n → 1, the Euclidean Rényi entanglement entropy reduces to the Euclidean entanglement entropy, (2.12)

Entanglement entropy for the single interval in 2d massless free fermion
The entanglement entropy of the free Dirac fermion after an inhomogeneous quench of a boundary state can be computed using bosonization similar to that in [72,73] but with the time evolution Hamiltonian replaced with the Möbius Hamiltonian.The n-th moment of the reduced density matrix for a single interval is where |Ψ 0 ⟩ is the boundary state that lives on the ends of the cylinder and there is a conformal factor that comes from rescaling the correlation function so that the spatial coordinate has a periodicity of L instead of 2π.Let y = τ − iσ where 0 ≤ τ ≤ 2ϵ and 0 ≤ σ ≤ 2π be the holomorphic coordinate on this rescaled cylinder.Applying the bosonization dictionary, the twist operators can be written as twisted vertex operators that depend on the boundary condition, The correlation function of vertex operators in the boundary state can be computed by writing the vertex operators and the boundary state in terms of the bosonic modes and by repeatedly applying the bosonic commutation and the Baker-Campbell-Haursdorff relations [72].The denominator is the partition function for the finite cylinder with boundary states |Ψ 0 ⟩ on both ends as is given by (2.15) Therefore, the correctly normalized Euclidean Rényi entanglement entropy to leading order in L ϵ is In this expression, as well as the subsequent expressions for Euclidean Rényi entropy and mutual information, the coordinates w New,α v i and wNew,α v i are analytically continued to real time τ 1 → it 1 .In the uniform θ = 0 case with spatial periodicity L = 2π, we recover the corresponding expressions in [72,73].The entanglement entropy for a boundary state at the initial time t = 0 for a subsystem of size |V| ≫ ϵ → v 1 − v 2 ≫ ϵ in the limit where L ≫ ϵ is given by which does not depend on the total system subsystem sizes since the boundary state possesses a low amount of entanglement [68].To compute the entanglement entropy of two intervals, the four-point function of the vertex operators is required.The calculation is a generalization of that in [72] and can be found in Appendix D.

If we have two intervals
then repeating the same calculation as for the single interval case gives (2. 18) where the universal part that does not depend on the boundary condition is and the non-universal part that depends on the boundary condition is where N and D stand for the Neumann and Dirichlet boundary conditions, respectively.
2.3 Entanglement entropy for the single interval in 2d holographic CFT

Method of image
In this paper, we employ the method of images to compute the Euclidean Rényi entanglement entropy for the holographic CFTs.Consider (2.11) for the single interval in this method.In this method, the two point function for the boundary state is given by the four point function of the original operators and their images.In the von Neumann limit, n → 1, the Euclidean entanglement entropy is given by where the length of the thermal cycle is 4ϵ, and the mirrors are located at τ = 0 and τ = 2ϵ.
Here, τ v i ,τ 1 ,α denotes the real part of w New,α v i and w New,α v i

, while
Lφv i ,τ 1 ,α 2π and denote the imaginary parts, respectively.The first term in the last equation of (2.21) is independent of the details of 2d CFTs, while the second term depends on those.We call the first and second terms the universal and non-universal pieces, respectively.In 2d holographic CFTs, the non-universal piece is determined by the geodesic length in on the background dual of the system considered.

Non-universal piece in 2d holographic CFT
Now, we focus on the non-universal piece of the entanglement entropy in 2d holographic CFT.By employing the method of images, the system to is equivalent to that on a spatial circle with circumference L in the thermal state with the inverse temperature 4ϵ (≪ L).In 2d holographic CFTs, the gravity dual of the system in the high temperature region, ϵ ≪ L, is given by the BTZ black hole, while that in the low temperature region, ϵ ≫ L, is given by the thermal AdS [74].Therefore, the non-universal piece in the high temperature region is given by the minimal length of geodesics in the BTZ black hole geometry [75,76]: where S con is the length of geodesics connecting the points on the different Euclidean time slices, while S dis is the length of geodesics connecting the points on the same Euclidean time slice.We present the details of S con and S dis , and they are given by 2. 4 The trajectory of the local operator during the time evolution induced by SSD/Möbius evolution slice.We present the details of S con and S dis , and they are given by

Location of the local operator during the time evolution induced by SSD/Möbius evolution
After performing the analytic continuation, ⌧ = it 1 , we consider the evolution of the twist and anti-twist operators during the time evolution induced by SSD/Möbius Hamiltonian.Define the spatial position of the these operators in the Heisenberg picture as where v i is the insertion point of these operators in the Schrödinger picture.If these operators are inserted at v i = 0, L/2, during the SSD time evolution, X New,↵ v i does not vary in time.Therefore, call them the fixed points, and let X 1 f and X 2 f denote 0 and L 2 , respectively.During the Möbius evolution, these operators in the Heisenberg picture periodically move in time between X 1 f and X ).Consequently, if the subsystem is a single interval including X 1 f , then during the SSD evolution, the e↵ective subsystem size in the Heisenberg picture defined as , monotonically grows with time.If the subsystem is a single interval excluding X 1 f , then during the SSD evolution, eventually shrinks with time.6 slice.We present the details of S con and S dis , and they are given by

Location of the local operator during the time evolution induced by SSD/Möbius evolution
After performing the analytic continuation, ⌧ = it 1 , we consider the evolution of the twist and anti-twist operators during the time evolution induced by SSD/Möbius Hamiltonian.Define the spatial position of the these operators in the Heisenberg picture as where v i is the insertion point of these operators in the Schrödinger picture.If these operators are inserted at v i = 0, L/2, during the SSD time evolution, X New,↵ v i does not vary in time.Therefore, call them the fixed points, and let X 1 f and X 2 f denote 0 and L 2 , respectively.During the Möbius evolution, these operators in the Heisenberg picture periodically move in time between X 1 f and X ).Consequently, if the subsystem is a single interval including X 1 f , then during the SSD evolution, the e↵ective subsystem size in the Heisenberg picture defined as , monotonically grows with time.If the subsystem is a single interval excluding X 1 f , then during the SSD evolution, eventually shrinks with time.

6
[a] Möbius evolution slice.We present the details of S con and S dis , an

Location of the local operator duced by SSD/Möbius evoluti
After performing the analytic continuation, ⌧ = and anti-twist operators during the time evolut Define the spatial position of the these operators where v i is the insertion point of these operators i are inserted at v i = 0, L/2, during the SSD tim Therefore, call them the fixed points, and let X 1 f a the Möbius evolution, these operators in the He between X 1 f and X 2 f (see [a] of Fig ??), while du and accumulate around that spatial point (see [a is a single interval including X 1 f , then during t size in the Heisenberg picture defined as eventually shrinks with t 6 slice.We present the details of S con and S S

Location of the local ope duced by SSD/Möbius ev
After performing the analytic continuatio and anti-twist operators during the time Define the spatial position of the these op where v i is the insertion point of these oper are inserted at v i = 0, L/2, during the S Therefore, call them the fixed points, and l the Möbius evolution, these operators in between X 1 f and X 2 f (see [a] of Fig ??), w and accumulate around that spatial point is a single interval including X 1 f , then d size in the Heisenberg picture defined as time.If the subsystem is a single interv The evolution of operators during the evolution induced by SSD/Möbius Hamiltonian.The black dots illustrate the fixed point of H SSD .The green (purple) dot illustrates the operator that is initially inserted at L/2 > x > 0 (L > x > L/2) and t 1 = 0, and its trajectory during the time evolution is illustrated by the green (purple) curve.In [a], we show the evolution of operators during the Möbius time evolution.In [b], we show the evolution of operators during the SSD time evolution.
After performing the analytic continuation, τ 1 = it 1 , we consider the trajectory of the twist and anti-twist operators during the time evolution induced by the SSD/Möbius Hamiltonian.Define the spatial position of these operators in the Heisenberg picture as where v i is the insertion point of these operators in the Schrödinger picture.If these operators are inserted at v i = 0, L/2, then during the SSD time evolution, X New,α does not vary in time.Therefore, call them fixed points, and let X 1 f and X 2 f denote 0 and L 2 , respectively.During the Möbius evolution, these operators in the Heisenberg picture periodically move in time between X 1 f and X 2 f (see [a] of Fig. 1), while during the SSD evolution, those move to X 2 f and accumulate around that spatial point (see [b] of Fig. 1).Consequently, if the subsystem is a single interval including X 1 f , then during the SSD evolution, the effective subsystem size in the Heisenberg picture defined as , monotonically grows with time.If the subsystem is a single interval excluding X 1 f , then during the SSD evolution, eventually shrinks with time.

The time dependence of entanglement entropy in 2d CFTs
In this section, we will report the time dependence of entanglement entropy in 2d CFTs.

The time evolution of entanglement entropy in 2d free fermion
As examples of non-chaotic dynamics, we compare the Rényi entropy of a single interval in the free fermionic CFT as well as in the quasiparticle picture which will be discussed later in Section 3.2.The plots for three different subsystems centered at three different points along the spatial circle are shown in Fig. 2. Let us just summarize the salient points of these plots.First, the CFT result and the quasiparticle entanglement agree to the leading order in 1 ϵ .The subleading difference is most apparent when the Rényi entropy after an SSD quench decays to the vacuum value for the CFT but it decays to zero for the quasiparticles since all quasiparticles will accumulate at the fixed point X 1 f at late times and hence the entanglement entropy will asymptotically decay to zero.The behavior of these free theories during Möbius evolution is very similar to that for the holographic CFTs except for the fact that the entanglement entropy of quasiparticles drops down to zero at integer multiples of L 2 cosh 2θ and the entanglement entropy of the free fermion CFT similarly drop downs to a small value.The entanglement entropy of quasiparticles must decay back to zero at integer multiples of L cosh 2θ because the quasiparticles return to their initial positions at those times.At half-integer multiples of L 2 cosh 2θ, the left and right moving partner of each Bell pair meet up and so give no contribution to the entanglement entropy.Secondly, the plots for both boundary conditions for the free fermion CFT appear to be identical.The boundary condition dependent terms in the free fermion CFT Rényi entropy are either zero or tiny compared to the total entropy in most cases except for the case where the interval ends exactly on the fixed point X 1 f in which case the boundary condition-dependent component is actually larger than the boundary condition-independent part for the larger values of θ.In any case, the upshot is that the plots appear identical for both boundary conditions in the free fermion CFT and hence do not appear to depend on the boundary condition imposed on the Dirac fermions.
Just as in the holographic case, to obtain a non-zero value of the entanglement entropy for the quasiparticles and free fermion CFTs at late times, place the interval so that one of the endpoints sits exactly on the SSD fixed point X 1 f so that exactly one member of each Bell pair in the quasiparticle picture is contained inside the subsystem at late enough times.

Quasiparticle Picture
The quasiparticle picture for the uniform global quench [77][78][79] can be extended to the inhomogeneous case by assuming that the quasiparticle are not moving with uniform speed but instead with a speed that is determined by the inhomogeneous envelope function that appears in the inhomogeneous Hamiltonian.This might be due to the fact that the CFTs considered in this paper are defined on curved backgrounds where the time component of metric, which determines the speed of moving objects, is given by the inhomogeneous envelope function.During the Möbius evolution, a quasiparticle that begins at position x 0 at time t 0 is located at x at time t 1 which is given by where the +(−) sign correspond to right(left) moving quasiparticles.For simplicity, set the initial time t 0 = 0.For a subsystem A, which could generally be a union of intervals, the entanglement entropy as predicted by the quasiparticle picture is given by the number of Bell pairs shared between A and its complement.Let x 0,i (x, t 1 ) be the initial position of a quasiparticle situated at position x at time t 1 , where i = R, L denotes the chirality of the quasiparticles.For additional details, see [32].Assuming that the quasiparticles are conserved, the rightmoving and left-moving quasiparticles that are inside A at a given time t 1 are therefore initially situated inside x 0,R (A, t 1 ) and x 0,L (A, t 1 ) at t = 0, respectively.A simple example for the uniform Hamiltonian is depicted in Fig. 3. Similarly, the left and right moving quasiparticles that end up in the complement of A, A, initially began in the subsystems x 0,L (A, t 1 ) and x 0,R (A, t 1 ), respectively.
While the distribution of quasiparticles after a time evolution with an inhomogeneous Hamiltonian is complicated [32], the quasiparticles are initially distributed uniformly.Since the entanglement entropy of A is given by the Bell pairs that it shares with its complement, Figure 2: Plots of the second Rényi entropy for the 2d free fermion CFT as well as the quasiparticle picture for a total system size of L = 100000, ϵ = 10 and a subsystem size of l A = 6000 with a center P c located at various positions.
Figure 3: For a given interval A, x 0,R (A, t 1 ) and x 0,L (A, t 1 ) are the intervals that will move respectively to the right and left and will coincide with A at time t 1 .Shown here is the simplest case where the Hamiltonian is uniform so the intervals are translated with unit speed.
the entanglement entropy can simply be written as where ρ 48ϵ can be fixed by equating the saturation value of S (n) A for the uniform Hamiltonian for a single finite interval on the real line with the known CFT result [77] to leading order in 1 ϵ and using the fact that for the uniform Hamiltonian, x 0,R (A, t 1 ) and x 0,L (A, t 1 ) are entirely contained inside x 0,L (A, t 1 ) and x 0,R (A, t 1 ) respectively at sufficiently late times.
The quasiparticle picture entanglement entropy appears to vanish when the time t 1 is a multiple of L 2 cosh 2θ.This is obvious when t 1 is a multiple of L cosh 2θ since all the quasiparticles would have returned to their original position so the entanglement entropy reverts back to its original value of 0. When t 1 is a half-integer multiple of L cosh 2θ, the quasiparticle entanglement entropy vanishes not because the quasiparticles have returned to their initial positions but because the left and right moving partners of each Bell pair have met up after traversing the spatial circle on opposite directions.
We can also compute the mutual information as predicted by the quasiparticle picture.The mutual information between two subsystems A and B is given by twice the number of Bell pairs that have one partner in each system since these Bell pairs contribute to both

Late time behaviour of free theories when the interval ends on X 1 f
The Rényi entropy for a subsystem that ends on the fixed point In the late time regime of interest t 1 ≫ L ≫ ϵ, the boundary condition-dependent terms in the free fermion CFT entanglement entropy are exponentially suppressed and the late time entanglement entropy of a single interval that ends on the fixed point when either boundary state is quenched by the SSD Hamiltonian is This is very similar to the late time behaviour of the entanglement entropy of a single interval that ends on the fixed point X 1 f for the quasiparticles undergoing going a SSD quench.The only difference is the additional factor of 2 in the power law 1 t 1 decay to the saturation value which is subleading in L ϵ .The saturation value also has the additional vacuum Rényi entropy which is zero in the quasiparticle case as discussed earlier.
The discrepancy between the quasiparticle entanglement entropy and the CFT Rényi entropy can be attributed to the order of the limits taken.The term that gives rise to the late time saturation is If we first send L ϵ → ∞ in (3.6) This is the same answer as for the quasiparticles (3.4).On the other hand, if we first send which corresponds to the asymptotic expression we found earlier.Therefore, we see that the discrepancy comes from the fact that the quasiparticles correspond to taking the ϵ → 0 limit of the CFT first and so would not truly capture the t 1 → ∞ behaviour of the CFT.

The time evolution of entanglement entropy in 2d holographic CFT
We focus on the analysis on the time dependence of entanglement entropy and mutual information in 2d holographic CFTs.

The analysis of non-universal piece
Now, we perform the analytic continuation to real time, τ 1 = it 1 , and report the time dependence of the non-universal piece of entanglement entropy in 2d holographic CFTs.As explained in Section 2.3, the time dependence of the non-universal piece is determined by that of the geodesic length on the BTZ black hole geometry.The t 1 -dependence of S V;con is determined by the length of the geodesic connecting the points at the different Euclidean time slices in the method of image.Without employing the method of the image, the t 1dependence of this geodesic length should be equal to that of the geodesic length ending at the end of the world brane (EoW).In the high temperature regime, the location of EoW in the radial direction is near the boundary.In the parameter regime, where v 1 − v 2 ≫ ϵ, the geodesic length is given by the one ending at the EoW brane.Consequently, in the small time region, t P > t 1 > 0, the t 1 -dependence of the non-universal piece of S A is determined by the motion of the EoW.Here, we define t P as the time when S V;dis exchanges the dominance with S V;con .We will present the detailed motion of the EoW in Section 5. We divide the system into A and A, the complement to A, let X i=1,2 denote the edges of A, and assume L > X 1 > X 2 ≥ 0. We will report on the time dependence of entanglement entropy during the Möbius/SSD time evolution in the four cases: (1) In the cases considered in this paper, S V;con monotonically grows in time according to the motion of EoW [59].For the large t 1 -regime, t 1 > t P , the t 1 -dependence of the non-universal piece is determined by the trajectory of twist and antitwist operators during the time evolution induced by the inhomogeneous Hamiltonians.As a simple example, let us consider the t 1 -dependence of the non-universal piece for the reduced density matrix associated with the subsystem including X 1 f during the evolution induced by H SSD .In t P > t 1 > 0, the t 1 -dependence of the non-universal piece is determined by that of S V;con .In t 1 > t P , the t 1 -dependence of the non-universal piece is determined by the smaller one of S V;dis,1 and S V;dis,2 .We define an effective subsystem size in the Heisenberg picture as In the time-regime where L/2 > V eff., the non-universal piece is given by S V;dis,1 , while in V eff.> L/2, it is determined by S V;dis,2 .

During the Möbius/SSD time evolution
Now, let us present the time dependence of S A during the Möbius/SSD time evolution.For all the cases considered, S A;con monotonically grows with time during the Möbius/SSD evolution (see Appendix B).The exchanging time t 1 = t P should be determined by f , respectively.The black dashed line illustrates the entanglement entropy of the thermal state with 4ϵ, the inverse temperature, for half of the total space.The gray and pink dashed lines illustrate S A of this thermal state and the vacuum state.
After t 1 = t P , the non-universal piece of S A is determined by S dis,i .
We depict S A in cases (1), (2), and (3) as a function of t 1 in Fig. 4.During the Möbius evolution in the cases (1), (2), and (3), the time-dependence of S A exhibits the two timeregimes: a heating one for 0 < t < t P , and an oscillating one of t P < t.In the heating regime, the growth of S A depends on the inhomogeneous parameter θ, the subsystem size l A , and the system size L. Regardless of time evolution considered, in the limit where π w New,α X i + w New,α X i /4iϵ ≫ 1, the early time behavior of S A should be determined by the non-universal piece of S A , where the details of φ and φ are reported in Appendix A.1.During the Möbius time evolution, φ X 1 ,τ 1 ,1 +φ X 1 ,τ 1 ,1 monotonically grows with time.In the time interval, 1 ≫ φ X i ,τ 1 ,0 +φ X i ,τ 1 ,0 ≫ 4πϵ L , the early-time growth of S A is approximated by Thus, in this time interval, S A linearly grows with t 1 as in the homogeneous quench [79,80].However, the coefficient of the linear growth is given by multiplying that of the homogeneous quench by an additional factor of the envelope function, 1 − tanh 2θ cos 2πX i L .In the oscillating regime, S A periodically behaves in time with the period L cosh 2θ.During the evolution induced by the SSD Hamiltonian (α = 0), the time dependence of S A exhibits the two time-regimes: a heating one for 0 < t < t P , and a cooling one for t P < t.As in the case of the time evolution in cases (1), (2), and (3), S A in the heating regime monotonically grows with t 1 .In the limit where π w New,0 X i + w New,0 X i /4iϵ ≫ 1, the early time behavior of S A is given by (3.11) for α = 0.During the SSD time evolution, φ X i ,τ 1 ,0 + φ X i ,τ 1 ,0 monotonically increases with t 1 .In the time interval, 1 ≫ φ X i ,τ 1 ,0 + φ X i ,τ 1 ,0 ≫ 4πϵ L , the early-time growth of S A is approximated by Thus, in this time interval, S A linearly grows with t 1 .The coefficient of the linear growth is given by multiplying that for homogeneous quench by the envelope function, 2 sin 2 πX i L .In the cooling regime, the universal and non-universal pieces of S A are asymptotically given by Thus, the universal and non-universal pieces logarithmically grow and decrease with t 1 , respectively.Since the logarithmic growth cancels with the logarithmic decrease, S A for large t 1 becomes independent of t 1 .As a consequence, S A for the large t 1 is approximated by the vacuum entanglement entropy, Unlike the case of the thermal state in [32], even if the subsystem includes x = X 1 f , the entanglement entropy saturates to that for the vacuum one, not the thermal one.

Thermal configuration
As in the cases (1)-( 3), if the edge of A is not at x = X 1 f , then S A is asymptotically approximated by the vacuum entanglement entropy.Now, we consider the case (4), the case where the edge of A is at x = X 1 f .During the Möbius evolution, the t 1 -dependence of S A is similar to the one in the cases (1)-( 3).During the SSD evolution, that of S A in case ( 4) is different from that in (1)-(3).In the SSD limit where θ → ∞, S A monotonically grows with t 1 , and then it is, for large t 1 , approximated by where the first term is the logarithmic function of t 1 , while the second term is the entanglement entropy of thermal entropy with 4ϵ for the half space.Thus, in case (4), S A is not asymptotically approximated by the vacuum entanglement entropy.In Fig. 5, we depict S A as a function of t 14 .

Cooling time
We close this section by defining the cooling time that describes how fast the subsystem evolves to the vacuum state during the SSD time evolution.In 2d holographic CFT, except for case (4), the entanglement entropy for the single intervals, A, is approximated by the vacuum one in the late time regime that is defined by where l A denotes the size of the subsystem, A. Thus, we call t * the cooling time.In In this figure, we take X 2 to be X 1 f .The black dashed line illustrates the entanglement entropy for half of the total space of the thermal state with 4ϵ, the inverse temperature.The gray and pink dashed lines illustrate S A of this thermal state and the vacuum state.entropy following this picture is smaller than O(1).This suggests that even during the SSD time evolution of 2d free fermion, t * characterizes the time for the subsystem to cool down to the vacuum state.
Incidentally, this is also the time scale it takes for the quasiparticle Rényi entropy to decay to an O(1) value.For intervals that are located away from the origin X 1 f so that 0 < X 2 < X 1 < L, at late times, when t 1 ≫ L, the Rényi entropy of the quasiparticles are approximately given by For a subsystem that is much smaller than the total system, the quasiparticle Rényi entropy becomes O(1) when t ∼ t * as defined in (3.17).

The time evolution of mutual information in 2d CFTs
Now, we consider the time dependence of mutual information during the SSD/Möbius evolution to see if the non-local correlation measured by the mutual information is also asymptotically approximated by that of the vacuum state.Divide the system into the subsystems, A and B, and A ∪ B, the compliment to the union of A and B, and then define the mutual information as the linear combination of entanglement entropies, where S V=A,B,A∪B denotes the entanglement entropies for the reduced density matrix associated with V = A, B, A ∪ B, respectively.Let X 1 and X 2 denote the edges of A, and let Y 1 and Y 2 denote the edges of B. Here, we assume that /c /c or L > X 1 > Y 1 > Y 2 > X 2 > 0, and A does not overlap with B. In this case, only the non-universal pieces of S V contribute to the mutual information because the universal pieces cancel out with each other.

Time dependence of I A,B in 2d free fermion
Consider the mutual information of the boundary state under these inhomogeneous quenches.The quasiparticle picture describes the second Rényi mutual information of the free Dirac CFT well as seen in Fig. 6.Just as for the second Rényi entropy, the second Rényi mutual information does not depend on the boundary condition.The mutual information also shows the periodicity of L cosh 2θ which goes to infinity in the SSD limit.For the setups shown in figure 6, the late time second Rényi mutual information of the free fermion boundary states after the SSD quench is given by the second Rényi mutual information for the vacuum state which is similar to the SSD quench of the spatially uniform thermal state [32].When both subsystems are symmetrically placed away from the fixed point X f 1 as seen in the first plot [a], the uniform quench produces non-zero mutual information when the Bell pairs that are initially nearly equidistant from both subsystems enter these subsystems.As these quasiparticles and their partners make their way around the spatial circle, they pass through both subsystems giving rise to two peaks in the first period.The quasiparticles that begin at around L/2 and X f 1 reach both subsystems at about the same time which is no longer the case for Möbius quenches where the Bell pairs that begin at L/2 reach the subsystems earlier than those that begin near X f 1 , leading to a small second peak.The members of the quasiparticle pairs that begin near X f 1 then go on to enter their second subsystem, leading to a third small peak.Finally, when the members of the quasiparticles that begin near L/2 enter their second subsystem, we see a fourth peak in the first period.In the SSD limit, we only observe a single peak since the quasiparticles are not able to go past the fixed point X 1 f .For both Möbius and SSD quenches, as the deformation parameter θ is increased, the speed about L/2 is greater so the first peak occurs earlier.The mutual information for the other two setups has peaks that are less symmetric but are nevertheless well-described by the quasiparticle picture.

Time dependence of I A,B in 2d holographic CFT
Now, we focus on the non-universal piece of S A∪B in 2d holographic CFT.We begin by computing the non-universal piece of S A∪B in the Euclidean space.In AdS/CFT correspondence, S A∪B is determined by the minimal geodesic length [75,81]: where L A∪B,con is the length of geodesic ending at the EoW brane, while L A∪B,dis is that of geodesic connecting two points at the same Euclidean time slice.These non-universal pieces, L A∪B,con and L A∪B,dis , are given by where L A;dis,i=1,2 and L A;con,i=1,2 are defined in Appendix C.After performing the analytic continuation, τ 1 = it 1 , the minimal geodesic length determines the time dependence of I A,B .In the early time-regime where the non-universal pieces of S V=A,B,A∪B are determined by S V,con , S A∪B cancels out with S A + S B .Consequently, in this early time-regime, I A,B is zero.
For the late time-regime where the non-universal pieces of S V are determined by S V,dis , the time-evolution of I A,B is determined by For the large t 1 -regime in the SSD limit, I A,B asymptotically reduces to the mutual information for the vacuum state, When the endpoint of A or B is at x = X 1 f , I A,B is given by (4.5) for X 1 f / ∈ A. In conclusion, when starting from a boundary state and time-evolving it with the SSD Hamiltonian, the entanglement entropy does not strictly approach that of the vacuum state.However, during the SSD time evolution, the mutual information may exactly approach that of the vacuum state.Furthermore, unless x = X 1 f at the edges of A and B, the reduced density matrices for V = A, B, A ∪ B are asymptotically approximated by the vacuum reduce density matrices where ρ Vacuum V is the vacuum reduced density matrices associated to V.

Gravitational description and cMERA interpretation
In this section, we will report on the gravity dual of the system considered in this paper.
In addition to it, we will discuss an interpretation for the SSD time evolution operator as a continuous multi-scale entanglement normalization ansatz (cMERA) [82][83][84].

Gravitational description
The early-time dependence of S A is determined by that of geodesic ending at the end of the world (EoW) brane.Here, we discuss the gravity dual of the deformed boundary state, especially the trajectory of the EoW brane.In the AdS/BCFT correspondence [85,86], the boundary effects we have discussed are explained by the insertion of the EoW brane into the bulk spacetime.The EoW brane is characterized by the brane tension T , which determines the boundary entropy in the calculation of the holographic entanglement entropy.We consider three-dimensional Einstein gravity on asymptotically AdS space M with the dynamical brane located on Q, where we fixed the AdS radius to be unity, and G N is Newton's constant.On the one hand, we impose the Dirichlet boundary conditions on the metric at the asymptotic boundary ∂M .On the other hand, we impose the Neumann boundary condition on Q which is necessary to consider the dynamical brane.Note that we have already fixed the matter profile on Q, characterized by the constant brane tension T , so that the boundary conformal symmetry is preserved.
As a solution of Einstein's equation, we obtain the BTZ black hole metric with the EoW brane, whose (Euclidean) trajectory is given by where −1 < T < 1.Throughout this subsection, we assume the length of circumference is 2π.In the CFT part, we have discussed only the case T = 0, while here we will discuss the more general value of T .For positive tension T > 0, the EoW brane is located on the other asymptotic boundary of a maximally extended solution.On the other hand, for negative tension T < 0, the EoW brane is outside of the horizon.In both cases, the brane eliminates the spacetime behind it, as the name suggests.In what follows, we mainly focus on the point of view of the boundary observer.Therefore, we shall discuss non-positive tension brane.
Let us describe the trajectory of EoW brane on the deformed black hole geometry.To this end, we follow the prescription discussed in [32].Namely, we rescale the radial coordinate r by the conformal factor and use the w New coordinates discussed in Appendix A.1.After this replacement, we perform the analytic continuation to the Lorentzian time t 1 .Consequently, the t 1 -dependence of radial location is determined by where τ New (t 1 , x) = (w New,α x + wNew,α x )/2.Note that since the new coordinates depend on both t 1 and x, the brane trajectory also acquires the spatial inhomogeneity.See Fig. 7 for a schematic picture of the motion of the EoW brane from the outside observer.In Figs. 8  and 9, we plot the spatial and time dependence of brane trajectories determined by (5.4) in SSD limit (α = 0, see Fig. 8) and Möbius Hamiltonian (α = 1, see Fig. 9) with small θ.It is worth noting that for sufficiently large θ, the time dependence reduces to the one in the SSD limit.
The existence of EoW brane becomes clear when we discuss holographic entanglement entropy with a large subsystem compared with the inverse temperature β = 2π/r + (= 4ϵ).In this case, the holographic entanglement entropy is calculated from the phase where the minimal surfaces end on the EoW brane.
As a reference, we also plot the geodesic distance between the location of the EoW brane with negative tension and horizon as L bh , although our true geometry ends at the EoW brane.See Fig. 10.These figures suggest that the size of the domain eliminated by the EoW brane comes to depend on the location.In particular, such regions are more likely to be eliminated in the early time.
Empty EoW the unentangled state to the vacuum state of the uniform Hamiltonian H.

Gravitational picture
Let us explain the motion of EoW in terms of cMERA language.The spacetime behind the EoW is empty.In other words, the spacetime in the region 0  r  rEoW is empty.At t = 0, the EoW is located at r = r0.As in Fig. ??, if the spacial location is closer and closer to x = L 2 , the radial position of EoW becomes smaller and smaller in the SSD evolution.The region where the entanglement structure of the time-evolved state is approximated by the vacuum one expands from x = L 2 thanks to cMERA of SSD evolution.In the late time region, the redial location of EoW around x = 0 is near the UV surface, so that the late-time entanglement structure near the x = 0 is di↵erent from the vacuum one.When the state su ciently evolve by SSD evolution operator, the distance between two peaks of EoW is at most O(✏).Since the e↵ective lattice spacing is given by ✏ in the coarse grained limit, the structure of entanglement at only x = 0 is di↵erent from the vacuum one.Therefore, if we finally remove the site at x = 0 somehow, the entanglement structure of the SSD-evolved state is the same as the vacuum state. 11 In [18], the authors has studied the cMERA from an unetangled state to the vacuum state.As in [20], the boundary state is an unentangled state, and the late-time entanglement structure of the late-time evolved state is almost the same as the vacuum state except for the origin.At least, in the late time region, the time evolution of entanglement entropy and mutual information is approximated by those for the vacuum state.In other words, even the entanglement structure depending on the detail of CFTs is approximated by that of the vacuum state.Therefore, if we interpret the SSD time direction as the renormalization direction, the SSD time evolution operator is approximately the sane as the cMERA from the unentangled state to the vacuum state of the uniform Hamiltonian H.

Gravitational picture
Let us explain the motion of EoW in terms of cMERA language.The spacetime behind the EoW is empty.In other words, the spacetime in the region 0  r  r EoW is empty.At t = 0, the EoW is located at r = r 0 .As in Fig. ??, if the spacial location is closer and closer to x = L 2 , the radial position of EoW becomes smaller and smaller in the SSD evolution.The region where the entanglement structure of the time-evolved state is approximated by the vacuum one expands from x = L 2 thanks to cMERA of SSD evolution.In the late time region, the redial location of EoW around x = 0 is near the UV surface, so that the late-time entanglement structure near the x = 0 is di↵erent from the vacuum one.When the state su ciently evolve by SSD evolution operator, the distance between two peaks of EoW is at most O(✏).Since the e↵ective lattice spacing is given by ✏ in the coarse grained limit, the structure of entanglement at only x = 0 is di↵erent from the vacuum one.Therefore, if we finally remove the site at x = 0 somehow, the entanglement structure of the SSD-evolved state is the same as the vacuum state. 11

AdS
[a] Initial region the unentangled state to the vacuum state of the uniform Hamiltonian H.

Gravitational picture
Let us explain the motion of EoW in terms of cMERA language.The spacetime behind the EoW is empty.In other words, the spacetime in the region 0  r  rEoW is empty.At t = 0, the EoW is located at r = r0.As in Fig. ??, if the spacial location is closer and closer to x = L 2 , the radial position of EoW becomes smaller and smaller in the SSD evolution.The region where the entanglement structure of the time-evolved state is approximated by the vacuum one expands from x = L 2 thanks to cMERA of SSD evolution.In the late time region, the redial location of EoW around x = 0 is near the UV surface, so that the late-time entanglement structure near the x = 0 is di↵erent from the vacuum one.When the state su ciently evolve by SSD evolution operator, the distance between two peaks of EoW is at most O(✏).Since the e↵ective lattice spacing is given by ✏ in the coarse grained limit, the structure of entanglement at only x = 0 is di↵erent from the vacuum one.Therefore, if we finally remove the site at x = 0 somehow, the entanglement structure of the SSD-evolved state is the same as the vacuum state. 11 In [18], the authors has studied the cMERA from an unetangled state to the vacuum state.As in [20], the boundary state is an unentangled state, and the late-time entanglement structure of the late-time evolved state is almost the same as the vacuum state except for the origin.At least, in the late time region, the time evolution of entanglement entropy and mutual information is approximated by those for the vacuum state.In other words, even the entanglement structure depending on the detail of CFTs is approximated by that of the vacuum state.Therefore, if we interpret the SSD time direction as the renormalization direction, the SSD time evolution operator is approximately the sane as the cMERA from the unentangled state to the vacuum state of the uniform Hamiltonian H.

Gravitational picture
Let us explain the motion of EoW in terms of cMERA language.The spacetime behind the EoW is empty.In other words, the spacetime in the region 0  r  r EoW is empty.At t = 0, the EoW is located at r = r 0 .As in Fig. ??, if the spacial location is closer and closer to x = L 2 , the radial position of EoW becomes smaller and smaller in the SSD evolution.The region where the entanglement structure of the time-evolved state is approximated by the vacuum one expands from x = L 2 thanks to cMERA of SSD evolution.In the late time region, the redial location of EoW around x = 0 is near the UV surface, so that the late-time entanglement structure near the x = 0 is di↵erent from the vacuum one.When the state su ciently evolve by SSD evolution operator, the distance between two peaks of EoW is at most O(✏).Since the e↵ective lattice spacing is given by ✏ in the coarse grained limit, the structure of entanglement at only x = 0 is di↵erent from the vacuum one.Therefore, if we finally remove the site at x = 0 somehow, the entanglement structure of the SSD-evolved state is the same as the vacuum state.Let us explain the motion of EoW in terms of cMERA language.The s EoW is empty.In other words, the spacetime in the region 0  r  t = 0, the EoW is located at r = r0.As in Fig. ??, if the spacial locatio to x = L 2 , the radial position of EoW becomes smaller and smaller in The region where the entanglement structure of the time-evolved state the vacuum one expands from x = L 2 thanks to cMERA of SSD evoluti region, the redial location of EoW around x = 0 is near the UV surface, entanglement structure near the x = 0 is di↵erent from the vacuum o su ciently evolve by SSD evolution operator, the distance between two most O(✏).Since the e↵ective lattice spacing is given by ✏ in the coar structure of entanglement at only x = 0 is di↵erent from the vacuum o finally remove the site at x = 0 somehow, the entanglement structure state is the same as the vacuum state. 11 In [18], the authors has studied the cMERA from an unetangled state to t state.As in [20], the boundary state is an unentangled state, and the late-time en structure of the late-time evolved state is almost the same as the vacuum state the origin.At least, in the late time region, the time evolution of entanglem and mutual information is approximated by those for the vacuum state.In o even the entanglement structure depending on the detail of CFTs is approximate the vacuum state.Therefore, if we interpret the SSD time direction as the reno direction, the SSD time evolution operator is approximately the sane as the cM the unentangled state to the vacuum state of the uniform Hamiltonian H.

Gravitational picture
Let us explain the motion of EoW in terms of cMERA language.The spacetime EoW is empty.In other words, the spacetime in the region 0  r  r EoW is t = 0, the EoW is located at r = r 0 .As in Fig. ??, if the spacial location is close to x = L 2 , the radial position of EoW becomes smaller and smaller in the SSD The region where the entanglement structure of the time-evolved state is appro the vacuum one expands from x = L 2 thanks to cMERA of SSD evolution.In th region, the redial location of EoW around x = 0 is near the UV surface, so that th entanglement structure near the x = 0 is di↵erent from the vacuum one.Whe su ciently evolve by SSD evolution operator, the distance between two peaks o most O(✏).Since the e↵ective lattice spacing is given by ✏ in the coarse graine structure of entanglement at only x = 0 is di↵erent from the vacuum one.Ther finally remove the site at x = 0 somehow, the entanglement structure of the S state is the same as the vacuum state.

11
[c] Late time region

Comments on cut off surface
As we have seen so far, the EoW brane approaches the asymptotic boundary.At sufficiently late time, t ≫ L, the location of "peak of horizon" is linearly grows in time [32], This is also the case for the brane because the distance between the brane and horizon becomes a constant value at the late time.
On the other hand, we should introduce a cutoff surface on a certain radial scale r ′ = r ∞ that determines the UV cutoff for the dual CFT.It means that the EoW will eventually collide with the cutoff surface at some time.We interpret this as the time scale on which the holographic calculation in Section 3.3.3becomes unreliable 5 .More concretely, we cannot trust our calculation after t ∼ L/r + ϵ U V .Here we introduced the cutoff length scale in the dual CFT ϵ U V such that r ∞ = 1/ϵ U V .

cMERA interpretation
Here, we discuss an interpretation of the Möbius/SSD time evolution operator as a continuous multi-scale entanglement normalization ansatz (cMERA).
cutoff surface locally around the entangling surface.Note that this is a natural prescription from the CFT side [87] and its gravity dual [88].

A brief review of MERA and cMERA
In this section, we begin by reviewing the MERA shortly, and subsequently review the cMERA.The MERA is the scheme of the renormalization group in terms of a tensor network constructed of the two kinds of tensors [89][90][91].This is suitable for the (numerical) computation in the discrete system at the critical point.The tensor network in the MERA has a discrete layered structure where we have the discrete energy scale direction perpendicular to the space-time direction.We label this discrete energy scale direction by u, and we assume that u = 0, −1, • • • , −∞.On each layer at u, the tensor network is constructed of two types of tensors, called isometry and (dis-)entangler.Let L and K denote the isometry and entangler.The isometry is a linear map where two nearest spins at u reduce to a single spin at u − 1.Since this resembles the coarse-graining or the scale transformation, the isometry is considered as these operations in MERA.If the dimension of Hilbert space at u = 0 is 2 L , where L is the system size, then the one at u is estimated with 2 2 u L .The entangler is the unitary operator acting the two nearest spins at u.This tensor endows the state with short-range entanglement.We start from an un-entangled state |Ω⟩ that is invariant under the scale transformation generator L, and then non-unitarily evolve the system from u = −∞ to u = 0 with the circuit constituting of the isometries and entanglers.We call |Ω⟩ the reference state, while we call the state at u = 0 the target state.We tune the parameters of isometries and entanglers so that the target state is the state of interest.Let us begin the review of cMERA.The cMERA is designed to be suitable for the (numerical) computation in the quantum field theories at the critical point.Define the state at u = −∞ as where L is the scale transformation generator as in the MERA.However, in the cMERA, the isometry is replaced with a unitary operator.Since the isometry and entangler preserve the dimension of Hilbert space at each step u, the tensor network in the cMERA is a unitary evolution operator.We assume that the energy scale is labeled by the continuous parameter u.Consequently, under the cMERA, the system unitarily evolves with the unitary evolution operator to the target state, where P is defined as the path-ordering operator arranging the operators in order of increasing u from right to left.The symbols, K(u) and L, are defined as the integrals of entangler and isometry densities along the spatial directions, respectively.Furthermore, define the state and entangler in the interaction picture, |Ψ I (u)⟩ and K(u), as (5.9) Consequently, the target state in the interaction picture is given by (5.10) Thus, |Ψ I (0)⟩ does not depend on the definition of L.

A cMERA interpretation on the Möbius/SSD time evolution
Now, we turn to the quenches induced by Möbius/SSD Hamiltonians, and discuss the interpretation for these quenches as the cMERA.As in [92], we employ the boundary state with the proper regularization as the reference state, where the Hamiltonian H is defined as H = (2π L 0 + L 0 )/L − (cπ)/6L, where L n and L n are chiral and anti-chiral Virasoro generators.In addition to them, the boundary state is defined as (5.12) In this paper, unlike the common procedure in cMERA, we define the operator that keeps the reference state invariant as L. We utilize the spin operator 2π L 0 − L 0 /L as L. Subsequently, we consider t 1 during the time evolution as the energy scale u in the tensor network as in [93,94].We start from the boundary state |Ω⟩ at t 1 = −U IR and evolve the system up to t 1 = 0.This is equivalent to the time evolution from t 1 = 0 to t = U IR as considered in Sections 3.1 and 3.3.The depth of the time evolution and tensor network in cMERA is determined by U IR .

Möbius/SSD time evolution
We begin by considering the quantum quench induced by the Möbius Hamiltonian.Divide the Möbius Hamiltonian into the entangler K(θ) and the isometry L, where we express H Möbius , K(θ), and L in terms of Virasoro generators.In the interaction picture, |Ψ I (0)⟩ is given by where T is defined as the time-ordering operator arranging the operators in order of increasing t 1 from right to left, and K(θ, t 1 ) is defined by K(θ, t 1 ) = e iLt 1 K(θ)e −iLt 1 .Thus, |Ψ I (0)⟩ does not depend on the definition of L. The entangler in the interaction picture is explicitly given by K(θ, t where H 0 and H(θ) are defined by .16)This entangler in the interaction picture posses the periodicity with t 1 , K(θ, t 1 + L) = K(θ, t 1 ). (5.17) This period is independent of the parameter θ.In the SSD limit where θ → ∞, K(θ) reduces to K, Consequently, the entangler in the interaction picture is given by replacing H Möbius in (5.15) with H SSD .

Expression in terms of spin variables
We will rewrite the entanglers in the interaction picture in terms of spin variables that suit experimental research.As explained in [95], these Virasoro generators can be realized in certain spin chains with N sites governed by a Hamiltonian that can be expressed as a sum of Temperley-Lieb generators e i , The generators e i satisfy the Temperley-Lieb algebra whose exact form is not necessary for our purposes.For a spin-1 2 chain, one possible representation of this algebra is given by where X i , Y i and Z i are Pauli matrices acting on site i and q is an arbitrary complex parameter although the physically interesting cases are obtained when q is a root of unity.With this representation of the Temperley-Lieb algebra, the Hamiltonian (5.19) is, up to an inconsequential constant, the XXZ Hamiltonian that contains some additional boundary terms The standard Heisenberg spin chain can be obtained by simply setting q = 1.It is conjectured, with substantial evidence, that the lattice operators [e k , e k+1 ] sin nkπ N + c 24 δ n,0 (5.22) approaches the Virasoro generators L n in the N → ∞ continuum limit.Here, v F = π sin γ γ is the Fermi velocity and is determined by the q parameter via 2 cos γ = q +q −1 .For treatments of the Ising model and the XX spin chain, see [96] and [97] respectively.
6 Discussion and Future Directions We will discuss the relation between the gravity dual in Heisenberg picture and renomalization group, and comment on future directions.

Renormalization group and SSD evolution
In section 5.1, we discussed the gravity dual in Schrödinger picture.In this picture, the EoW brane moves and is deformed during the Möbius/SSD time evolution.We can instead consider the gravity dual in the Heisenberg picture.In this picture, the location of EoW is pinned at the horizon of the BTZ black hole, while the location of the surface (UV surface) where CFT lives moves and is deformed in time.Let r UV denote the radial location of this UV surface at t 1 = 0.The trajectory of r UV during the Möbius/SSD time evolution is determined by r Thus, the location of the UV surface depends on (x, t 1 ).
In the AdS/CFT correspondence, the radial direction in the gravity dual is considered as the energy scale in the 2d CFT.In the coordinate considered in this paper, the larger r becomes, the larger the energy scale becomes.The spatial and temporal dependence of the r UV suggests that the energy scale in 2d CFT depends on (x, t 1 ).Except for the case discussed in Section 3.3.3,during SSD time evolution, S A is eventually approximated by the vacuum entanglement entropy.This suggests that for the large t 1 , the location of the UV surface is further from the EoW than at t 1 = 0.In other words, in the large time regime, the energy scale is larger than the initial one.The location of the UV surface near x = X 2 f gets further from the EoW faster.We can see from the spatial and time dependence of the UV surface that the reverse of SSD time evolution considered in this paper may be used as the renormalization group where the energy scale depends on the spatial location.

Future direction
We close this section with comments on the entanglement entropy for the large time regime.In this paper and [32], we propose the quasiparticle picture, an effective picture, describing the time dependence of entanglement entropy at O( 1 ϵ ).However, this does not describe the late-time entanglement entropy, the vacuum entanglement entropy, because the vacuum one is at O(1).The findings in the inhomogeneous quench show that inhomogeneous evolution may endow the states with two types of entanglement structure: One of them is a dynamical entanglement structure that can be described by the propagation of quasiparticles, while the other is a static entanglement structure that still remains after quasiparticles pass away [98].Study on the static entanglement structure may lead to a deeper understanding of entanglement dynamics.We leave this as a future problem.
where if x = X 1 f is nether in A nor B, L (i=1,2) V=A,B;dis,j=1,2 are respectively given by L A;dis,1 = log sin π 4ϵ (w New,α A;dis,1 = log sin π 4ϵ (w New,α A;dis,2 = log sin A;dis,2 = log sin B;dis,2 = log sin π 4ϵ (w New,α (C.2) V=A,B;dis,j=1 is the same as (C.2), and L (i=1,2) V=A,B;dis,j=2 are given by L A;dis,2 = log sin In this section, we generalize the computation of the vertex operator two-point function in [72] to the four-point function which we use to compute the mutual information.
We normal order the vertex operator in the same way as in [72] with the position and momenta of the zero mode in the same exponent, The zero mode is ordered differently from [99] where the position and momentum coordinates are split between different exponentials.For Neumann boundary conditions, k L = −k R = a N , while k L = k R = a N for Dirichlet boundary conditions.The symbols, s L and s R , are arbitrary signs that we have introduced in front of the zero mode momentum such that under a spatial translation σ → σ + 2π, if s L = s R = s, the boson winds around the target manifold as X(σ + 2π) = X(σ) + 2sπwR so that s = 1 has the same periodicity as in [100].This is equivalent to flipping the sign of the zero mode in the Laurent expansion of the current i∂X which is an equally legitimate Laurent expansion.Following [72], we also do not include any cocycle factors as explained in [100].Since these cocycle factors do not depend on the coordinates y, ȳ, commuting them past the position operators can only give phases that are independent of the spacetime coordinate.Furthermore, as explained in [100], the cocycle factors only affect the relative signs of certain amplitudes but we are only considering a single correlation function.
Set s L = s R = 1.For Neumann boundary conditions, set µ = −1, −q R .A calculation similar to the one in [72] gives

2 f 2 f
(see [a] of Fig ??), while during the SSD evolution, those move to X and accumulate around that spatial point (see [a] of Fig ??
P c = X 1 f [b] P c = X 2

Figure 4 :
Figure 4: The t 1 -dependence of S A during the evolution induced by the Möbius and SSD Hamiltonians.The panels, [a], [b], and [c], correspond to cases (1), (2), and (3).Here, P c denotes the center of A. For simplicity, in [a], [b], and [c], P c is taken to be P c = X 1 f , L/4, and X 2f , respectively.The black dashed line illustrates the entanglement entropy of the thermal state with 4ϵ, the inverse temperature, for half of the total space.The gray and pink dashed lines illustrate S A of this thermal state and the vacuum state.

Figure 5 :
Figure 5: The t 1 -dependence of S A during the evolution induced by the Möbius and SSD Hamiltonians.In this figure, we take X 2 to be X 1 f .The black dashed line illustrates the entanglement entropy for half of the total space of the thermal state with 4ϵ, the inverse temperature.The gray and pink dashed lines illustrate S A of this thermal state and the vacuum state.

Figure 6 :
Figure 6: Plots of the second Rényi mutual information for the free Diract fermion as well as the quasiparticles for different placements of the subsystems of equal size 2X.The total system size is fixed at L = 100000 while the regulator is set at ϵ = 10.The continuous curves correspond to the CFT result while the dashed plots are the quasiparticle plots.
region the unentangled state to the vacuum state of the uniform Hamiltonian 2.8.1 Gravitational picture

Figure 7 :
Figure 7: A schematic view of motion of EoW brane under the SSD evolution (α = 0) with non-positive brane tension T ≤ 0. In this picture, time flows from the right panel to the left panel.The reduced density matrix in the red region is approximated by that for the vacuum state.The size of the red region grows with time following the deformation of EoW brane.

m>0 e k L α −m m e my +k R α−m m e mȳ m>0 e
−k L αm m e −my −k R αm m e −mȳ .(D.2)