# Boundary-induced transitions
in Möbius quenches of holographic BCFT

Alice Bernamonti⋆,∗, Federico Galli∗, Dongsheng Ge†

###### Abstract

Boundary effects play an interesting role in finite-size physical systems. In
this work, we study the boundary-induced properties of 1+1-dimensional
critical systems driven by inhomogeneous Möbius-like quenches. We focus on the
entanglement entropy in BCFTs with a large central charge and a sparse
spectrum of low-dimensional operators. We find that the choice of boundary
conditions leads to different scenarios of dynamical phase transitions. We
also derive these results in a holographic description in terms of
intersecting branes in AdS3, and find a precise match.

OU-HET-1222

⋆Dipartimento di Fisica e Astronomia, Universitá di Firenze;

Via G. Sansone 1; I-50019 Sesto Fiorentino, Italy

∗INFN, Sezione di Firenze;

Via G. Sansone 1; I-50019 Sesto Fiorentino, Italy

$\,{}^{\dagger}$Department of Physics, Osaka University,

Machikaneyama-Cho 1-1, Toyonaka 560-0043, Japan

###### Contents

  1. 1 Introduction
  2. 2 BCFT with mixed boundary conditions
    1. 2.1 Strip with mixed boundary conditions
    2. 2.2 Twist operator correlator
    3. 2.3 Universal contribution from the stress tensor
  3. 3 Entanglement entropy evolution under SSD and Möbius Hamiltonians
    1. 3.1 SSD and Möbius Hamiltonian on the strip
    2. 3.2 Entanglement entropy after the quench
  4. 4 Quench with generally deformed Hamiltonians
    1. 4.1 A general class of deformed Hamiltonians
    2. 4.2 Entanglement entropy after the quench
  5. 5 Holographic dual
    1. 5.1 Gravitational action and equations of motion
    2. 5.2 Intersecting branes in AdS3
    3. 5.3 Holographic entanglement entropy
  6. 6 Discussion
  7. A Stress tensor in the UHP

##  1 Introduction

Ordinary physical systems have boundaries. These introduce boundary effects,
which can be probed by physical quantities, such as the partition and
correlation functions. Of particular interest are conformally invariant
systems with conformal boundaries, described by boundary conformal field
theory (BCFT). In 1+1 dimensions this subject was mainly laid out by Cardy [1,
2, 3], and it is the main focus of this work.

In 2d BCFT, boundary effects are captured by the boundary entropy
$s_{\mathcal{B}}=\ln g_{\mathcal{B}}\equiv\ln\langle 0\ket{\mathcal{B}}$,
which measures the ground state degeneracy for a given conformal boundary
condition $\mathcal{B}$ [4, 5, 6]. This can be evaluated explicitly in
rational conformal field theories once the solutions to Cardy’s equation are
found [1, 7, 8]. For an infinite strip of width $L$, the boundary conditions
for its two edges can be chosen differently. As a result, the distinct
boundary conditions lead to two different boundary entropies. In rational CFT
they can assume a limited number of values due to the presence of a finite
number of boundary primaries [8].

Here we are concerned mainly with holographic CFTs, which have a large central
charge $c$ and a sparse spectrum of light operators [9, 10]. Large part of the
recent work in holographic BCFT is connected with the developments in
understanding entanglement in gravitational systems, via the entanglement
island proposal [11, 12, 13, 14, 15]. This has sparked renewed interest in
refining the understanding of the AdS/BCFT correspondence [16, 17]. In the
simplest AdS/BCFT models, boundary conditions are implemented via the
insertion of end of the world (EOW) branes anchored on the spacetime boundary
and extending into the AdS bulk [16, 17]. A number of features emerge for
holographic BCFTs realized in this way [16, 17, 27, 26, 23, 25, 24, 22, 21,
18, 19, 20]. In particular, they have a continuous spectrum of heavy
operators, with conformal weight proportional to the central charge, up to the
black hole threshold $c/24$. Also, the tension $T_{i}$ of the EOW branes in
the AdS bulk determines the boundary entropy
$s_{i}=\frac{c}{6}\text{arctanh}\,T_{i}$ for the dual boundary state [16, 17].
The brane tension can take values continuously in the range $|T_{i}|\leq 1$
and thus yields for these models a boundary entropy taking arbitrary values
and scaling with the CFT central charge. Indeed, in the holographic
computation of the entanglement entropy in AdS/BCFT models, the boundary
entropy arises from the Ryu-Takayanagi formula applied to bulk geometries
bounded by EOW branes [16, 17].

In holographic BCFT, different competing phases in the evaluation of the
entanglement entropy may arise, and the boundary entropy plays a role in
determining the dominant one. These situations include holographic
computations in terms of the Ryu-Takayanagi surfaces [16, 17], but also first
principles CFT calculations in specific limits, and applications to double
holographic models via the entanglement island formula as in, e.g., [29, 26,
23, 30, 11, 12, 13, 14].

In far-from-equilibrium settings, such as those generated by quantum quenches,
the system may experience dynamical transitions between these competing
phases. In this work, we consider spatially inhomogeneous quenches in 2d BCFT.
In particular, we focus on a family of processes in which the Hamiltonian is
abruptly changed to the so called Möbius Hamiltonian [31, 33, 34, 32, 35, 36,
37, 38, 39]. A well-studied limit of this class is known as sine-square-
deformed (SSD) Hamiltonian, which was originally introduced to suppress
boundary effects and efficiently isolate bulk properties [31, 32, 33, 41, 42,
40, 34]. In fact, it was observed that in 1+1-dimensional CFT the SSD
Hamiltonian shares the same ground state of a uniform system with periodic
boundary conditions and that, to a certain extent, the system size is
effectively infinite in this limit [35, 36, 43, 44, 45].

As we summarize below, here we deal with an instance that combines the above
ingredients. For a BCFT on an interval with mixed boundary conditions, we
study the entanglement entropy evolution following an inhomogeneous quench by
a Möbius-like deformation. We will show that two competing phases in the
entanglement entropy arise from the presence of mixed boundary conditions, and
that a dynamical transition from one phase to the other is controlled by the
quench protocol. This is a new feature as compared to previous studies on this
class of quenches, where boundary effects were found to be suppressed [31, 33,
34, 32, 35].

##### Summary

The dynamical process we consider is the following. We prepare a state
$\ket{\psi_{AB}}$ on a finite-size system of length $L$, with distinct
boundary conditions $A$ and $B$, as illustrated in fig. 1.

Figure 1: Left: The physical system is defined on a strip of spatial width
$L$, with conformal boundary conditions $A$ and $B$ a the two edges. We
consider a bipartition of the system defined considering a spatial interval of
size $\sigma$ adjacent to the boundary with conformal boundary condition $A$.
Via the replica trick the evaluation of the entanglement entropy for this
bipartition is mapped to calculating the one point fucntion of a replica twist
operator $\Phi_{n}$ inserted at the endpoint of the interval. Right: The
system mapped to the upper half plane. The change in boundary conditions is
sustained by the insertion of a boundary condition changing operator
$\psi_{AB}$.

This is achieved inserting boundary condition changing (BCC) operators, which
mediate the change in boundary conditions [46]. BCC operators are not local
operators of the boundary CFT in the conventional sense, but are treated as
boundary primaries with similar scaling properties as bulk primaries [3]. In
particular $\ket{\psi_{AB}}$ can be seen as an eigenstate of the BCFT
Hamiltonian.

At $t=0$ the state is quenched by turning on a deformation of the BCFT
Hamiltonian. For $t>0$ the system is driven with such a deformed Hamiltonian
and undergoes a non-trivial evolution.111In spirit, this is the same thing
happening in the prototypical global quench protocol of Calabrese and Cardy
[47]. In both cases the initial state is not an eigenstate of the Hamiltonian
that drives the dynamics for $t>0$. We analyse the dynamics following from
different deformations: the SSD, Möbius and generalized deformation of section
4. A key technical advantage is that these allow to explicitly obtain the time
evolution in a closed form [35, 36, 48].

As a probe of the dynamics, we consider the real-time evolution of the
entanglement entropy for an interval adjacent to one end of the system. Via
the replica trick, the computation of the entanglement entropy is recast in
evaluating a three-point function of two BCC operators and a replica twist in
the bulk. The details of such a correlator depend on the specific theory. We
extract and analyse the contribution coming from the conformal family of the
stress tensor. The reason is twofold. On the one hand, this contribution is
universal. It is present in every 2d CFT with the identity operator in its
normalizable spectrum, and it does not depend on CFT data other than the
central charge and the ground state degeneracy. On the other hand, in
holographic CFTs the contribution of the identity conformal family is expected
to dominate, and to match dual holographic calculations in three dimensional
semiclassical gravity.

With mixed boundary conditions the approximate result obtained from the
conformal family of the identity operator is not unique. There are different
boundary operator expansions of a bulk CFT operator one can consider [49]. In
particular, one associated with boundary condition $A$ and one with boundary
condition $B$. These give competing channels in the evaluation of the
correlator. Already in the initial state what channel, or phase, dominates
depends on the size of the interval considered and difference in boundary
entropies $s_{A,B}$ [23, 22, 21]. More interestingly, we show that the quench
dynamics can drive a transition from one phase to the other (and back) as time
evolves.

We find that the dynamical phase transition pattern is determined by the
relative amount of boundary degrees of freedom associated to the two
boundaries. Depending on the difference between $s_{A}$ and $s_{B}$, the
entanglement entropy can exhibit or not a transition between phase $A$ and
phase $B$. Under the SSD dynamics the transition can only happen once, while
it becomes recurrent for the Möbius-like quenches, with characteristic
timescales determined by the details of the deformation.

This finds an exact match and a simple interpretation in the holographic
gravitational bulk description obtained in terms of AdS3 bounded by two
intersecting EOW branes [21, 22]. We show that the entanglement entropy
obtained in the BCFT computation for the two phases is in one to one
correspondence with the length of minimal geodesics. Indeed for a point on the
AdS3 boundary, there exists in general two competing Ryu-Takayanagi geodesics,
each ending on one of the two EOW branes. We also show that the entanglement
dynamics can be matched exploiting the local equivalence of AdS3 geometries
and extending the map relating the initial and time evolved state in the BCFT
to a diffeomorphism in the bulk.

The rest of the paper is organized as follows, in section 2, we introduce the
setup and relevant BCFT technicalities needed for our analysis. In particular,
we show how the initial state of our system is defined and compute the
corresponding entanglement entropy. In section 3, we introduce the SSD and
Möbius Hamiltonian on the strip, and analyze the time evolution of the
entanglement entropy for a quench with these Hamiltonians. In section 4, we
consider the generalization of this analysis to the case of deformed
Hamiltonians obtained from $sl(2,\mathbb{R})$ subalgebras of Virasoro
generators at arbitrary levels. Section 5 describes the holographic dual of
our setup and analysis. Using a bulk description in terms of AdS3 with
intersecting EOW branes, we find an exact match of the BCFT computation with
the holographic result. We conclude in section 6, where we comment on the
broader applicability of our analysis and results, and give future
perspectives.

##  2 BCFT with mixed boundary conditions

In this section we review how to characterize a CFT on an interval with
different conformal boundary conditions, $A$ and $B$, at the two ends [49, 46,
3]. The state with mixed boundary conditions we will consider,
$|\psi_{AB}\rangle$, will either be the ground state or a high energy
eigenstate for the BCFT with distinct boundary conditions.

We will review how to evaluate the entanglement entropy in this state for an
interval adjacent to one of the boundaries. Via the replica approach, this
reduces to evaluating a replica twist one-point function on the strip with
mixed boundary conditions. Specializing to the case of holographic CFTs, we
will approximate the correlator with the universal contribution of the stress
tensor and its conformal family. This is motivated by the assumptions in an
holographic CFT on the sparseness of the spectrum of low dimensional operators
and on the dominance of the identity Virasoro conformal block (see, e.g., [9,
50, 51, 56, 52, 10, 53, 55, 57, 54] for examples and discussions). Let us
remark that this is a working assumption and here we are not proving the
validity of this approximation. We will however show explicitly in section 5
that the result obtained in this way exactly captures the result following
from the corresponding dual computation in terms of Ryu-Takayanagi surfaces
[58, 16, 17, 59].

###  2.1 Strip with mixed boundary conditions

We consider a state in a CFT on an interval with two different conformally
invariant boundary conditions. In particular, in the Euclidean plane
$w=\tau+i\sigma$, we consider an infinite strip $-\infty<\tau<\infty$ of width
$0\leq\sigma\leq L$. Two conformally invariant boundary conditions $A,B$ are
imposed along $\sigma=0$ and $\sigma=L$ respectively. With

$z=e^{\frac{\pi}{L}w}$ (2.1)

the strip is mapped into the upper half plane (UHP), with the two boundaries
$\sigma=0,L$ mapped respectively to the positive and negative real axis $z=x$.
In the UHP the change in conformal boundary conditions along the boundary at
$x=0,\infty$ is mediated by boundary condition changing (BCC) operators
$\psi_{AB}(x)$ (see fig. 1) [49, 3]. More precisely, starting in the UHP with
a state with homogeneous boundary condition $A$, the mixed boundary conditions
$A$ and $B$ state is obtained as222For notational simplicity and since it is
clear from the context, we do not distinguish between the BCC operators
$\psi_{AB}$ and $\psi_{BA}$ [49].

$\ket{\psi_{AB}}=\psi_{AB}(0)\ket{0}_{AA}\qquad\bra{\psi_{AB}}=\lim_{x\to\infty}\frac{x^{2h_{\psi}}}{c_{\psi\psi}}\,{}_{AA}\bra{0}\psi_{AB}(x)\equiv{}_{AA}\bra{0}\psi_{AB}(\infty)\,.$
(2.2)

BCC operators act non-locally to implement the change of boundary conditions,
but can be treated as primaries with weight $h_{\psi}$ in the CFT defined on
the boundary [49, 3]. $c_{\psi\psi}$ is the coefficient of the two-point
function of the BCC operator, so that the state is normalized
$\bra{\psi_{AB}}\psi_{AB}\rangle=1$. The conformal dimension $h_{\psi}$ is
determined in terms of gap in energy between the initial state, with
homogeneous boundary conditions $A$, and the state with mixed boundary
conditions $A$ and $B$. In particular [46]

$h_{\psi}=\frac{L\,\Delta E}{\pi}=\frac{L}{\pi}\left(E_{AB}-E_{AA}\right)\,,$
(2.3)

where $E_{AA}$ is the energy of the ground state on the strip with boundary
conditions $A$. $E_{AB}$ instead is the energy of the lowest energy eigenstate
on the strip with mixed boundary conditions with a non-zero overlap with
$\ket{\psi_{AB}}$. Notice that depending on the operator $\psi_{AB}$, this may
correspond to the ground state or an excited state of the BCFT with mixed
boundary conditions [46].

We are interested in holographic BCFTs, which have a dual description in terms
of semiclassical gravitational configurations in AdS3. For these, the change
in boundary conditions holographically translates into a change of the
asymptotically AdS3 geometry [16, 17, 60]. The corresponding change in energy
is thus proportional to the central charge $c$. We will then be concerned with
the case where the BCC has a conformal weight proportional to the central
charge $c$ of the BCFT [23, 22, 21]. As we will review in section 5, in
bottom-up models of holographic CFTs the BCC conformal operators have a
continuous spectrum in the range $0<h_{\psi}/c\leq 1/24$ [22, 21]. When
talking of holographic BCFT in the rest of this work we will then assume
implicitly this range of values.

###  2.2 Twist operator correlator

We want to evaluate the entanglement entropy for a spatial region consisting
of a portion of the strip that includes one of the boundaries. Without loss of
generality, we pick the entangling region to be a portion of dimension
$0<\sigma<L$ of the entire system, adjacent to the origin. The boundary
included in this region is therefore the one with boundary condition $A$, see
fig. 1.

Using the replica trick, the entanglement entropy $S$ can be obtained in terms
of the the Rényi entropies $S^{(n)}_{\sigma}$ (for a review see [6]). For any
integer $n\geq 2$, these are defined from the reduced density matrix
$\rho_{\sigma}$ associated to the geometric region we are considering as

$S^{(n)}_{\sigma}=\frac{1}{1-n}\log{\rm Tr}\rho_{\sigma}^{n}\,.$ (2.4)

The computation of $S^{(n)}_{\sigma}$ can be recast into the evaluation of a
correlator of twist operators in the strip geometry with mixed boundary
conditions. For the case at hand this is a one-point function333Here are
considering a time independent state, so we are setting $\tau=0$ without loss
of generality.

$S^{(n)}_{\sigma}=\frac{1}{1-n}\log\bra{\psi_{AB}}\Phi_{n}(w=i\sigma,\bar{w}=-i\sigma)\ket{\psi_{AB}}\,,$
(2.5)

with the twist operator $\Phi_{n}$ inserted at the endpoint of the entangling
interval (see fig. 1). The entanglement entropy $S$ is then obtained as the
analytic continuation to $n=1$. Schematically,

$S=\lim_{n\to 1}S^{(n)}_{\sigma}=\lim_{n\to
1}\frac{1}{1-n}\log\bra{\psi_{AB}}\Phi_{n}(w,\bar{w})\ket{\psi_{AB}}\,.$ (2.6)

The above correlator is not defined in the original CFT, but in the cyclic
orbifold theory CFT${}^{n}/\mathbb{Z}_{n}$ resulting from using the replica
trick.444In defining the state $\ket{\psi_{AB}}$ in (2.5) one would actually
need to consider the operator $\Psi_{AB}=\psi_{AB}^{\otimes n}$ in the
orbifold CFT${}^{n}/\mathbb{Z}_{n}$, instead of $\psi_{AB}$. Nonetheless, for
all practical purposes, in the limit (2.6) one can simply keep defining the
state with $\psi_{AB}$. Any appearance of $\Psi_{AB}$ via $h_{\Psi}=nh_{\psi}$
would reduce to the effect of having $\psi_{AB}$ with $h_{\psi}$. See, e.g.,
[53, 30] for similar computations. The corresponding replica twist field has
conformal dimension

$h_{h}=\bar{h}_{n}=\frac{c}{24}\left(n-\frac{1}{n}\right)\,.$ (2.7)

The strip correlator on the right hand side of (2.6) is conveniently evaluated
via a map to the UHP. Using $z=e^{\frac{\pi}{L}w}$ and owing to the
normalization of the state we have555The UHP expectation value is homogeneous
with uniform boundary condition $A$, that is
$\langle\dots\rangle_{\text{UHP}}={}_{AA}\bra{0}\dots\ket{0}_{AA}$.

$\displaystyle\bra{\psi_{AB}}\Phi_{n}(w,\bar{w})\ket{\psi_{AB}}=\left(\frac{\partial
z}{\partial
w}\right)^{h_{n}}\left(\frac{\partial{\bar{z}}}{\partial{\bar{w}}}\right)^{h_{n}}\langle\psi_{AB}(0)\Phi_{n}(z,\bar{z})\psi_{AB}(\infty)\rangle_{\text{UHP}}\,.$
(2.8)

This three point function in the UHP involves two boundary and one bulk
operator and is thus not completely fixed by conformal invariance. Similarly
to a four-point function in the full complex plane, it can be expanded in the
sum of conformal Virasoro blocks. There are in fact two different expansions,
or channels, one can consider here [49]. One can perform the boundary
expansion of a primary bulk operator $\Phi$, in our case a bulk twist
operator, assuming it approaches the boundary where the boundary condition is
fixed to be $A$. Alternatively, $\Phi$ can expanded in terms of boundary
operators assuming the condition on the boundary is $B$. The two corresponding
conformal block expansions can be written as [49]

$\langle\psi_{AB}(0)\Phi(z,\bar{z})\psi_{AB}(\infty)\rangle_{\text{UHP}}=(z-{\bar{z}})^{-2h_{\Phi}}(1-\eta)^{2h_{\Phi}}\sum_{p}C^{A}_{p}{\cal
F}_{p}(1-\eta)\,,$ (2.9)

and

$\langle\psi_{AB}(0)\Phi(z,\bar{z})\psi_{AB}(\infty)\rangle_{\text{UHP}}=(z-{\bar{z}})^{-2h_{\Phi}}(1-\eta)^{2h_{\Phi}}\sum_{p}C^{B}_{p}{\cal
F}_{p}(1-e^{-2\pi i}\eta)\,$ (2.10)

where we defined the ratio

$\eta=\frac{z}{\bar{z}}\,.$ (2.11)

In the above expressions, the sum is over exchanged primaries of the CFT
defined on the boundary, with $C^{A}_{p}$ and $C^{B}_{p}$ constants related to
operator expansion coefficients (see [49] for details). ${\cal F}_{p}$ is the
same function defining the holomorphic Virasoro conformal block in the full
complex plane. More precisely, take in the full complex plane a primary $\phi$
having the same conformal weight as $\Phi$ and a primary $\psi$ with the same
conformal weight as $\psi_{AB}$. ${\cal F}_{p}(1-\eta)$ is then the
holomorphic Virasoro conformal block appearing in the t-channel expansion

$\langle\psi(0)\phi(\eta,\bar{\eta})\phi(1)\psi(\infty)\rangle_{\text{plane}}=\sum_{p}a_{p}{\cal
F}_{p}(1-\eta)\bar{\cal F}_{p}(1-\bar{\eta})\,.$ (2.12)

Notice that despite the full correlator being single valued in the complex
plane, ${\cal F}_{p}$ is in general multivalued in the $\eta$ plane and
requires a choice of branch cut. Here we are taking the branch cut of ${\cal
F}_{p}(1-\eta)$ in such a way as to recover the expected short distance
singularity in the OPE limit $\phi(\eta,\bar{\eta})\to\phi(1)$. Having fixed
that, the explicit phase in front of $\eta$ in (2.10) differentiates the
single terms in the expansion from those in (2.9). Of course, the exact
correlator obtained from the full sums is single valued.

As discussed in section 2.1, we will consider the case of heavy BCC operators
$\psi_{AB}$ with conformal weight $h_{\psi}$ scaling with the central charge
$c$. In our computation the bulk operator above is identified with a replica
twist $\Phi_{n}$. Its weight $h_{n}$ also scales with the central charge of
the CFT, but $h_{n}/c\to 0$ in the limit $n\to 1$. In the holographic CFT
literature this is referred to as a (perturbatively) light operator (see,
e.g., [51, 56]). The relevant Virasoro conformal blocks ${\cal F}_{p}$ are
therefore those for an Heavy-Heavy-Light-Light four point function (2.12). At
the leading order in a large $c$ expansion these are known in a closed form
[51, 56].

###  2.3 Universal contribution from the stress tensor

The two expansions (2.9) and (2.10) give completely equivalent results when
evaluating the full correlator
$\langle\psi(0)\Phi_{n}(z,\bar{z})\psi(\infty)\rangle_{\text{UHP}}$. However,
for holographic CFTs the full CFT data are not known and we can only provide
an approximate result. We will consider the universal contribution of the
stress tensor coming from the identity Virasoro block and assume that in an
holographic CFT the correlator is well approximated by this contribution alone
(see, e.g., [50, 51, 56, 52, 53, 57, 54] for work in similar context and more
detailed discussions of this approximation).

Roughly speaking, this approximation is motivated by the fact that for
holographic CFTs the sum over Virasoro conformal blocks is a sum of
exponentials, weighted by the conformal dimension of the exchanged operator
[61, 62]. Applying a saddle-point approach, the sum is then approximated by
the largest term. For holographic CFTs, which have a sparse spectrum of low-
dimensional operators, this is given by the identity Virasoro block. In making
this approximation one is also assuming that other contributions from light
operators are suppressed in the large $c$ expansion, and ignoring possible
non-perturbative terms. In addition, the universal contribution of the stress
energy tensor alone is expected to encode the geometric features of the dual
holographic description, as we will show in section 5.

The different expansions channels in (2.9) and (2.10) provide competing
identity Virasoro block approximations. Following the saddle point logic, one
is then lead to consider these channels as competing, and to retain the
dominant saddle point contribution. Here we shall notice that we expect each
of the identity Virasoro contributions to provide a good approximation to the
full correlator only for some range of the parameters. When knowing the full
correlator one would observe a smooth transition connecting these regimes
[63]. In our working approach, where we only retain two competing saddles
provided by the identity Virasoro block, we will instead observe a sharp
transition.

To evaluate the correlator, we then consider the explicit expression for the
Virasoro identity block at leading order in a large $c$ expansion [51, 56]

${\cal
F}_{0}(1-\eta)=\eta^{(\alpha-1)h_{n}}\left(\frac{1-\eta^{\alpha}}{\alpha}\right)^{-2h_{n}}\,.$
(2.13)

Here

$\alpha=\sqrt{1-\frac{24h_{\psi}}{c}}\,,$ (2.14)

and $h_{n}$ is the conformal weight (2.7) of the twist operator $\Phi_{n}$.
Notice that (2.13) is only valid for $h_{n}/c\ll 1$, i.e., for $n$ close to
one.

At leading order in the $\eta\to 1$ limit

${\cal F}_{0}(1-\eta)\approx(1-\eta)^{-2h_{n}}\,.$ (2.15)

Taking this limit in (2.9) selects the identity contribution in the bulk to
boundary OPE of the bulk operator $\Phi_{n}$. This shows how the coefficient
$C^{A}_{0}$ in (2.9) is related to the expectation value of $\Phi_{n}$ in the
UHP with homogeneous boundary conditions $A$. Explicitly

$\langle\Phi_{n}(z,{\bar{z}})\rangle_{\text{UHP}}=\frac{A_{\Phi_{n}}}{|z-{\bar{z}}|^{2h_{n}}}=\frac{C^{A}_{0}}{(z-{\bar{z}})^{2h_{n}}}\,,$
(2.16)

where we are neglecting the regularization associated to the insertion of a
twist operator, which will be restored in the final result for the
entanglement entropy.

Approximating the full three-point function with the identity contribution in
the $A$-channel (2.9) then gives

$\displaystyle\bra{\psi_{AB}}\Phi_{n}(z,{\bar{z}})\ket{\psi_{AB}}_{\text{UHP}}$
$\displaystyle\approx
C_{0}^{A}\alpha^{2h_{n}}(z{\bar{z}})^{(\alpha-1)h_{n}}(z^{\alpha}-{\bar{z}}^{\alpha})^{-2h_{n}}$
(2.17)
$\displaystyle=C_{0}^{A}\alpha^{2h_{n}}(z{\bar{z}})^{-h_{n}}\left[2\sinh\frac{\alpha}{2}\log\frac{z}{{\bar{z}}}\right]^{-2h_{n}}\,,$

which via (2.8) in terms of the physical strip setup yields

$\bra{\psi_{AB}}\Phi_{n}(w,{\bar{w}})\ket{\psi_{AB}}\approx
A_{\Phi_{n}}\left(\frac{\alpha\pi}{2L}\right)^{2h_{n}}\left[\sin\alpha\pi\frac{\sigma}{L}\right]^{-2h_{n}}\,.$
(2.18)

In the last expression we have used the fact that the twist insertion is at
$w=i\sigma$. In the limit $\sigma\to 0$ this result reproduces the result for
a small entangling region adjacent to the boundary at $\sigma=0$ [6], which is
insensitive to boundary condition $B$.

Performing the same computation for the expansion in the $B$-channel (2.10)
gives

$\bra{\psi_{AB}}\Phi_{n}(z,{\bar{z}})\ket{\psi_{AB}}_{\text{UHP}}\approx
C_{0}^{B}\alpha^{2h_{n}}(z{\bar{z}})^{-h_{n}}\left[2\sinh\frac{\alpha}{2}\left(\log\frac{z}{{\bar{z}}}-2\pi
i\right)\right]^{-2h_{n}}\,.$ (2.19)

The coefficient $C_{0}^{B}$, up to a phase, is the expectation value
$B_{\Phi_{n}}$ of $\Phi_{n}$ in the UHP with boundary conditions $B$ [49].
More precisely, mapping to the strip

$\bra{\psi_{AB}}\Phi_{n}(w,{\bar{w}})\ket{\psi_{AB}}\approx
B_{\Phi_{n}}\left(\frac{\alpha\pi}{2L}\right)^{2h_{n}}\left[\sin\alpha\pi\frac{L-\sigma}{L}\right]^{-2h_{n}}\,.$
(2.20)

Here for $\sigma\to L$ one recognises the result for a small entangling region
adjacent to the boundary at $\sigma=L$ with boundary conditions $B$ [6].

The correlator is then evaluated taking the largest between the approximate
results (2.18) and (2.20), which yields

$\bra{\psi_{AB}}\Phi_{n}(w,{\bar{w}})\ket{\psi_{AB}}=\left(\frac{\alpha\pi}{2L}\right)^{2h_{n}}\max\left\\{A_{\Phi_{n}}\left[\sin\alpha\pi\frac{\sigma}{L}\right]^{-2h_{n}};B_{\Phi_{n}}\left[\sin\alpha\pi\frac{L-\sigma}{L}\right]^{-2h_{n}}\right\\}\,.$
(2.21)

Substituting into (2.6) and continuing the result to $n=1$ gives the
entanglement entropy. The maximization in equation (2.21) translates into a
minimization over entanglement entropy contributions computed using the two
channels

$S=\frac{c}{6}\log\frac{2L}{\pi\alpha\epsilon}+\min\left\\{\frac{c}{6}\log\sin\frac{\alpha\pi\sigma}{L}+s_{A};\frac{c}{6}\log\sin\frac{\alpha\pi(L-\sigma)}{L}+s_{B}\right\\}\,.$
(2.22)

In writing this result we have made explicit the UV regulator $\epsilon$
associated with the insertion of a replica twist operator $\Phi_{n}$. $s_{A}$
is the boundary entropy associated to the boundary with boundary conditions
$A$ [4, 5, 6]

$s_{A}\equiv\log g_{A}=\lim_{n\to 1}\frac{1}{n-1}\log A_{\Phi_{n}}\,,$ (2.23)

and a similar definition holds for $s_{B}$. Here $g_{A,B}$ is the ground-state
degeneracy of the corresponding boundary condition $A,B$ as defined in [4].
Notice that the boundary entropy $s_{A,B}$ can be both positive or negative.
In the holographic realisation of BCFTs it actually takes continuous values in
the reals [16, 17] (see also the discussion in sec. 5).

The result in (2.22) reproduces what found in [23] using a map to the twisted
UHP where the effect of the stress energy tensor is trivialized. The case with
homogeneous boundary condition $A$ and no BCC insertions [48] is recovered for
$s_{A}=s_{B}$ and $\alpha=1$. In this limit the two contributions coincide.
Indeed, the correlator reduces to the twist operator one point function with
boundary condition $A$, which is fixed by conformal invariance. Taking
$s_{A}=s_{B}$ and $0<\alpha<1$ in (2.22) can be regarded as the case of a high
energy eigenstates on the strip with homogeneous boundary condition $A$. This
is an excited state prepared by the insertion of a primary boundary operator
$\psi$, rather than a BCC. The conformal weight $h_{\psi}$ corresponds to the
energy gap between this excited state and the ground state with homogeneous
boundary conditions $A$. For this state, the exchange of dominance between the
two contributions as the size of the interval is changed happens at
$\sigma=L/2$.

We are interested in the general case where $s_{A}\neq s_{B}$, and in
holographic CFTs. For these the boundary entropy scales like the central
charge, $s_{A,B}\sim c$ and $0<h_{\psi}/c\leq 1/24$ giving $0<\alpha\leq 1$
[22, 21]. The dominant channel is then determined by the boundary conditions
$A$ and $B$ both via the boundary entropy $s_{A,B}$ and the specific state,
through the energy gap (2.3) encoded in $\alpha$. In particular, as the size
$\sigma$ of the interval adjacent to $A$ is increased, $S$ undergoes a
transition from channel $A$ to channel $B$ when

$s_{B}-s_{A}=\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\pi\sigma}{L}\right)}{\sin\left(\frac{\alpha\pi(L-\sigma)}{L}\right)}\,.$
(2.24)

This happens for any boundary state with $s_{A}\neq s_{B}$ and for any value
$0<\alpha<1$. Notice that the transition can happen for arbitrary small values
of $\sigma$. That is, channel $B$ can become dominant for $\sigma/L\ll 1$ as
long as $s_{B}-s_{A}$ takes a large negative value. Similarly, for
$s_{B}-s_{A}$ positive and sufficiently large one can remain in channel $A$
for $(L-\sigma)/L\ll 1$.

##  3 Entanglement entropy evolution under SSD and Möbius Hamiltonians

In the previous section we described the initial state of our system and the
corresponding entanglement entropy computation. We now turn to the study of a
particular class of quantum quenches.

A quench is a sudden change in the system that produces a time-dependent
excited state. Depending on the protocol, the change can affect the system
locally or globally, homogeneously or inhomogeneously. Here we will focus on a
class of inhomogeneous quenches that affect the system globally (for other
studies on the topic see, e.g., [64, 65, 66, 67, 68, 69, 70, 71, 72, 37, 38,
39, 73]). They are obtained evolving the initial state with a class of
Hamiltonians known as sine-square-deformed (SSD) and Möbius Hamiltonians [31,
33, 34, 32, 35, 36]. These are deformations of the usual Hamiltonian of the
CFT defined on the strip, $H_{0}$. The state $\ket{\psi_{AB}}$ is an
eigenstate of $H_{0}$ and represents an out-of-equilibrium state for the
evolution dictated by these deformed Hamiltonians.

###  3.1 SSD and Möbius Hamiltonian on the strip

The SSD Hamiltonian consists of an inhomogeneous spatial deformation of the
standard Hamiltonian with a sine squared enveloping function [48]

$H_{\rm
SSD}\equiv\int_{0}^{L}d\sigma\,2\sin^{2}\left(\frac{\pi\sigma}{L}\right)\,T_{\tau\tau}(\sigma)\,.$
(3.1)

This can be expressed as

$H_{\rm SSD}=H_{0}-\frac{1}{2}\left(H_{+}+H_{-}\right)$ (3.2)

where

$H_{0}=\int_{0}^{L}d\sigma\,T_{\tau\tau}(\sigma)=\int_{0}^{L}\frac{dw}{2\pi}\,\left(T(w)+{\bar{T}}({\bar{w}})\right)$
(3.3)

is the standard CFT Hamiltonian on the strip, and the deformations $H_{\pm}$
are defined as

$H_{\pm}=\int_{0}^{L}\frac{dw}{2\pi}\left(e^{\pm 2\pi w/L}T(w)+e^{\mp
2\pi{\bar{w}}/L}{\bar{T}}({\bar{w}})\right)\,.$ (3.4)

The Möbius Hamiltonian generalizes this to a one parameter family of
deformations

$H_{\theta}\equiv H_{0}-\frac{\tanh(2\theta)}{2}\left(H_{+}+H_{-}\right)\,,$
(3.5)

where the SSD Hamiltonian is recovered for $\theta\to\infty$.

Mapping the strip to the UHP by $z=e^{\frac{\pi w}{L}}$ and using the
properties of the stress tensor of a CFT in the UHP [3], one can write

$\displaystyle H_{0}$ $\displaystyle=\frac{\pi}{L}\oint
z\,T(z)-\frac{\pi}{L}\frac{c}{24}=\frac{\pi}{L}\left(L_{0}-\frac{c}{24}\right)\,,$
(3.6) $\displaystyle H_{+}+H_{-}$
$\displaystyle=\frac{\pi}{L}\oint\left(z^{3}T(z)+z^{-1}T(z)\right)=\frac{\pi}{L}\left(L_{2}+L_{-2}\right)\,,$

with the contour integral defined in the full complex plane (see appendix A
for details).

Next to the dilatation generator $L_{0}$, which defines with the Casimir
energy the standard Hamiltonian, $H_{\theta}$ involves the level-two Virasoro
generators $L_{\pm 2}$. The subset $\\{L_{0},L_{\pm 2}\\}$ forms a closed
$sl(2,\mathbb{R})$ algebra [74, 75]. It is possible to write in an explicit
closed form the action of the Möbius Hamiltonian $H_{\theta}$ on a primary
operator $\mathcal{O}$ [35, 36, 48]. Its Heisenberg evolution can be expressed
as a conformal transformation to new coordinates
$(z_{{t_{E}}},{\bar{z}}_{{t_{E}}})$ such that

${\cal O}_{{t_{E}}}(z,\bar{z})\equiv e^{H_{\theta}{{t_{E}}}}{\cal
O}(z,\bar{z})e^{-H_{\theta}{{t_{E}}}}=\left(\frac{\partial
z_{{t_{E}}}}{\partial z}\right)^{h_{\cal
O}}\left(\frac{\partial{\bar{z}}_{{t_{E}}}}{\partial{\bar{z}}}\right)^{\bar{h}_{\cal
O}}{\cal O}(z_{{t_{E}}},\bar{z}_{{t_{E}}})\,.$ (3.7)

The explicit form of the transformation can be obtained finding a map to a
geometry where $H_{\theta}$ has a simple action [35, 36, 48]. Using

$\tilde{z}^{2}=-\frac{z^{2}\cosh\theta-\sinh\theta}{z^{2}\sinh\theta-\cosh\theta}\,$
(3.8)

and a similar one for the ${\bar{z}}$ coordinate, one gets to

$\displaystyle H_{\theta}$
$\displaystyle=\frac{\pi}{L}\left[\frac{1}{\cosh(2\theta)}\oint\tilde{z}T(\tilde{z})-\frac{c}{24}\left(1-\frac{1}{\cosh(2\theta)}\right)\right]$
(3.9)
$\displaystyle=\frac{\pi}{L}\left[\frac{1}{\cosh(2\theta)}\tilde{L}_{0}-\frac{c}{24}\left(1-\frac{1}{\cosh(2\theta)}\right)\right]\,.$

Here $\tilde{L}_{0}$ the dilatation generator in the $\tilde{z}$-plane. The
action of $H_{\theta}$ in the $\tilde{z}-$plane is then just a dilatation.
Assuming for simplicity ${h_{\cal O}}={\bar{h}_{\cal O}}$, which will be the
relevant case for us here, one has

$e^{H_{\theta}{{t_{E}}}}{\cal
O}(\tilde{z},\bar{\tilde{z}})e^{-H_{\theta}{{t_{E}}}}=\left|\frac{\partial\tilde{z}_{{t_{E}}}}{\partial\tilde{z}}\right|^{2h_{\cal
O}}{\cal O}(\tilde{z}_{{t_{E}}},\bar{\tilde{z}}_{{t_{E}}})=\lambda^{2h_{\cal
O}}{\cal O}(\lambda\tilde{z},\lambda\bar{\tilde{z}})\,,$ (3.10)

where

$\tilde{z}_{t_{E}}=\lambda\tilde{z}\qquad\qquad\bar{\tilde{z}}_{t_{E}}=\lambda\bar{\tilde{z}}\,\,,$
(3.11)

and with the scaling parameter $\lambda$ depending on the deformation $\theta$
and the Euclidean time ${{t_{E}}}$

$\lambda=\exp{\frac{\pi\,{{t_{E}}}}{L\cosh(2\theta)}}\,.$ (3.12)

From here one can go back to the $z$ coordinate using (3.8) on both sides of
(3.11)

$\left(-\frac{z^{2}\cosh\theta-\sinh\theta}{z^{2}\sinh\theta-\cosh\theta}\right)^{\frac{1}{2}}=\lambda\left(-\frac{z_{{t_{E}}}^{2}\cosh\theta-\sinh\theta}{z_{{t_{E}}}^{2}\sinh\theta-\cosh\theta}\right)^{\frac{1}{2}}\,.$
(3.13)

The map for implementing the time dependence in the UHP is then more
conveniently expressed as666As a consistency check, one can take the
$t_{E}$-derivative on the two sides of equation (3.7), and using the
commutation relation between the Virasoro generators and the primary operator,
one can match the coefficients mode by mode as in, e.g., [76].

$z_{{t_{E}}}^{2}=\frac{[(1-\lambda^{2})\cosh(2\theta)-(1+\lambda^{2})]z^{2}-(1-\lambda^{2})\sinh(2\theta)}{(1-\lambda^{2})\sinh(2\theta)z^{2}-(1-\lambda^{2})\cosh(2\theta)-(1+\lambda^{2})}\,.$
(3.14)

With a completely similar procedure it is immediate to write the result for
the strip

$w_{{t_{E}}}=\frac{L}{2\pi}\log\left[\frac{[(1-\lambda^{2})\cosh(2\theta)-(1+\lambda^{2})]e^{\frac{2\pi}{L}w}-(1-\lambda^{2})\sinh(2\theta)}{(1-\lambda^{2})\sinh(2\theta)e^{\frac{2\pi}{L}w}-(1-\lambda^{2})\cosh(2\theta)-(1+\lambda^{2})}\right]\,,$
(3.15)

and analogous expressions hold for ${\bar{z}}_{{t_{E}}}$ and
${\bar{w}}_{{t_{E}}}$.

For later convenience, we write explicitly the form of the transformation in
the SSD case, obtained as the $\theta\to\infty$ limit. The result is
conveniently written in terms of ${{t_{E}}}$ as

$z_{{t_{E}}}^{2}=\frac{\frac{\pi{{t_{E}}}}{L}(z^{2}-1)+z^{2}}{\frac{\pi{{t_{E}}}}{L}(z^{2}-1)+1}\,,$
(3.16)

or equivalently for the strip

$w_{{t_{E}}}=\frac{L}{2\pi}\log\left[\frac{\frac{\pi{{t_{E}}}}{L}(e^{\frac{2\pi}{L}w}-1)+e^{\frac{2\pi}{L}w}}{\frac{\pi{{t_{E}}}}{L}(e^{\frac{2\pi}{L}w}-1)+1}\right]\,.$
(3.17)

###  3.2 Entanglement entropy after the quench

The Euclidean post quench state is described by the density matrix

$\rho({{t_{E}}})=e^{-H_{\theta}{{t_{E}}}}\ket{\psi_{AB}}\bra{\psi_{AB}}e^{H_{\theta}{{t_{E}}}}\,.$
(3.18)

The entanglement entropy for the interval $w\in[0,\sigma]$ in this state can
be evaluated via the one-point function of a replica twist operator

$S^{(n)}_{\sigma}=\frac{1}{1-n}\log{\rm
Tr}\rho_{\sigma}^{n}=\frac{1}{1-n}\log\bra{\psi_{AB}}e^{H_{\theta}{{t_{E}}}}\Phi_{n}(w=i\sigma,\bar{w}=-i\sigma)e^{-H_{\theta}{{t_{E}}}}\ket{\psi_{AB}}\,,$
(3.19)

analytically continuing the result to $n=1$ as in (2.6).

The evaluation of the time evolved one-point twist correlator in (3.19) can be
reduced to the computation in the UHP discussed in section 2. Mapping to the
UHP and implementing the time evolution with (3.7) one gets

$\displaystyle\bra{\psi_{AB}}e^{H_{\theta}{{t_{E}}}}\Phi_{n}(w,\bar{w})e^{-H_{\theta}{{t_{E}}}}\ket{\psi_{AB}}=$
$\displaystyle\left|\frac{\partial z_{{t_{E}}}}{\partial
w}\right|^{2h_{n}}\langle\psi_{AB}(\infty)\Phi_{n}(z_{{t_{E}}},\bar{z}_{{t_{E}}})\psi_{BA}(0)\rangle_{\rm
UHP}\,.$ (3.20)

This is analogus to (2.8), with the crucial difference that here we are
considering a time dependent state. This is reflected in the time dependent
insertion point $(z_{{t_{E}}},\bar{z}_{{t_{E}}})$ for the twist field and in
the conformal factor, which is now the composition of the map implementing the
time dependence with the one relating the strip to the UHP.

Following the approach used in the time independent case, we will approximate
$\langle\psi_{AB}(\infty)\Phi_{n}(z_{{t_{E}}},\bar{z}_{{t_{E}}})\psi_{BA}(0)\rangle_{\rm
UHP}$ with the contribution coming from the identity Virasoro block in the two
channels (2.9) and (2.10). After continuing to real time ${{t_{E}}}\to it$,
the correlator will be evaluated as the dominant channel at each time $t$. The
entanglement entropy, is then obtained as the analytic continuation of (3.19)
to $n=1$.

In the next subsection we work out the explicit expression for the correlator
(3.20) and its time dependence. We do this first for the simpler SSD case and
then for the Möbius evolution. The reader interested in the physical results
for the entanglement entropy can move directly to section 3.2.2.

####  3.2.1 Evaluation of the time dependent correlator

##### SSD deformation.

We consider first the correlator (3.20) approximated by the identity
contribution to the $A$-channel expansion. Reading from (2.17) the UHP result
in channel $A$ and analytically continuing to Lorentzian time with
${{t_{E}}}\to it$, we have

$\displaystyle\bra{\psi_{AB}}e^{iH_{\text{SSD}}t}\Phi_{n}(w,\bar{w})$
$\displaystyle e^{-iH_{\text{SSD}}t}\ket{\psi_{AB}}$ $\displaystyle\approx
C_{0}^{A}\alpha^{2h_{n}}\left(\frac{\partial z_{t}}{\partial
w}\right)^{h_{n}}\left(\frac{\partial\bar{z}_{t}}{\partial\bar{w}}\right)^{h_{n}}(z_{t}{\bar{z}}_{t})^{(\alpha-1)h_{n}}(z_{t}^{\alpha}-{\bar{z}}_{t}^{\alpha})^{-2h_{n}}\,.$
(3.21)

Making use of the explicit form of $z_{t}$ and ${\bar{z}}_{t}$ obtained from
(3.16) with $z=e^{\frac{i\pi\sigma}{L}}$ and ${{t_{E}}}\to it$

$\displaystyle z_{t}=\frac{L\cos\frac{\pi\sigma}{L}-2\pi
t\sin\frac{\pi\sigma}{L}+iL\sin\frac{\pi\sigma}{L}}{\sqrt{(L\cos\frac{\pi\sigma}{L}-2\pi
t\sin\frac{\pi\sigma}{L})^{2}+L^{2}\sin^{2}\frac{\pi\sigma}{L}}}\,,$ (3.22)
$\displaystyle\bar{z}_{t}=\frac{L\cos\frac{\pi\sigma}{L}+2\pi
t\sin\frac{\pi\sigma}{L}-iL\sin\frac{\pi\sigma}{L}}{\sqrt{(L\cos\frac{\pi\sigma}{L}+2\pi
t\sin\frac{\pi\sigma}{L})^{2}-L^{2}\sin^{2}\frac{\pi\sigma}{L}}}\,,$

gives

$\displaystyle\bra{\psi_{AB}}e^{iH_{\text{SSD}}t}$
$\displaystyle\Phi_{n}(w,\bar{w})e^{-iH_{\text{SSD}}t}\ket{\psi_{AB}}$ (3.23)
$\displaystyle=A_{\Phi_{n}}\left(\frac{\pi\alpha}{2L}\right)^{2h_{n}}\left[\left(f(t)^{2}+\sin^{2}\left(\frac{2\pi\sigma}{L}\right)\right)\sin^{2}\left(\frac{\alpha}{2}~{}\delta(t)\right)\right]^{-h_{n}}\,,$

where we used the explicit relation (2.16) between $C^{A}_{0}$ and $A_{\Phi}$.

In the above we have defined the function

$f(t)\equiv-\frac{2\pi^{2}t^{2}}{L^{2}}+\left(1+\frac{2\pi^{2}t^{2}}{L^{2}}\right)\cos\left(\frac{2\pi\sigma}{L}\right)\,,$
(3.24)

and the phase $\delta(t)$ in (3.23) is defined in terms of the ratio

$e^{i\delta}\equiv\frac{z_{t}}{{\bar{z}}_{t}}\,.$ (3.25)

The Lorentzian insertions $z_{t}$ and $\bar{z}_{t}$ in (3.22) move on the unit
circle defining each a time dependent phase, which difference gives
$\delta(t)$. In particular, $z_{t}$ goes from the initial value
$z_{t=0}=e^{\frac{i\pi\sigma}{L}}$ anti-clock-wisely to $e^{i\pi_{+}}$ for
$t\to\infty$. $\bar{z}_{t}$ goes from
${\bar{z}}_{t=0}=e^{-\frac{i\pi\sigma}{L}}$ anti-clock-wisely to $e^{i0_{-}}$.
The resulting ratio phase $\delta(t)$ goes from $2\pi\sigma/L$ to $\pi/L$
monotonically from below (above) for $\sigma<L/2$ ($\sigma>L/2$ ). This is
illustrated in fig. 2. The result can also be expressed in terms of the
function $f(t)$ defined in (3.24) as

$e^{i\delta}=\frac{f(t)+i\sin\left(\frac{2\pi\sigma}{L}\right)}{\sqrt{f(t)^{2}+\sin^{2}\left(\frac{2\pi\sigma}{L}\right)}}\,.$
(3.26)

Notice that in the limiting case $\sigma=L/2$ there is actually no time
evolution for the phase $\delta(t)$. Despite $z_{t}$ and ${\bar{z}}_{t}$
changing in time, their relative distance along the unit circle remains
constant and equal to $\pi$. The factor
$f(t)^{2}+\sin^{2}\left(\frac{2\pi\sigma}{L}\right)$ in (3.23) has
nevertheless a non-trivial evolution, which yields the time dependence for the
full correlator.

Figure 2: SSD quench. Real time evolution of the phases of $z_{t}$ in dashed
red, ${\bar{z}}_{t}$ in dashed blue, and $\delta(t)$ in solid orange. From
left to right, $\sigma/L=\frac{1}{4},\frac{1}{2},\frac{3}{4}$.

As discussed in section 2, one can perform a different expansion for the
three-point function and approximate the result with the $B$-channel Virasoro
identity block (2.19). The $A$-channel and $B$-channel blocks differ by a
monodromy, which however does not affect the conformal factor in (3.20). The
results for the $B$-channel follows in a straightforward manner from (2.19)
and is obtained performing the replacements $\delta(t)\to 2\pi-\delta(t)$ and
$A_{\Phi_{n}}\to B_{\Phi_{n}}$ in (3.23)

$\displaystyle\bra{\psi_{AB}}$ $\displaystyle
e^{iH_{\text{SSD}}t}\Phi(w,\bar{w})e^{-iH_{\text{SSD}}t}\ket{\psi_{AB}}$
$\displaystyle=B_{\Phi_{n}}\left(\frac{\pi\alpha}{2L}\right)^{2h_{n}}\left[\left(f(t)^{2}+\sin^{2}\left(\frac{2\pi\sigma}{L}\right)\right)\sin^{2}\left(\frac{\alpha}{2}\left(2\pi-\delta(t)\right)\right)\right]^{-h_{n}}\,.$
(3.27)

##### Möbius deformation.

For the Möbius case, the evaluation of the correlator (3.20) is performed with
a similar computation.

The relevant coordinate transformation implementing the Euclidean evolution is
now (3.14). Continuing to Lorentzian time ${{t_{E}}}\to it$ and defining for
compactness the rescaled time variable

$T\equiv\frac{\pi t}{L\cosh(2\theta)}\,,$ (3.28)

one obtains

$z_{t}=\frac{\cos(T)\cos\left(\frac{\pi\sigma}{L}\right)-e^{2\theta}\sin(T)\sin\left(\frac{\pi\sigma}{L}\right)+i\left(e^{-2\theta}\cos\left(\frac{\pi\sigma}{L}\right)\sin(T)+\cos(T)\sin\left(\frac{\pi\sigma}{L}\right)\right)}{\sqrt{\cos^{2}(T)-\sin(2T)\sin\left(\frac{2\pi\sigma}{L}\right)\sinh(2\theta)+\sin^{2}(T)\left(\cosh(4\theta)-\cos\left(\frac{2\pi\sigma}{L}\right)\sinh(4\theta)\right)}}\,.$
(3.29)

with the expression for ${\bar{z}}_{t}$ obtained via the replacement
$\sigma\to-\sigma$.

The correlator approximated by the identity block contribution in the
$A$-channel can be expressed in the form

$\displaystyle\bra{\psi_{AB}}$ $\displaystyle
e^{iH_{\theta}t}\Phi(w,\bar{w})e^{-iH_{\theta}t}\ket{\psi_{AB}}$
$\displaystyle=A_{\Phi_{n}}\left(\frac{\pi\alpha}{2L}\right)^{2h_{n}}\left[\left(f_{\theta}(T)^{2}+\sin^{2}\left(\frac{2\pi\sigma}{L}\right)\right)\sin^{2}\left(\frac{\alpha}{2}\delta_{\theta}(T)\right)\right]^{-h_{n}}\,,$
(3.30)

while the in $B$-channel it reads

$\displaystyle\bra{\psi_{AB}}$ $\displaystyle
e^{iH_{\theta}t}\Phi(w,\bar{w})e^{-iH_{\theta}t}\ket{\psi_{AB}}$
$\displaystyle=B_{\Phi_{n}}\left(\frac{\pi\alpha}{2L}\right)^{2h_{n}}\left[\left(f_{\theta}(T)^{2}+\sin^{2}\left(\frac{2\pi\sigma}{L}\right)\right)\sin^{2}\left(\frac{\alpha}{2}\left(2\pi-\delta_{\theta}(T)\right)\right)\right]^{-h_{n}}\,.$
(3.31)

The difference with respect to the SSD case completely resides in the explicit
form of the function $f_{\theta}$ and phase $\delta_{\theta}$. In particular,

$f_{\theta}(T)\equiv-\sin^{2}(T)\sinh(4\theta)+\left(\cos^{2}(T)+\cosh(4\theta)\sin^{2}(T)\right)\cos\left(\frac{2\pi\sigma}{L}\right)$
(3.32)

is the finite $\theta$ generalization of the function $f(t)$ defined above for
the SSD case and recovered as the $\theta\to\infty$ limit. The phase
$\delta_{\theta}(T)$ is similarly defined as

$e^{i\delta_{\theta}}\equiv\frac{z_{t}}{{\bar{z}}_{t}}\,$ (3.33)

with $z_{t}$ and ${\bar{z}}_{t}$ given in (3.29).

As time evolves, $z_{t}$ and ${\bar{z}}_{t}$ move around the unit circle with
$T$ periodicity $2\pi$. Within half a period, $z_{t}$ goes from the initial
value $z_{t=0}=e^{\frac{i\sigma\pi}{L}}$ to $e^{i\pi+\frac{i\sigma\pi}{L}}$,
and ${\bar{z}}_{t}$ from ${\bar{z}}_{t=0}=e^{-\frac{i\sigma\pi}{L}}$ to
$e^{i\pi-\frac{i\sigma\pi}{L}}$, as portrayed in fig. 3.

Figure 3: Möbius quench. Real time evolution of the phases of $z_{t}$ in
dashed red, ${\bar{z}}_{t}$ in dashed blue, and $\delta(t)$ in solid orange.
From left to right, $\sigma/L=\frac{1}{4},\frac{1}{2},\frac{3}{4}$. In all
plots $\theta=0.4$.

Despite the individual phases of $z_{t}$ and ${\bar{z}}_{t}$ always increase
in time, their difference yields an oscillatory behaviour for
$\delta_{\theta}(T)$ with periodicity $\pi$, as shown explicitly in fig. 3.
The time evolution of $\delta_{\theta}(T)$ can be expressed as

$e^{i\delta_{\theta}}=\frac{f_{\theta}(T)+i\sin\left(\frac{2\pi\sigma}{L}\right)}{\sqrt{f_{\theta}(T)^{2}+\sin^{2}\left(\frac{2\pi\sigma}{L}\right)}}\,.$
(3.34)

$\delta_{\theta}$ starts from the initial value $\frac{2\pi\sigma}{L}$ and
reaches an extremal value

$\delta_{\text{ext}}=\frac{2\pi\sigma}{L}+2\arctan\left(\frac{\sin\left(\frac{2\pi\sigma}{L}\right)}{\coth(2\theta)-\cos\left(\frac{2\pi\sigma}{L}\right)}\right)\,$
(3.35)

for $T=\pi/2$, before going back to the initial value. $\delta_{\text{ext}}$
represents a maximum for $\sigma<L/2$ and a minimum for $\sigma>L/2$, as
visible in fig. 3.

Figure 4: Möbius quench. The oscillatory behaviour for $\delta_{\theta}$ is
depicted as a function of the effective time $T$ and physical time $t$, for
$\sigma/L=0.45$ (left) and $\sigma/L=0.65$ (right). $\theta$ = 0.3 (green),
0.9 (blue), 1.5 (red).

For fixed $\sigma$, increasing values of $\theta$ give values of
$\delta_{\text{ext}}$ closer to $\pi$ (see fig. 4). In terms of the physical
time $t$, the oscillatory period increases with $\theta$ yielding an effective
description where the physical size of the system is rescaled to $L_{\rm
eff}\sim L\cosh(2\theta)$ (see, e.g., [48]). In the $\theta\to\infty$ limit
the effective size goes to infinity, suppressing the return phase, and
$\delta_{\text{ext}}\to\pi$ giving back the SSD evolution in fig. 2.

####  3.2.2 Evolution of the entanglement entropy

We summarize the result obtained in the previous subsection for the twist one-
point function (3.20) in the time dependent state and discuss the ensuing
entanglement entropy dynamics. We start considering the SSD case and then move
to the Möbius quench.

##### SSD quench.

Under the working assumption that the correlator (3.20) is well approximated
by the identity Virasoro block in the $A$\- or $B$-channel expansion, in
Lorentzian time we have

$\displaystyle\bra{\psi_{AB}}\Phi_{n}(w,{\bar{w}})\ket{\psi_{AB}}=$
$\displaystyle\left(\frac{\alpha\pi}{2L}\right)^{2h_{n}}\left(f(t)^{2}+\sin^{2}\left(\frac{2\pi\sigma}{L}\right)\right)^{-h_{n}}$
(3.36)
$\displaystyle\times\max\left\\{A_{\Phi_{n}}\left[\sin^{2}\left(\frac{\alpha}{2}\delta(t)\right)\right]^{-h_{n}};B_{\Phi_{n}}\left[\sin^{2}\left(\frac{\alpha}{2}(2\pi-\delta(t))\right)\right]^{-h_{n}}\right\\}\,,$

with $f(t)$ given in (3.24) and $\delta(t)$ in (3.26).

Plugging (3.36) into the expression for the Rényi entropies (3.19) and
continuing to $n=1$ gives the entanglement entropy for $t>0$

$\displaystyle S(t)=\frac{c}{12}\log$
$\displaystyle\left[\left(\frac{2L}{\pi\alpha\epsilon}\right)^{2}\left(f^{2}(t)+\sin^{2}\left(\frac{2\pi\sigma}{L}\right)\right)\right]$
(3.37)
$\displaystyle+\min\left\\{\frac{c}{12}\log\sin^{2}\left(\frac{\alpha}{2}\delta(t)\right)+s_{A};\frac{c}{12}\log\sin^{2}\left(\frac{\alpha}{2}\left(2\pi-\delta(t)\right)\right)+s_{B}\right\\}\,,$

where we have reinstated the UV regulator $\epsilon$.

$S(t)$ is thus obtained at each time minimizing over the two competing terms
inside the brackets, corresponding to the two channels. Depending on the
values of the different parameters, one channel can dominate for all times or
there can be a transition from one to the other contribution at some finite
time $t^{*}$.

Consider first a situation where $s_{A}=s_{B}$ and $0<\alpha<1$. This is
interpreted as the case where the initial state is a high energy eigenstate of
the undeformed Hamiltonian on the strip with equal boundary conditions (see
section 2.3). In this case, the initial value $\delta(t=0)=2\pi\sigma/L$
determines which of the two contributions in (3.37) dominates for all times.
In fact, the condition to have an exchange of dominance and a transition from
the initial channel to the other is to cross the value $\delta(t)=\pi$ at some
finite time $t^{*}$. However this is only reached asymptotically for
$t\to\infty$ in the SSD case (see fig. 2). The resulting evolution is then
qualitatively similar to the case of the ground state with equal boundary
conditions on the two sides of the strip, studied in [48].777This is recovered
for $\alpha=1$, in which case the two contributions in (3.37) coincide. Indeed
the late time growth is $S(t)\sim\frac{c}{3}\log t$ and does not exhibit any
revival, compatibly with an effective infinite length for the deformed
dynamics. It also admits a quasi-particle interpretation, as detailed in [48].

For general values of $s_{A}$ and $s_{B}$, the late time behaviour is
unchanged, but at intermediate times the situation is richer and a dynamical
exchange of dominance between channel $A$ and $B$ may occur. Figure 5
illustrates an example of how different values of the boundary entropy give
different patterns.

Figure 5: Plot of the entanglement entropy (3.37), rescaled by $12/c$, with
the initial value subtracted. The solid line is the entanglement entropy
obtained at each time with the minimization procedure over the two competing
channels in (3.37). The dot-dashed lines correspond to the two competing
channels in their non-dominant phase, red for channel $A$ and green for
channel $B$. The plots show the case of $\sigma/L=0.15$ and $\alpha=0.5$, for
$\frac{12}{c}(s_{B}-s_{A})=1,-1,-2,-3$ from top-left to bottom-right.

In order to have a transition the two contributions in (3.37) should exchange
dominance at some time $t^{*}$. In other words,

$\frac{c}{6}\log\frac{\sin\left(\frac{\alpha}{2}\delta(t)\right)}{\sin\left(\frac{\alpha}{2}\left(2\pi-\delta(t)\right)\right)}=s_{B}-s_{A}\,.$
(3.38)

must have a solution at some value of time $t^{*}$. In the SSD case this
condition is not satisfied for arbitrary values of $s_{A}$ and $s_{B}$.

The left hand side of (3.38) is always negative and a monotonically increasing
function of $\delta$. For $0<\sigma\leq L/2$, $\delta(t)$ increases
monotonically from $2\pi\sigma/L$ at $t=0$ to $\pi$ for $t\to\infty$. To
observe a transition at some finite time $t^{*}$, the boundary entropies must
then fall within the range

$\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\pi\sigma}{L}\right)}{\sin\left(\frac{\alpha\pi(L-\sigma)}{L}\right)}<s_{B}-s_{A}<0\
.$ (3.39)

The monotonicity of $\delta(t)$ also implies that when this condition is
satisfied, the initial value of the entropy corresponds to the contribution
given by channel $A$. Therefore the transition will be from channel $A$ to
channel $B$ as time increases. On the other hand, when the inequality is
violated there is no transition and the dominant channels are

$\displaystyle A\text{-channel}$
$\displaystyle~{}~{}\text{for}~{}~{}s_{B}-s_{A}>0\,$ (3.40) $\displaystyle
B\text{-channel}$
$\displaystyle~{}~{}\text{for}~{}~{}s_{B}-s_{A}<\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\pi\sigma}{L}\right)}{\sin\left(\frac{\alpha\pi(L-\sigma)}{L}\right)}.$

Notice that in the limiting case $\sigma=L/2$ the window of values (3.39) for
the boundary entropies closes and there is no dynamical transition despite
having a time evolving entanglement entropy.

For $L/2<\sigma<L$ the situation is similar, but $\delta(t)$ now decreases
monotonically in time from $2\pi\sigma/L$ to $\pi$. In practical terms, the
dynamics can be described inverting $A$ and $B$ and a transition from channel
B to channel A happens for

$0<s_{B}-s_{A}<\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\pi\sigma}{L}\right)}{\sin\left(\frac{\alpha\pi(L-\sigma)}{L}\right)}\,.$
(3.41)

The complementary range of $s_{B}-s_{A}$, where there is no transition, can
instead be summarized as

$\displaystyle B\text{-channel}$
$\displaystyle~{}~{}\text{for}~{}~{}s_{B}-s_{A}<0\,$ (3.42) $\displaystyle
A\text{-channel}$
$\displaystyle~{}~{}\text{for}~{}~{}s_{B}-s_{A}>\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\pi\sigma}{L}\right)}{\sin\left(\frac{\alpha\pi(L-\sigma)}{L}\right)}.$

As shown in fig. 5, for $0<\sigma\leq L/2$, the transition happens at earlier
times for smaller values of $s_{B}-s_{A}$. In fact, $\delta(t)$ is a
monotonically increasing function of time and solving (3.38) for
$\delta(t^{*})$ gives the relation

$\cot\left(\frac{\alpha}{2}\delta(t^{*})\right)=\cot(\pi\alpha)+\frac{e^{6(s_{B}-s_{A})/c}}{\sin(\pi\alpha)}\,,$
(3.43)

On the contrary, for $L/2<\sigma<L$ the transition happens at larger times for
decreasing values of $s_{B}-s_{A}$.

##### Möbius deformation.

The Möbius result for the time dependent three point function and the
corresponding entanglement entropy can be written in the same form as for the
SSD quench

$\displaystyle\bra{\psi_{AB}}\Phi_{n}(w,{\bar{w}})\ket{\psi_{AB}}=$
$\displaystyle\left(\frac{\alpha\pi}{2L}\right)^{2h_{n}}\left(f_{\theta}(T)^{2}+\sin^{2}\left(\frac{2\pi\sigma}{L}\right)\right)^{-h_{n}}$
(3.44)
$\displaystyle\times\max\left\\{A_{\Phi_{n}}\left[\sin^{2}\left(\frac{\alpha}{2}\delta_{\theta}(T)\right)\right]^{-h_{n}};B_{\Phi_{n}}\left[\sin^{2}\left(\frac{\alpha}{2}(2\pi-\delta_{\theta}(T))\right)\right]^{-h_{n}}\right\\}\,,$

and

$\displaystyle S(T)=\frac{c}{12}\log$
$\displaystyle\left[\left(\frac{2L}{\pi\alpha\epsilon}\right)^{2}\left(f_{\theta}(T)^{2}+\sin^{2}\left(\frac{2\pi\sigma}{L}\right)\right)\right]$
(3.45)
$\displaystyle+\min\left\\{\frac{c}{12}\log\sin^{2}\left(\frac{\alpha}{2}\delta_{\theta}(T)\right)+s_{A};\frac{c}{12}\log\sin^{2}\left(\frac{\alpha}{2}\left(2\pi-\delta_{\theta}(T)\right)\right)+s_{B}\right\\}\,.$

The difference resides in the definition of the function $f_{\theta}$, in
(3.32), and in the time oscillating phase $\delta_{\theta}(T)$ (see equation
(3.34) and discussion around it).

Let us illustrate the case $0<\sigma\leq L/2$ for simplicity. The physics in
the complementary range of $\sigma$ is easily obtained and substantially
amounts to interchanging $A$ and $B$ in the following discussion.

The phase $\delta_{\theta}(T)$ oscillates with periodicity $\pi$ between the
initial value $2\pi\sigma/L$ and the extremal, maximum, value
$\delta_{\text{ext}}<\pi$ in (3.35) (see fig. 4). To have a dynamical phase
transition between the two channels in (3.45) the boundary entropies
difference must fall within the range

$\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\pi\sigma}{L}\right)}{\sin\left(\frac{\alpha\pi(L-\sigma)}{L}\right)}<s_{B}-s_{A}<\frac{c}{6}\log\frac{\sin\left(\frac{\alpha}{2}\delta_{\text{ext}}\right)}{\sin\left(\frac{\alpha}{2}\left(2\pi-\delta_{\text{ext}}\right)\right)}<0\
.$ (3.46)

As compared to the SSD case, the range of $s_{B}-s_{A}$ which leads to a
transition is reduced. Smaller values of the deformation parameter $\theta$
correspond to a smaller range of $s_{B}-s_{A}$ available for the transition to
happen. In the limit $\theta\to 0$, the transition window in (3.46) closes
reflecting the fact that there is no dynamics when the deformation is off.

An example of the entanglement entropy evolution following from the
minimization procedure is depicted in fig. 6. The different panels show how
depending on the value of $s_{B}-s_{A}$ there is or not a transition.

Figure 6: Plot of the entanglement entropy, rescaled by $12/c$, with the
initial value subtracted. In all plots $\sigma/L=0.15,\alpha=0.5$ and
$\theta=0.3$. From left to right $\frac{12}{c}(s_{B}-s_{A})=-1,-2,-3$. The
dot-dashed lines correspond to channel A (red) and B (green) when in their
non-dominant phase. The solid line is the entanglement entropy. The dominant
channel is $A$ in the left plot and $B$ in the right plot. In the middle there
is a transition from channel $A$ to $B$ and back to $A$ happening within each
period.

Similarly to the SSD case, outside the range (3.46) there is no transition and
the term within brackets in (3.45) is minimized by the

$\displaystyle A\text{-channel}$
$\displaystyle~{}~{}\text{for}~{}~{}s_{B}-s_{A}>\frac{c}{6}\log\frac{\sin\left(\frac{\alpha}{2}\delta_{\text{ext}}\right)}{\sin\left(\frac{\alpha}{2}\left(2\pi-\delta_{\text{ext}}\right)\right)}\,,$
(3.47) $\displaystyle B\text{-channel}$
$\displaystyle~{}~{}\text{for}~{}~{}s_{B}-s_{A}<\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\pi\sigma}{L}\right)}{\sin\left(\frac{\alpha\pi(L-\sigma)}{L}\right)}.$

In fig. 7 we also report sample plots of the time dependence of the
entanglement entropy for different values of $\theta$.

Figure 7: Plot of the entanglement entropy with the initial value subtracted
and rescaled by $12/c$. The plots are for $\sigma/L=0.15,\alpha=0.5$ and
$s_{B}-s_{A}=-\frac{c}{6}$. $\theta$ = 0.3 (green), 0.9 (blue), 1.5 (red). The
plot in terms of the physical time $t$ displays an evolution compatible with
an effective spatial size $L_{\text{eff}}\sim L\cosh(2\theta)$.

One important difference with respect to the SSD quench is the oscillatory
character of the dynamics that characterizes the Möbius case. When present,
the transition happens at times $T^{*}$ where

$\frac{c}{6}\log\frac{\sin\left(\frac{\alpha}{2}\delta_{\theta}(T^{*})\right)}{\sin\left(\frac{\alpha}{2}\left(2\pi-\delta_{\theta}(T^{*})\right)\right)}=s_{B}-s_{A}\,.$
(3.48)

As seen explicitly in the central panel of fig. 6, when a first transition
from channel $A$ to channel $B$ happens at a certain time $T^{*}$ this will be
necessarily followed by a transition back to channel $A$ at a time
$\pi-T^{*}$, and so forth for each period of the effective time $T$.

##  4 Quench with generally deformed Hamiltonians

We have discussed quenches induced by the SSD and Möbius Hamiltonians. On the
strip these are constructed through a deformation of the usual CFT Hamiltonian
given in terms of level-two Virasoro generators $L_{\pm 2}$.

This construction can be extended to define a general class of deformed
Hamiltonians obtained from any subset of the Virasoro generators
$\\{L_{0},L_{\pm k}\\}$ with $k\in\mathbb{Z}^{+}$ (see, e.g., [77, 78, 79,
80]), which also form a $sl(2,\mathbb{R})$ algebra [74, 75]. In this section
we extend our analysis to this general class of deformed Hamiltonians.

###  4.1 A general class of deformed Hamiltonians

In analogy with the Möbius Hamiltonian (3.5), the general class of deformed
Hamiltonians constructed from $\\{L_{0},L_{\pm k}\\}$ can be written as

$H_{k}\equiv H_{0}-\frac{\tanh(2\theta)}{2}(H^{+}_{k}+H^{-}_{k})$ (4.1)

where $H_{0}$ is the undeformed Hamiltonian (3.3) on an interval of width $L$,
and the deformations $H^{\pm}_{k}$ are defined as

$H^{\pm}_{k}=\int_{0}^{L}\frac{dw}{2\pi}\left(e^{\pm k\pi w/L}T(w)+e^{\mp
k\pi{\bar{w}}/L}\bar{T}({\bar{w}})\right)\,,$ (4.2)

with $T(w)$ and $\bar{T}(\bar{w})$ defined on the strip.

Mapping the strip to the UHP via $z=e^{\frac{\pi}{L}w}$, we can then write

$\displaystyle
H_{k}=\frac{\pi}{L\cosh(2\theta)}\oint\left[\cosh(2\theta)z-\frac{\sinh(2\theta)}{2}(z^{-k+1}+z^{k+1})\right]T(z)-\frac{c\pi}{24L}\,,$
(4.3)

where the contour integral is in the full complex plane (see appendix A for
details).

This makes it explicit that $H_{k}^{\pm}$ can be expressed in terms of
Virasoro generators in the complex plane, giving

$H^{+}_{k}+H^{-}_{k}=\frac{\pi}{L}\oint\left(z^{k+1}T(z)+z^{-k+1}T(z)\right)=\frac{\pi}{L}\left(L_{k}+L_{-k}\right)\,.$
(4.4)

Employing the map

$\tilde{z}^{k}=-\frac{z^{k}\cosh\theta-\sinh\theta}{z^{k}\sinh\theta-\cosh\theta}\,,$
(4.5)

allows to bring the total Hamiltonian into the form

$\displaystyle H_{k}$
$\displaystyle=\frac{\pi}{L}\left[\frac{1}{\cosh(2\theta)}\oint\tilde{z}\,T(\tilde{z})-\frac{c}{24}\left(1-\frac{k^{2}-1}{\cosh(2\theta)}\right)\right]$
(4.6)
$\displaystyle=\frac{\pi}{L}\left[\frac{1}{\cosh(2\theta)}\tilde{L}_{0}-\frac{c}{24}\left(1-\frac{k^{2}-1}{\cosh(2\theta)}\right)\right]\,,$

where $\tilde{L}_{0}$ indicates the dilatation generator in the
$\tilde{z}$-coordinate. Therefore, for a primary ${\cal O}$

$e^{H_{k}{{t_{E}}}}{\cal
O}(\tilde{z},\bar{\tilde{z}})e^{-H_{k}{{t_{E}}}}=\lambda^{2h_{\cal O}}{\cal
O}(\lambda\tilde{z},\lambda\bar{\tilde{z}})\,.$ (4.7)

with

$\lambda=\exp{\frac{\pi{{t_{E}}}}{L\cosh(2\theta)}}\,.$ (4.8)

Mapping back to the UHP, the Euclidean evolution of a CFT primary operator
${\cal O}$ under the deformed Hamiltonian can be expressed using (3.7) with
the coordinate transformation

$z_{{t_{E}}}^{k}=\frac{\left[(1-\lambda^{k})\cosh(2\theta)-(1+\lambda^{k})\right]z^{k}-(1-\lambda^{k})\sinh(2\theta)}{(1-\lambda^{k})\sinh(2\theta)z^{k}-(1-\lambda^{k})\cosh(2\theta)-(1+\lambda^{k})}\,.$
(4.9)

This can be seen as a $SL(2,\mathbb{R})$ transformation in the $k$-covering
space of the original coordinate $z$ and generalizes the result for the
Hamiltonians discussed in the previous section, which is readily recovered for
$k=2$.

###  4.2 Entanglement entropy after the quench

The time evolution of the entanglement entropy after a quench to the generally
deformed Hamiltonian $H_{k}$ is obtained with the analytic continuation to
$n=1$ of the Rényi calculation (2.5)

$S^{(n)}_{\sigma}=\frac{1}{1-n}\log{\rm
Tr}\rho_{\sigma}^{n}=\frac{1}{1-n}\log\bra{\psi_{AB}}e^{H_{k}{{t_{E}}}}\Phi_{n}(w=i\sigma,\bar{w}=-i\sigma)e^{-H_{k}{{t_{E}}}}\ket{\psi_{AB}}\,.$
(4.10)

Employing $z_{{t_{E}}}$ from (4.9) in the evolution equation (3.7), one
implements the time dependence as

$\displaystyle\bra{\psi_{AB}}e^{H_{k}t_{E}}$
$\displaystyle\Phi_{n}(w,\bar{w})e^{-H_{k}t_{E}}\ket{\psi_{AB}}$
$\displaystyle=\left(\frac{\partial z_{t_{E}}}{\partial
w}\right)^{h_{n}}\left(\frac{\partial{\bar{z}}_{t_{E}}}{\partial{\bar{w}}}\right)^{h_{n}}\langle\psi_{AB}(\infty)\Phi_{n}(z_{t_{E}},\bar{z}_{t_{E}})\psi_{BA}(0)\rangle_{\rm
UHP}\,.$ (4.11)

####  4.2.1 Evaluation of the time dependent correlator

Continuing (4.9) to real time $t_{E}\to it$ gives

$z^{k}_{t}=\frac{\left(\cos(T_{k})\cos\left(\frac{k\pi\sigma}{2L}\right)-e^{2\theta}\sin(T_{k})\sin\left(\frac{k\pi\sigma}{2L}\right)+i\left(e^{-2\theta}\cos\left(\frac{k\pi\sigma}{2L}\right)\sin(T_{k})+\cos(T_{k})\sin\left(\frac{k\pi\sigma}{2L}\right)\right)\right)^{2}}{{\cos^{2}(T_{k})-\sin(2T_{k})\sin\left(\frac{k\pi\sigma}{L}\right)\sinh(2\theta)+\sin^{2}(T_{k})\left(\cosh(4\theta)-\cos\left(\frac{k\pi\sigma}{L}\right)\sinh(4\theta)\right)}}$
(4.12)

with ${\bar{z}}_{t}$ obtained via the replacement $\sigma\to-\sigma$. The
rescaled time variable $T_{k}$ is defined as

$T_{k}\equiv\frac{k\pi t}{2L\cosh(2\theta)}=\frac{kT_{\theta}}{2}\,.$ (4.13)

The Lorentzian insertions $z_{t}$ and $\bar{z}_{t}$ start from their
respective initial values, $z_{t=0}=e^{i\frac{\pi\sigma}{L}}$ and
$\bar{z}_{t=0}=e^{-i\frac{\pi\sigma}{L}}$, and move on the unit circle with
phases increasing in time. They go around the circle once over a rescaled
$T_{k}$ period $k\pi$ (see fig. 8).

Figure 8: Real time evolution of the phases of $z_{t}$ dashed red,
${\bar{z}}_{t}$ dashed blue, and $\delta_{k}$. From left to right, $k=3,6,9$.
In all plots $\theta=0.4$, $\sigma/L=\frac{1}{5}$.

Using (4.12) and evaluating the UHP correlator in (4.2) as the identity
Virasoro block contribution in channel $A$ or $B$ gives

$\displaystyle\bra{\psi_{AB}}e^{H_{k}t_{E}}\Phi_{n}(w,\bar{w})$ $\displaystyle
e^{-H_{k}t_{E}}\ket{\psi_{AB}}=\left(\frac{\alpha\pi}{2L}\right)^{2h_{n}}\left(f_{k}(T_{k})^{2}+\sin^{2}\left(\frac{k\pi\sigma}{L}\right)\right)^{-h_{n}}$
(4.14)
$\displaystyle\times\max\left\\{A_{\Phi_{n}}\left[\sin^{2}\left(\frac{\alpha}{2}\delta_{k}(T_{k})\right)\right]^{-h_{n}};\,B_{\Phi_{n}}\left[\sin^{2}\left(\frac{\alpha}{2}(2\pi-\delta_{k}(T_{k}))\right)\right]^{-h_{n}}\right\\}\,.$

This is the analogue of (3.36) and (3.44) for the SSD and Möbius cases. In
particular, $f_{k}$ is a generalization of the function $f_{\theta}$ defined
in equation (3.32) for the Möbius Hamiltonian888Notice that $f_{k}$ also
depends on the deformation parameter $\theta$.

$f_{k}(T_{k})\equiv-\sin^{2}(T_{k})\sinh(4\theta)+\left(\cos^{2}(T_{k})+\cosh(4\theta)\sin^{2}(T_{k})\right)\cos\left(\frac{k\pi\sigma}{L}\right)\,.$
(4.15)

$\delta_{k}$ is the phase difference between $z_{t}$ and $\bar{z}_{t}$, which
can be expressed as

$e^{i\delta_{k}}\equiv\frac{z_{t}}{{\bar{z}}_{t}}=\left(\frac{\left(f_{k}(T_{k})+i\sin\left(\frac{k\pi\sigma}{L}\right)\right)^{2}}{f_{k}(T_{k})^{2}+\sin^{2}\left(\frac{k\pi\sigma}{L}\right)}\right)^{\frac{1}{k}}\,.$
(4.16)

As displayed in fig. 8, $\delta_{k}(T_{k})$ oscillates with periodicity $\pi$
between the initial value $\frac{2\pi\sigma}{L}$ and the extremal value

$\delta_{k,\text{ext}}=\frac{2\pi\sigma}{L}+\frac{4}{k}\arctan\left(\frac{\sin\left(\frac{k\pi\sigma}{L}\right)}{\coth(2\theta)-\cos\left(\frac{k\pi\sigma}{L}\right)}\right)\,.$
(4.17)

$\delta_{k,\text{ext}}$ represents a maximum or a minimum depending on the
initial value, i.e., depending on the size $\sigma$ of the entangling region.
For $\sin\left(\frac{k\pi\sigma}{L}\right)$ positive (negative),
$\delta_{k,\text{ext}}$ is the maximum (minimum) value reached at half periods
of the effective time $T_{k}=\pi/2,3\pi/2,\dots$ . A few cases for different
values of $k$ are illustrated in fig. 9.

Figure 9: Evolution of the phase $\delta_{k}$ with $\sigma/L=0.45$. The solid
lines are for $k=2,5$ going from the bottom-up at $\delta_{k,\text{ext}}$
which is a maximum, i.e., $\frac{2m}{k}<\frac{\sigma}{L}<\frac{2m+1}{k}$. The
dashed-dotted lines are for $\delta_{k,\text{ext}}$ corresponding to a
minimum, i.e., $\frac{2m+1}{k}<\frac{\sigma}{L}<\frac{2(m+1)}{k}$, with
$k=3,7$ from the bottom-up. $\theta=0.3$ on the left and $\theta=0.6$ on the
right, showing how the deformation affects the amplitude of the oscillations.

In the Möbius case $(k=2)$ the initial value was always the minimum (maximum)
value for $\sigma$ smaller (larger) than half of the system (see fig. 4). Here
instead the extra parameter $k$ allows for both cases for any value of
$\sigma$.

We briefly comment on the equivalent of the SSD limit, $\theta\to\infty$. Also
here, in the limit, the returning phase which in the Möbius case distinguished
the finite $\theta$ deformation from the SSD case disappears. In particular
the phase $\delta_{k}$ interpolates monotonically between the initial value
and $\delta_{k,\text{ext}}$ in (4.17) in the limit $\theta\to\infty$. This is
attained asymptotically for $t\to\infty$ and reads

$\delta_{k,\text{ext}}=2\pi\frac{(2m+1)}{k}~{}~{}~{}\text{where $m$ is such
that}~{}~{}~{}\frac{2m}{k}\leq\frac{\sigma}{L}<\frac{2(m+1)}{k}\,.$ (4.18)

The $k=2$ case corresponds to the Möbius and SSD case. There $m=0$ for any
$\sigma$, giving $\delta_{2,\text{ext}}=\pi$, consistently with the
$t\to\infty$ limit of the phase in the SSD case discussed in section 3.2.

####  4.2.2 Boundary-induced phase transition

The expression for the entanglement entropy follows in a straightforward
manner using the result (4.14) in (4.10) and continuing to $n=1$

$\displaystyle S(T)=\frac{c}{12}\log$
$\displaystyle\left[\left(\frac{2L}{\pi\alpha\epsilon}\right)^{2}\left(f_{k}(T_{k})^{2}+\sin^{2}\left(\frac{k\pi\sigma}{L}\right)\right)\right]$
(4.19)
$\displaystyle+\min\left\\{\frac{c}{12}\log\sin^{2}\left(\frac{\alpha}{2}\delta_{k}(T_{k})\right)+s_{A};\frac{c}{12}\log\sin^{2}\left(\frac{\alpha}{2}\left(2\pi-\delta_{k}(T_{k})\right)\right)+s_{B}\right\\}\,.$

$f_{k}(T_{k})$ is defined in equation (4.15) and $\delta_{k}$ oscillates
between the initial value $\delta_{k}(T_{k}=0)=2\pi\sigma/L$ and the extremal
value $\delta_{k,\text{ext}}$ in (4.17). Minimizing between the two channels
at each instant of time gives the entanglement entropy evolution. Figure 10
illustrate some examples for different choices of the parameters.

Figure 10: Plot of the entanglement entropy evolution, rescaled by $12/c$ with
the initial value subtracted. The solid line is the entanglement entropy
obtained from (4.19). The dot-dashed lines correspond to channel $A$ (red) and
$B$ (green) when in their non-dominant phase. In all plots
$\sigma/L=0.15,\alpha=0.5$ and $\theta=0.3$. Top: $k=3$ and it corresponds to
the case $\frac{2m}{k}<\frac{\sigma}{L}<\frac{2m+1}{k}$, with $m=0$. From left
to right $\frac{12}{c}(s_{B}-s_{A})=-1,-2,-3$. Bottom: $k=8$ and this
situation corresponds to the case
$\frac{2m+1}{k}<\frac{\sigma}{L}<\frac{2(m+1)}{k}$, with $m=0$. From left to
right $\frac{12}{c}(s_{B}-s_{A})=-2,-3,-4$.

Similarly to the Möbius quench, a dynamical transition between the two
channels can happen only for specific ranges of the boundary entropies. This
condition can be summarized as

$\displaystyle\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\pi\sigma}{L}\right)}{\sin\left(\frac{\alpha\pi(L-\sigma)}{L}\right)}<s_{B}-s_{A}<\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\delta_{k,\text{ext}}}{2}\right)}{\sin\left(\frac{\alpha(2\pi-\delta_{k,\text{ext}})}{2}\right)}~{}$
for $\displaystyle\frac{2m}{k}<\frac{\sigma}{L}<\frac{2m+1}{k}\,,$ (4.20)
$\displaystyle\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\delta_{k,\text{ext}}}{2}\right)}{\sin\left(\frac{\alpha(2\pi-\delta_{k,\text{ext}})}{2}\right)}<s_{B}-s_{A}<\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\pi\sigma}{L}\right)}{\sin\left(\frac{\alpha\pi(L-\sigma)}{L}\right)}~{}$
for $\displaystyle\frac{2m+1}{k}<\frac{\sigma}{L}<\frac{2(m+1)}{k}$

with $m=0,1,2,\dots,[\frac{k}{2}-1]$, with $[\cdot]$ indicating the integer
part.

For a given set of parameters, when $s_{B}-s_{A}$ falls in the above range one
observes a phase transition between the entanglement evaluated in the two
channels at each time $T_{k}^{*}$ such that

$\frac{c}{6}\log\frac{\sin\left(\frac{\alpha}{2}\delta_{k}(T_{k}^{*})\right)}{\sin\left(\frac{\alpha}{2}\left(2\pi-\delta_{k}(T_{k}^{*})\right)\right)}=s_{B}-s_{A}\,.$
(4.21)

The periodic behavior induced by the Möbius type deformation at finite
$\theta$ implies that the transition happens twice during a period of $T_{k}$.
One goes from the initial channel to the other at some time $T_{k}^{*}$ and
then back to the initial channel at $\pi-T_{k}^{*}$.

When the difference of boundary entropies is outside the range (4.20), one
channel always dominates over the other. More precisely, when
$\frac{2m}{k}<\frac{\sigma}{L}<\frac{2m+1}{k}$, the dominant channels are

$\displaystyle A\text{-channel}$
$\displaystyle~{}~{}\text{for}~{}~{}s_{B}-s_{A}\geq\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\delta_{k,\text{ext}}}{2}\right)}{\sin\left(\frac{\alpha(2\pi-\delta_{k,\text{ext}})}{2}\right)}\,,$
(4.22) $\displaystyle B\text{-channel}$
$\displaystyle~{}~{}\text{for}~{}~{}s_{B}-s_{A}\leq\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\pi\sigma}{L}\right)}{\sin\left(\frac{\alpha\pi(L-\sigma)}{L}\right)}\,.$

In the complementary range $\frac{2m+1}{k}<\frac{\sigma}{L}<\frac{2(m+1)}{k}$,
$A$ and $B$ simply get exchanged

$\displaystyle B\text{-channel}$
$\displaystyle~{}~{}\text{for}~{}~{}s_{B}-s_{A}\leq\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\delta_{k,\text{ext}}}{2}\right)}{\sin\left(\frac{\alpha(2\pi-\delta_{k,\text{ext}})}{2}\right)}\,,$
(4.23) $\displaystyle A\text{-channel}$
$\displaystyle~{}~{}\text{for}~{}~{}s_{B}-s_{A}\geq\frac{c}{6}\log\frac{\sin\left(\frac{\alpha\pi\sigma}{L}\right)}{\sin\left(\frac{\alpha\pi(L-\sigma)}{L}\right)}\,.$

##  5 Holographic dual

In this section we present the gravity dual to the BCFT setup with distinct
boundary conditions of section 2.1 and evaluate the holographic entanglement
entropy. We then show how to reproduce holographically the result for the time
dependent entanglement entropy in the post quench state.

Discussions of aspects of the holographic dual of the BCFT setup and related
AdS/BCFT constructions appeared in [23, 21, 22]. Here we will closely follow
the approach of [21], while modifying their construction as to adapt it to a
BCFT on an interval at zero temperature.

###  5.1 Gravitational action and equations of motion

The holographic description of the BCFT setup of section 2.1 involves two end
of the world (EOW) branes with different tensions, intersecting at a defect in
the bulk of global AdS3. The branes single out a region of the AdS3 spacetime
which provides the dual geometry for the BCFT on an interval with mixed
boundary conditions (see fig. 11).

Figure 11: On a given Cauchy slice, the two EOW branes corresponding to the
two conformal boundary conditions imposed in the CFT anchor at $\phi_{0}$ and
$-\phi_{0}$ on the asymptotic boundary and intersect in the bulk. The shaded
region is the gravitational dual to the CFT we are considering. A defect in
the bulk supports the intersection of the two branes. For a given boundary
interval $\Delta\phi$ there will be in general two geodesic configurations
$\gamma_{A}$ and $\gamma_{B}$ emanating from the asymptotic boundary and
anchoring on the corresponding EOW brane.

The total gravitational action of the holographic dual is given by [21]

$I=I_{\rm EH}+\sum_{i=A,B}I_{\rm EOW,i}+I_{\rm DEFECT}+I_{\rm CT}\,.$ (5.1)

The first contribution is the Einstein-Hilbert action with negative
cosmological constant $\Lambda$ plus a Gibbons-Hawking-York term at the AdS
boundary $\partial$AdS

$I_{\rm EH}=-\frac{1}{16\pi G_{N}}\int_{\rm
AdS}\sqrt{-g}\left(R-2\Lambda\right)-\frac{1}{8\pi G_{N}}\int_{\partial{\rm
AdS}}\sqrt{-\gamma}K\,.$ (5.2)

To implement holographically the two distinct boundary conditions of the BCFT,
we add in the AdS3 bulk two EOW branes $\Sigma_{i}$.999While strictly speaking
these objects are strings in AdS3, we will use the nomenclature that is most
common in this literature and refer to them as branes, as in the higher
dimensional settings. The anchoring points on the boundary $\partial$AdS
corresponds to the BCFT interval endpoints. For purely tensional branes, the
action is

$I_{\rm EOW,i}=\frac{1}{8\pi
G_{N}}\int_{\Sigma_{i}}\sqrt{-h}T_{i}-\frac{1}{8\pi
G_{N}}\int_{\Sigma_{i}}\sqrt{-h}K-\frac{1}{8\pi G_{N}}\int_{\partial{\rm
AdS}\cap\Sigma_{i}}\sqrt{-\delta}\left(\pi-\Theta_{i}\right)\,.$ (5.3)

Here $h_{\alpha\beta}$ is the induced metric on the brane and
$K_{\alpha\beta}$ its extrinsic curvature. For each brane the equations of
motion read

$K_{\alpha\beta}=(K-T_{i})h_{\alpha\beta}\,.$ (5.4)

This sets the shape of the EOW brane $\Sigma_{i}$ in terms of its tension
$T_{i}$ and boundary conditions.

The last term in (5.3) is a Hayward term at the corner $\partial{\rm
AdS}\cap\Sigma_{i}$ between the AdS boundary and each brane $\Sigma_{i}$.
$\delta$ is the induced metric and the internal angle $\Theta_{i}=\pi$ on
shell.

The two branes have in general different tensions $T_{i}$ and will intersect
in the bulk on a worldline $\Gamma$. There one needs to include an interaction
term in the form of a defect to hold together two distinct branes. The
simplest such action is a slight modification of a Hayward term given by

$I_{\rm DEFECT}=-\frac{1}{16\pi
G_{N}}\int_{\Gamma}\sqrt{-h_{\Gamma}}\left(\theta_{0}-\hat{\theta}\right)\,.$
(5.5)

$\theta_{0}-\pi$ can be regarded as the effective tension of the defect and
$\hat{\theta}$ is the internal angle at the branes intersection. On shell this
is related to the defect tension via the condition

$\hat{\theta}=\theta_{0}\,.$ (5.6)

Finally, the action includes appropriate counterterms $I_{\rm CT}$ to
renormalize the divergences coming from near the AdS boundary [21].

###  5.2 Intersecting branes in AdS3

We first briefly review the AdS dual to a BCFT on an interval with equal
boundary conditions at the endpoints.

Consider empty AdS3 with metric

$ds^{2}=-\frac{r^{2}+1}{R^{2}}dt^{2}+\frac{dr^{2}}{r^{2}+1}+r^{2}d\phi^{2}\,,$
(5.7)

with $r\in[0,\infty)$, $t\in(-\infty,\infty)$ and $\phi\in[-\pi,\pi]$. The AdS
radius is set to one and $R$ denotes the radius of the boundary circle.

Take a purely tensional brane of tension $T$ emanating from a point
$0\leq\phi_{0}<\pi$ on $\partial$AdS3. On shell, the brane has the following
static profile

$r=\frac{k}{\sin\left(\phi-\phi_{0}\right)}\,,$ (5.8)

where we have defined the combination

$-\infty<k\equiv\frac{T}{\sqrt{1-T^{2}}}<\infty\,.$ (5.9)

The brane connects antipodal points on $\partial$AdS3. If $k<0$ the brane
spans in the bulk the angular interval $-\pi+\phi_{0}\leq\phi\leq\phi_{0}$
while for $k>0$ it spans the interval
$\phi_{0}\leq\phi\leq\pi\cup-\pi\leq\phi\leq-\pi+\phi_{0}$. Retaining the AdS
bulk geometry comprised between the brane and the portion of boundary with
angular range $-\pi+\phi_{0}\leq\phi\leq\phi_{0}$ gives the dual for the BCFT
on the strip with equal boundary conditions (see for example [16, 17]).

To implement two distinct boundary conditions as in section 2.1, we consider
two static branes anchored at symmetric points about $\phi=0$, i.e., at
$\pm\phi_{0}$ with $0<\phi_{0}<\frac{\pi}{2}$. In the intersecting
configuration, we retain the AdS bulk portion that has angular boundary range
$-\phi_{0}\leq\phi\leq\phi_{0}$, as illustrated in fig. 11.101010One could in
principle also consider the complementary range
$-\pi\leq\phi\leq-\phi_{0}\cup\phi_{0}\leq\phi\leq\pi$ or, equivalently,
$\frac{\pi}{2}<\phi_{0}<\pi$ [21].

We parametrize the two branes profiles as

$r=\frac{k_{A}}{\sin\left(\phi-\phi_{0}\right)}\qquad\quad\text{and}\quad\qquad
r=-\frac{k_{B}}{\sin\left(\phi+\phi_{0}\right)}\,,$ (5.10)

without making any assumption about the signs and values of the tension
parameters. With this choice, for equal tensions $k_{A}=k_{B}$ and
$\phi_{0}=\pi/2$ the two branes coincide. Notice that for $k_{A}>0$
($k_{A}<0$) the first brane spans the angular coordinate range
$\phi_{0}\leq\phi\leq\pi\cup-\pi\leq\phi\leq-\pi+\phi_{0}$ (
$-\pi+\phi_{0}\leq\phi\leq\phi_{0}$). If $k_{B}>0$ ($k_{B}<0$) the second
brane spans the angular coordinate range
$-\pi\leq\phi\leq-\phi_{0}\cup-\phi_{0}+\pi\leq\phi\leq\pi$
($-\phi_{0}\leq\phi\leq-\phi_{0}+\pi$) (see fig. 11).

On any fixed time slice, the branes intersect at a point $(r_{*},\phi_{*})$
such that

$\displaystyle r_{*}^{2}$ $\displaystyle=$
$\displaystyle\frac{k_{A}^{2}+k_{B}^{2}+2k_{A}k_{B}\cos
2\phi_{0}}{\sin^{2}2\phi_{0}}$ (5.11) $\displaystyle\tan\phi_{*}$
$\displaystyle=$ $\displaystyle\frac{k_{B}-k_{A}}{k_{A}+k_{B}}\tan\phi_{0}\,.$
(5.12)

In particular we have

$\displaystyle\phi_{*}\in\left\\{\begin{array}[]{cl}\left[\phi_{0},\pi-\phi_{0}\right]&\mbox{if}~{}~{}k_{A}>0,k_{B}<0\\\
\left[\pi-\phi_{0},\pi\right]\cup\left[-\pi,-\pi+\phi_{0}\right]&\mbox{if}~{}~{}k_{A}>0,k_{B}>0\\\
\left[-\phi_{0},\phi_{0}\right]&\mbox{if}~{}~{}k_{A}<0,k_{B}<0\\\
\left[-\pi+\phi_{0},-\phi_{0}\right]&\mbox{if}~{}~{}k_{A}<0,k_{B}>0\end{array}\right..$
(5.17)

Also, notice that in the limit $\phi_{0}\to\pi/2$, the radial coordinate
$r_{*}$ of the branes intersection goes to infinity, but this limiting value
can be approached for any choice of $T_{A}$ and $T_{B}$.

The angle $\hat{\theta}$ between the branes at the intersection point is
obtained in terms of the unit normals $n_{A},n_{B}$. Defining them to be
outward-pointing with respect to the bulk region between the branes gives

$\displaystyle n_{A\mu}$
$\displaystyle=\pm\sqrt{\frac{\sin^{2}(\phi-\phi_{0})}{1+k_{A}^{2}}}\left(0,1,\frac{k_{A}}{\sin(\phi-\phi_{0})\tan(\phi-\phi_{0})}\right)\,,$
(5.18) $\displaystyle n_{B\mu}$
$\displaystyle=\pm\sqrt{\frac{\sin^{2}(\phi+\phi_{0})}{1+k_{B}^{2}}}\left(0,1,-\frac{k_{B}}{\sin(\phi+\phi_{0})\tan(\phi+\phi_{0})}\right)\,,$

with the $\pm$ sign respectively for positive and negative values of the
tension parameters. Keeping track of the sign of the tensions and using
equation (5.10), (5.11) and (5.12), one obtains the relation

$\cos(\pi-\hat{\theta})=n_{A\mu}n_{B\nu}g^{\mu\nu}=\frac{k_{A}k_{B}-\cos
2\phi_{0}}{\sqrt{(1+k_{A}^{2})(1+k_{B}^{2})}}\,$ (5.19)

for any sign of $k_{A}$ and $k_{B}$.

On shell, the brane intersection angle coincides with the tension parameter of
the defect sustaining the intersection, $\theta_{0}=\hat{\theta}$, which is
then related to the opening angle $\phi_{0}$ as

$\cos 2\phi_{0}=k_{A}k_{B}+\sqrt{(1+k_{A}^{2})(1+k_{B}^{2})}\cos\theta_{0}\,.$
(5.20)

The case in which the branes have the same tension and there is no
intersection nor defect, $\theta_{0}=\pi$, yields $\phi_{0}=\pi/2$
consistently with the parametrization (5.10).

####  5.2.1 Matching the BCFT setup

We now match the BCFT parameters of the setup in section 2.1 to the
holographic ones. The length $L$ of the CFT interval is given by the opening
between the two branes at the AdS boundary

$L=2\phi_{0}R\,.$ (5.21)

The conformal dimension $h_{\psi}$ of the BCC operator is directly related to
the opening angle $\phi_{0}$. The relation is obtained remembering that
$h_{\psi}$ is determined by the gap in energy between the state with and
without BCC operator, i.e., between the state with mixed boundary conditions
and the ground state with homogeneous boundary conditions (see equation
(2.3)). Holographically this is the energy difference between the
configuration with two intersecting branes and the one with a single
brane.111111This is actually independent from the tension of the brane, e.g.,
[81]. Evaluating the corresponding bulk actions, the shift in energy is found
to be [21]

$h_{\psi}=\frac{L\,\Delta
E}{\pi}=\frac{c}{24}\left(1-\frac{4\phi_{0}^{2}\,}{\pi^{2}}\right)$ (5.22)

giving

$\frac{2\phi_{0}}{\pi}=\sqrt{1-\frac{24h_{\psi}}{c}}\,.$ (5.23)

Notice that $h_{\psi}\propto c$ and the spectrum goes from arbitrarily close
to $0$ to $c/24$ [21, 22]. Therefore, in the holographic realization there
exists a BCC operator implementing the change in boundary conditions with an
arbitrary small change in energy. That is, for any set of mixed boundary
conditions holographically specified by $T_{A}$ and $T_{B}$, the lowest energy
state has at leading order in $c$ the same energy of a the ground state with
homogeneous boundary conditions.

###  5.3 Holographic entanglement entropy

To reproduce holographically the entanglement entropy result for the strip
with different boundary conditions, equation (2.22), we shall evaluate the
Ryu-Takayanagi [58] formula

$S=\frac{\text{Area}\,\gamma}{4G_{N}}$ (5.24)

in the static geometry with EOW branes [16, 17]. In AdS3, the surfaces
$\gamma$ are regulated geodesics and the area term is evaluated as the
geodesic proper length, $\text{Area}\,\gamma=d_{\gamma}$.

In our setup we need to consider the minimal length geodesic extending from a
point on the boundary to the EOW branes. There are in general two competing,
locally extremal, curves one should take into account. The minimal length
geodesic extending into the bulk and ending on the brane with tension
parameter $k_{A}$ and the one ending on the brane with tension $k_{B}$, see
fig. 11.

####  5.3.1 Geodesic computation

Let us start reviewing the case of a geodesic stretching between a boundary
point and a single brane in AdS3. The proper length $d_{\gamma}$ of a constant
time geodesic $\gamma$ in AdS3 between two generic points is given by

$\cosh
d_{\gamma}=\sqrt{(r_{1}^{2}+1)(r_{2}^{2}+1)}-r_{1}r_{2}\cos\left(\phi_{1}-\phi_{2}\right)\,.$
(5.25)

We take one extrema to be at a generic point $(r_{b},\phi_{b})$ on the EOW
brane with profile

$r_{b}=\frac{k}{\sin\phi_{b}}\,.$ (5.26)

The other is taken on the regulated AdS boundary at $r_{\infty}$ displaced by
$\tilde{\phi}$ from the brane anchoring point at $\phi=0$, that is at
$(r_{\infty},\tilde{\phi})$. For the length $d_{\gamma}$ this gives

$\cosh d_{\gamma}\approx
r_{\infty}\left(\sqrt{\frac{k^{2}}{\sin^{2}\phi_{b}}+1}-\frac{k}{\sin\phi_{b}}\cos(\tilde{\phi}-\phi_{b})\right)\,,$
(5.27)

where we made a large $r_{\infty}$ expansion and used (5.26) to express
$r_{b}$ in terms of $\phi_{b}$.

Extremizing the length over the possible ending points on the brane
sets121212Notice that for $\pi/2<|\tilde{\phi}|<\pi$ this means that
$\pi/2<|\phi_{b}|<\pi$.

$\sin\phi_{b,\rm
ext}=\frac{k\sin|\tilde{\phi}|}{\sqrt{k^{2}+\cos^{2}\tilde{\phi}}}\,$ (5.28)

and selects the geodesic with minimal length amongst all geodesics ending on
the brane. This has length

$d_{\gamma}=\log\left[2r_{\infty}\left(-k\sin\tilde{\phi}+\sqrt{1+k^{2}}\,\sin|\tilde{\phi}|\right)\right]\,.$
(5.29)

The single brane analysis is immediately extended to the situation with two
intersecting EOW branes as in equation (5.10). We parametrize the boundary
point from where the geodesic emanates in terms of the distance
$0<\Delta\phi<2\phi_{0}$ from brane $A$. The point is then at the angular
location $\phi_{0}-\Delta\phi$.

There are now two locally minimal geodesics to consider. One is the geodesic
$\gamma_{A}$ ending on the EOW brane $A$, which has length

$d_{\gamma_{A}}=\log\left[2r_{\infty}\sin\Delta\phi\right]+{\rm
arcsinh}\,k_{A}\,.$ (5.30)

The other geodesic $\gamma_{B}$ is the minimal length one between those ending
on the brane $B$. This has length

$d_{\gamma_{B}}=\log\left[2r_{\infty}\sin(2\phi_{0}-\Delta\phi)\right]+{\rm
arcsinh}\,k_{B}\,.$ (5.31)

Notice that this is obtained from (5.29) taking into account that the equation
giving the brane profile $B$ in (5.10) has an overall extra minus sign as
compared to (5.26).

For any given boundary interval $\Delta\phi$, the geodesic which has minimal
length between (5.30) and (5.31) is the one selected by the Ryu-Takayanagi
formula (5.24)

$S=\frac{\min\\{d_{\gamma_{A}},d_{\gamma_{B}}\\}}{4G_{N}}\,.$ (5.32)

As a function of $\Delta\phi$, a dominance transition from one geodesic to the
other happens for a value $\Delta\phi_{t}$ corresponding to
$d_{\gamma_{A}}=d_{\gamma_{B}}$. Namely

$\cot\Delta\phi_{t}=\csc
2\phi_{0}\,\frac{k_{A}+\sqrt{1+k^{2}_{A}}}{k_{B}+\sqrt{1+k^{2}_{B}}}+\cot
2\phi_{0}\,,$ (5.33)

which is consistently within the range $0<\Delta\phi_{t}<2\phi_{0}$.

Notice that the two geodesics may not always exist at the same time in our
setup. For instance, $\gamma_{A}$ may cease to exist for a large enough
$\Delta\phi>\Delta\phi_{A}$. The limiting value corresponds to the situation
where the endpoint of $\gamma_{A}$ on the brane $\phi_{b,\rm ext}$ coincides
with the branes intersection point $\phi_{*}$. That is, when

$\sin(\phi_{*}-\phi_{0})=\frac{k_{A}\sin\Delta\phi_{A}}{\sqrt{k_{A}^{2}+\cos^{2}\Delta\phi_{A}}}\,.$
(5.34)

Substituting the explicit expression (5.12) for $\phi_{*}$ gives131313This can
formally give a value of $\Delta\phi_{A}$ outside the physical range
$0<\Delta\phi<2\phi_{0}$. In such a case $\gamma_{A}$ always exists in the
setup we are looking at. A similar consideration applies to $\Delta\phi_{B}$.

$\cot\Delta\phi_{A}=-\frac{\csc 2\phi_{0}\left(k_{A}\cos
2\phi_{0}+k_{B}\right)}{\sqrt{k_{A}^{2}+1}}\,.$ (5.35)

Similarly $\gamma_{B}$ may not exist for a small enough value of
$\Delta\phi<\Delta\phi_{B}$. Here $\Delta\phi_{B}$ is given by

$\cot(2\phi_{0}-\Delta\phi_{B})=-\frac{\csc
2\phi_{0}\left(k_{B}\cos(2\phi_{0})+k_{A}\right)}{\sqrt{k_{B}^{2}+1}}\,.$
(5.36)

However, one can check explicitly that

$\Delta\phi_{B}<\Delta\phi_{t}<\Delta\phi_{A}\,,$ (5.37)

so that at the transition point both geodesic configurations do actually exist
in our setup.

####  5.3.2 Matching the BCFT computation

To show that the holographic entanglement entropy in (5.32) matches the static
result in section 2.3, we need to express it in terms of the dual CFT
parameters. For the BCFT cutoff $\epsilon$ and size of the entangling interval
$\sigma$, we have the relations

$\frac{r_{\infty}}{R}=\frac{1}{\epsilon}\,\qquad\qquad R\Delta\phi=\sigma\,.$
(5.38)

These directly follow from matching the conformal boundary of AdS metric (5.7)
with the BCFT geometry. Expressing everything in terms of the physical size
$L$ through (5.21) and using the relation (5.23) yields

$r_{\infty}=\frac{L}{\alpha\pi\epsilon}\,\qquad\qquad\Delta\phi=\frac{\sigma
L}{\alpha\pi}\,,$ (5.39)

where we have used $\alpha=\sqrt{1-24h_{\psi}/c}$ (see equation (2.14)).

Substituting these into the expressions for the geodesics length (5.30) and
(5.31) and using the relation $c={3/2G_{N}}$, we get from (5.32)

$S=\frac{c}{6}\log\frac{2L}{\pi\alpha\epsilon}+\text{min}\left\\{\frac{c}{6}\log\sin\frac{\alpha\pi\sigma}{L}+\frac{c}{6}\,{\rm
arcsinh}\,k_{A};~{}\frac{c}{6}\log\sin\frac{\alpha\pi(L-\sigma)}{L}+\frac{c}{6}\,{\rm
arcsinh}\,k_{B}\right\\}\,,$ (5.40)

By matching the brane ${A,B}$ with the conformal boundary condition $A,B$ in
the CFT via the identification [16, 17]

$s_{A,B}=\frac{c}{6}\,{\rm arcsinh}\,k_{A,B}\,,$ (5.41)

one exactly reproduces the result (2.22) for the static case derived in
section 2.3.

The holographic computation of the entanglement entropy in the post quench
state involves considering geodesics in a AdS3 time dependent geometry [58,
59, 16, 17]. In principle it should be possible to work explicitly the form of
this dual geometry, which one expects to be constructed from two EOW branes
with a non-trivial time dependent profile.141414See [37, 38] for this type of
analysis performed in the case of a CFT on a circle. Here we will simply show
how to reproduce the time dependent entanglement entropy.

The static state yielding the above result and the time dependent post quench
state are related on the CFT side by a transformation of the conformal
coordinates. This map can be extended to a diffeomorphism in AdS3 that relates
the geometries dual to the CFT states described in the two sets of coordinates
[82, 83].151515A subtlety is that the time governing the evolution ${{t_{E}}}$
appears as a parameter in the map and not as a component of the complex CFT
coordinates. The map relates for each ${{t_{E}}}$ the initial state static
geometry to an auxiliary geometry with its own time slicing where the geodesic
length giving the time dependent result can be computed. See, e.g., [84, 85]
for related discussions. The length of geodesics relevant for the post quench
state can then be obtained using this local equivalence of AdS3 geometries
(see, e.g., [86, 53, 28]).

Let us start expressing the static result (5.40) using coordinates that match
those defined on the strip in section 2. For this we can simply focus on the
asymptotic form of the AdS3 metric (5.7), which we can write in Euclidean as

$ds^{2}\approx\frac{dr^{2}}{r^{2}}+\frac{r^{2}}{R^{2}}(d\tau^{2}+R^{2}d\phi^{2})=\frac{dwd{\bar{w}}+du^{2}}{u^{2}}\
.$ (5.42)

We made the identifications

$u=\frac{R}{r}\qquad w=\tau+iR\phi\qquad{\bar{w}}=\tau-iR\phi\,.$ (5.43)

and $(w,{\bar{w}})$ match the Euclidean coordinates on the strip.

The entanglement entropy result obtained from the static geometry analysis is
then rewritten as

$S=\text{min}\left\\{\frac{c}{6}\log\frac{2L}{\pi\alpha
u_{\infty}}\sin\frac{\alpha\pi|w_{\infty}-{\bar{w}}_{\infty}|}{2L}+s_{A};~{}\frac{c}{6}\log\frac{2L}{\pi\alpha
u_{\infty}}\sin\frac{\alpha\pi(2L-|w_{\infty}-{\bar{w}}_{\infty}|)}{2L}+s_{B}\right\\}\,$
(5.44)

and using the identifications

$\displaystyle w_{\infty}$ $\displaystyle=i\sigma$ (5.45)
$\displaystyle\bar{w}_{\infty}$ $\displaystyle=-i\sigma$ $\displaystyle
u_{\infty}$ $\displaystyle=\epsilon\,$

reproduces the result (5.40).

The holographic entanglement entropy for the time dependent state is instead
obtained with a different identification

$\displaystyle w_{\infty}$ $\displaystyle=w_{{t_{E}}}(w=i\sigma)$ (5.46)
$\displaystyle\bar{w}_{\infty}$
$\displaystyle=\bar{w}_{{t_{E}}}(\bar{w}=-i\sigma)$ $\displaystyle u_{\infty}$
$\displaystyle=\epsilon\sqrt{w_{{t_{E}}}^{\prime}(w=i\sigma){\bar{w}}_{{t_{E}}}^{\prime}({\bar{w}}=-i\sigma)}\,.$

This is the asymptotic form of the bulk diffeomorphism corresponding to the
CFT map relating the initial state to the post quench state [82, 83]. It pulls
back the point $w_{\infty},\bar{w}_{\infty},u_{\infty}$ in the time
independent geometry to the point $\sigma$ and bulk cutoff $\epsilon$ in the
geometry dual to the time dependent state at time $t_{E}$. This can be though
of as the bulk counterpart of the fact that on the CFT side an operator in the
time dependent state at a given time ${{t_{E}}}$ can be expressed in terms of
the initial static state operator through the map $w_{{t_{E}}}$.

Using the identifitcation (5.46) in (5.44), and continuing to Lorentzian time
${{t_{E}}}\to it$, it is quite immediate to see how this reproduces the post
quench result. To illustrate this more explicitly we consider the example of
the SSD quench, other cases follows with similar computations. Using in (5.46)
the explicit form of the SSD time dependent map (3.17) and continuing to
Lorentzian time ${{t_{E}}}\to it$, one can easily check that

$u_{\infty}=\epsilon\sqrt{w_{t}^{\prime}(w=i\sigma){\bar{w}}_{t}^{\prime}({\bar{w}}=-i\sigma)}=\epsilon\sqrt{\frac{1}{f(t)^{2}+\sin^{2}\left(\frac{2\pi\sigma}{L}\right)}}$
(5.47)

which matches the time depedent factor accompanying the cutoff in the
expressions for the SSD entanglement entropy (3.37). Similarly, in the
argument of the $\sin$ in the holographic formula (5.44) one recognizes
$\delta(t)$, the phase defined in (3.37)

$|\omega_{t}-\bar{\omega}_{t}|=\frac{L}{\pi}\Big{|}\log\frac{z_{t}}{\bar{z}_{t}}\Big{|}=\frac{L}{\pi}\delta(t)\,.$
(5.48)

To be explicit we used the relation between the CFT coordinates on the strip
and in the UHP to write the middle expression.

##  6 Discussion

We studied the role of boundary effects for the entanglement entropy in a CFT
defined on an interval with mixed boundary conditions after an inhomogeneous
quench. We considered a quench protocol where the initial state is an
eigenstate of $L_{0}$ and the evolution is driven by a deformation of the
standard CFT Hamiltonian. In particular, we analyzed the case of the SSD and
Möbius deformation, as well as a general class of Hamiltonian deformations
defined in terms of $sl(2,\mathbb{R})$ subalgebras of the full Virasoro
algebra. We probed the entanglement dynamics in terms of the entanglement
entropy for a portion of the system that includes one of the boundaries.

Although the setup and the techniques employed are in large part general, our
analysis focused on the case of holographic CFTs. In this context it is
natural to focus on the universal contribution to the entanglement entropy
coming from the identity Virasoro block. This is expected to give the leading
contribution at large $c$ and to capture the physics of semiclassical gravity
in the holographic AdS3 dual.

Our analysis reveals an entanglement dynamics with a characteristic phase
transition pattern governed by boundary effects. The mixed conformal boundary
conditions, $A$ and $B$, in our setup give rise to two competing phases for
the entanglement entropy obtained from the conformal family of the stress
tensor, as described in section 2.3. For the initial static state, one or the
other phase dominates depending on the size of the entangling region and the
relative value of the boundary entropies. The post-quench evolution, on the
other hand, turns out to allow for a dynamical transition between these
phases. For a fixed entangling region, the transition pattern is determined by
the relative amount of degrees of freedom associated with the two conformal
boundaries, as encoded in the boundary entropies $s_{A,B}$. When the number of
degrees of freedom encoded in $s_{A}$ and $s_{B}$ are close enough to each
other, in a sense made quantitive in section 3.2 and 4.2.2, the quench
dynamics induces a transition from the initial phase to the other. For
boundary entropies values that differ significantly, the initial phase instead
dominates for all time. This is a common feature of all the deformations we
considered in this work, but the details depend on the specific deformation
considered. In particular the precise range of $s_{B}-s_{A}$ for which the
transition occur depends on the deformation.

At the kinematic level the entanglement dynamics of each phase is analogous to
the case of homogeneous boundary conditions [48]. For the Möbius and
generalized $sl(2,\mathbb{R})$ deformations the entanglement entropy exhibits
a kinematics compatible with a finite size system, characterized by an
effective length which details are determined by the deformation parameters.
This implies also a recurrent behaviour for the phase transition pattern.
Within one period the phases alternate, going from, e.g., phase $A$ to $B$,
and back to the initial phase $A$, as depicted in fig. 7 and 10. In the SSD
case the kinematic is compatible with the one of an infinite dimensional
system, with no return phase and a $\sim\log t$ growth of the entanglement
entropy at lates times. Still, the presence of a phase transition signals the
finiteness of the system with different conformal boundary conditions.

As we discussed in section 5, our CFT analysis finds a direct match on the
geometric AdS3 description. The holographic dual of the boundary CFT setup
involves two EOW branes anchored on the boundary and extending in the AdS3
bulk, where they intersect. This construction selects a portion of the AdS3
geometry comprised between the AdS3 boundary and the two intersecting branes.
In this setup the brane tensions $T_{A}$ and $T_{B}$ are in one to one
correspondence with the conformal boundary conditions $A$ and $B$ and
determine the value of the boundary entropies. The two CFT channels described
in section 2.3 have a direct correspondence in the two competing Ryu-
Takayanagi surfaces one can draw in the asymptotically AdS3 geometry. These
are given by geodesics extending from the boundary of AdS to the EOW branes,
as depicted in fig. 11.

In section 5.3, we showed explicitly how the holographic entanglement entropy
exactly matches the CFT result. We obtained this result exploiting directly
the local equivalence of asymptotically AdS3 geometries and the dictionary
relating CFT maps to bulk diffeomorphisms. We did not analyze explicitly the
details of the time dependent geometries dual to the post quench state for the
various deformations. It should be possible to do so along the lines of [37,
38, 39] which studied the case of a CFT on a circle.

##### Double holography and quantum extremal surface

Our analysis lends itself for an interpretation in the context of double
holography. In fact, there is a third complementary description which is
available for our system, the so called brane perspective, as we schematically
discuss in the following.

There, the bulk spacetime enclosed between the asymptotic AdS boundary and the
EOW branes is integrated out leaving behind an effective gravitational theory
on each brane [87, 88, 89]. In the case of intersecting branes with a defect
supporting the intersection, we have two gravitational spacetimes, given by
portions of AdS2 with different effective cosmological constants. These are
separated from one another by the defect on one side and in contact with the
same non-gravitational CFT region on the other side.

Our entanglement entropy result can be interpreted in this perspective as a
generalized entropy computation [90, 11, 12, 13, 14, 15]. In the BCFT
perspective, the region we considered includes the boundary with conformally
invariant boundary condition $A$. In the brane description, this corresponds
to considering the entanglement entropy for the interval $[0,\sigma]$ in the
2d CFT and an adjacent portion of the gravitational space. Which portion of
the AdS2 exactly enters is determined by the generalized entropy formula
itself, which requires finding the quantum extremal surface (QES) [90]. In
double holography the location of the latter corresponds to the intersection
of the Ryu-Takayanagi surface with the brane (see, e.g., [13, 91, 92, 93,
94]). The region enclosed by the QES in the brane perspective then goes from
$\sigma$ to this point, as depicted in fig. 12.

Figure 12: A pictorial representation of the quantum extremal surface in the
double holographic interpretation of our model. The RT surfaces of the
AdS/BCFT setup select the QES in the brane perspective. The phases $A$ and $B$
are depicted respectively on the left and on the right. In the $B$ phase the
QES selects a region which comprises the entire AdS2 region associated with
$A$ and part of the interior of the AdS2 region associated with $B$.

In the simplest case of a SSD quench, one can interpret the dynamical phase
transition as follows:161616Here we are thinking in a quasi-static manner. A
complete picture would require knowing the exact time dependent post quench
geometry and the time dependent profile of the branes at the initial time of
the post quench evolution the QES is inside universe $A$. As time evolves this
goes deeper into the AdS2 bulk of universe $A$. At the phase transition time,
the QES surface jumps to universe $B$ selecting a region that extends from
$\sigma$ all the way to cover the entire universe $A$, the bulk defect and
part of the interior of universe $B$.

##### Driving the system with a sequence of Hamiltonians

In our analysis we focused on quenching the system with one specific
Hamiltonian. As discussed for instance in [77, 78, 79, 80, 95, 96] for the
case of homogeneous boundary conditions, having an entire class of deformed
Hamiltonian and a periodic behaviour allows for more complicated dynamical
patterns. For finite $\theta$, the system exhibits a periodic behaviour and
gets revived to its initial state at the end of each period. One can then
drive the system with a different Hamiltonian at the end of each time period
in a Floquet-like manner. The simplest case here would be an evolution
governed by two different Hamiltonians $H_{k_{1}}$ and $H_{k_{2}}$ at fixed
$\theta$, with $n$-cycles

$U=\left(e^{-iH_{k_{2}}t_{2}}e^{-iH_{k_{1}}t_{1}}\right)^{n}\,$ (6.1)

where $H_{k_{i}}$ drives the system for a period
$t_{i}=\frac{2L\cosh(2\theta)}{k_{i}}$.

Choosing appropriately the parameters of our setup, it would then be possible
to engineer a variety of situations. For instance one could have a dynamics
where a phase transition from phase $A$ to $B$ and back would happen for
either, both or none the evolution intervals associated with $H_{k_{1}}$ and
$H_{k_{2}}$.

##### Generalizations

Our explicit analysis is limited to the case of holographic CFTs. Even though
the complete set of conformal data is unknown, the holographic geometric
construction in terms of EOW branes leads to consider arbitrary values for the
boundary entropy. In turn this allows to always find a window of parameters
where a transition occurs. In the case of rational CFTs, on the other hand,
the boundary entropy can only take a finite number of values given by the
$g$-functions [3, 4, 7, 8]. It would then be interesting to revisit our
analysis for the rational case and see if this dynamical phase transition is
linked to the irrational character of holographic CFTs, as it is the case for
chaotic and scrambling behaviour (see, e.g., [97, 98, 99]).

Also, we only considered the case where the interval is adjacent to one of the
boundaries. An interval which does not include any of the boundaries will give
rise to a more complicated pattern for the evolution of the entanglement
entropy after the quench. An interesting observation that has been made for
finite systems with periodic boundary conditions is that the SSD quench
dynamics can be used to bring different initial states to a state where the
entanglement entropy of an interval approximately matches that of the vacuum
state at late times [37, 39].

Another interesting direction is extending the discussions to the thermal
case. The relevant Euclidean CFT computation in this case is not on the
infinite strip, but on the finite one with two sides identified, or
equivalently on the annulus. Holographically it would correspond to
considering an AdS black hole geometry with two branes selecting a part of the
bulk. Despite a first principle CFT computation might be technically
challenging in this case, the same techniques of section 5 would allow to
conduct an holographic study. We hope to report on this in the future.

## Acknowledgement

We thank Pawel Caputa, Yingfei Gu, Hosho Katsura, Masamichi Miyaji, Yu
Nakayama, Dominik Neuenfeld, Tatsuma Nishioka, Jiaxin Qiao, Erik Tonni and
Satoshi Yamaguchi for useful discussions and comments at various stages of
this work, and especially Tadashi Takayanagi for comments on a draft of the
paper. The authors are grateful for the hospitality of Galileo Galilei
Institute for Theoretical Physics (GGI). AB and FG are grateful for the
hospitality of Perimeter Institute where part of this work was carried out. DG
thanks the University of Florence and INFN for partial support during the
completion of this work, and IAS Tsinghua and YITP Kyoto where the work was
partially presented. DG also thanks YITP for providing regular visiting
opportunities. DG is supported by Grant-in-Aid for Transformative Research
Areas(A) “Extreme Universe” No. 21H05190. FG would like to thank the Isaac
Newton Institute for Mathematical Sciences, Cambridge, for support and
hospitality during the programme Black holes: bridges between number theory
and holographic quantum information where work on this paper was undertaken.
This research was supported in part by the Simons Foundation through the
Simons Foundation Emmy Noether Fellows Program at Perimeter Institute.
Research at Perimeter Institute is supported by the Government of Canada
through the Department of Innovation, Science and Economic Development and by
the Province of Ontario through the Ministry of Research, Innovation and
Science. This work was also supported in part by EPSRC grant no EP/R014604/1.

##  Appendix A Stress tensor in the UHP

On the infinite strip of width $L$ with conformal boundary conditions on the
edges only one set of Virasoro symmetries survives. To see that, one can use
the conformal map

$z=e^{\frac{\pi}{L}w}\,,~{}~{}~{}w=\tau+i\sigma\,,~{}~{}~{}\sigma\in[0,L]$
(A.1)

to bring the strip to the UHP. See, e.g., [3, 100].

In the UHP Virasoro generators are still defined in terms of the stress energy
tensor, though its holomorphic part and anti-holomorphic part are no longer
independent from each other. More precisely, they conformal boundary
conditions imply $T(z)=\bar{T}({\bar{z}})$ when $z=\bar{z}$, i.e., on the real
axis. One can analytically continue the holomorphic stress tensor to the lower
half plane defining $T(z^{*})=\bar{T}(z)$, where $z$ in the UHP. Similarly for
the anti-holomorphic part of the stress tensor $\bar{T}(z^{*})=T(z)$.

Then the Virasoro generators can be expressed in the UHP in a similar way as
in the full complex plane

$L_{n}=\frac{1}{2\pi i}\oint_{C^{\prime}}dzz^{n+1}T(z)-\frac{1}{2\pi
i}\oint_{C^{\prime}}dz^{*}{z^{*}}^{n+1}\bar{T}(z)$ (A.2)

where the contour $C^{\prime}$ is a semicircle and a segment along the real
axis including the origin. Using the analytic properties of the stress tensor
this can also be expressed as a contour integral in the full complex plane

$L_{n}=\frac{1}{2\pi i}\oint_{C}dzz^{n+1}T(z)=-\frac{1}{2\pi
i}\oint_{C}d\bar{z}\bar{z}^{n+1}\bar{T}(\bar{z})=\bar{L}_{n}\,,$ (A.3)

In the main text we will express the Virasoro generators as contour integrals
in the full complex plane using the holomorphic part and use the short-hand
notation

$\oint z^{n+1}T(z)\equiv\frac{1}{2\pi i}\oint_{C}dzz^{n+1}T(z)=L_{n}\,.$ (A.4)

Notice that $L_{n}$ now acts both on the holomorphic and anti-holomorphic
coordinates of the bulk primary operator as

$[L_{n},\mathcal{O}(z,\bar{z})]=h(n+1)z^{n}\mathcal{O}(z,\bar{z})+z^{n+1}\partial_{z}\mathcal{O}(z,\bar{z})+\bar{h}(n+1)\bar{z}^{n}\mathcal{O}(z,\bar{z})+\bar{z}^{n+1}\partial_{\bar{z}}\mathcal{O}(z,\bar{z})\,.$
(A.5)

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