aainstitutetext: School of Basic Sciences, Indian Institute of Technology
Bhubaneswar, Bhubaneswar 752050, India

# Quantum information scrambling and quantum chaos in little string theory

Sandip Mahish and Karunava Sil<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In the current manuscript we perform a systematic investigation about the
effects of nonlocal interaction to the spread of quantum information in many
body system. In particular, we have studied how nonlocality influence the
existing bound on the growth rate of the commutator involving two local
operators, the butterfly velocity. For this purpose, we consider the nonlocal
theory on the worldvolume of $N\gg 1$, NS$5$ branes arising in the limit of
vanishing string coupling, the ‘little string theory’. A direct evidence of
nonlocality can be realized from the ‘volume law’ behavior for the most
dominant part of holographic entanglement entropy. We obtain the butterfly
velocity by studying the dynamics of the near horizon geometry backreacted by
a high energy quanta in the form of a shockwave resulting from an early
perturbation on the corresponding thermofield double state. We observe that
the butterfly velocity increases with the nonlocal scale of little string
theory, the inverse Hagedorn temperature $\beta_{h}$, indicating a faster rate
of information spread due to the nonlocal interaction. The same conclusion
follows as the disruption of two sided mutual information is observed to occur
at a faster rate for higher values of $\beta_{h}$. Finally, we realize a
direct connection between the parameters of quantum chaos and the quasinormal
modes for collective excitations through the phenomenon of ‘pole skipping’.

##  1 Introduction

Recently, there has been a growing interest in quantifying the nature of
quantum information spread in strongly interacting many body system. In
quantum many body system, the spread of information effectively depends on the
nature of commutator between two local generic operators. In particular, the
commutator behaves differently when evaluated in chaotic and integrable
systems. There exists fundamental bounds on such commutators as evaluated for
different quantum theories which in turn reflects how fast quantum information
can scramble from being accessible to a few local degrees of freedom to the
entire system.111We appreciate the nicely written lecture notes BS  which
helps us to understand the basic idea and also recent developments regarding
the quantum information scrambling in many body quantum theory.

As an example, a relativistic system that is governed by the rules of Lorentz
symmetry does not allow any correlation between two space like separated local
operators. In this case causality sets the bound to the information spread by
the velocity of light. In a non-relativistic system, on the other hand two
local operators can have finite overlap even if they are largely separated in
space as compared to the elapsed time. However, a bound on the information
propagation still exists for the non-relativistic system, known as Lieb-
Robinson bound as observed in a discrete lattice system Lieb:1972wy . Here the
corresponding ‘Lieb-Robinson velocity’ defines a bound for the growth of
operator norm of two Heisenberg operators $W(t,x)$ and $V(t,x)$. Lieb-Robinson
bound depends on the details of the UV physics and hence it is independent of
the state of the system. However, at lower energy scale such state independent
bound is not relevant. There exists much tighter and slower speed limit, for
example the butterfly velocity Shenker:2013pqa ; Roberts:2016wdl  that bounds
the propagation of all kinds of physical excitations which are important at
the low energy density or temperature. Butterfly velocity is a state dependent
quantity as it is given in terms of the IR scale of the underlying theory.
Instead of operator norm butterfly velocity is governed by the growth of
commutator square involving generic operators. The butterfly velocity in
general local quantum theory follows several bounds and inequalities. In this
paper we will particularly be interested in how these bounds and inequalities
gets modified due to the inherent non locality that the little string theory
possess. Moreover, we will mainly focus on the connection between information
scrambling and quantum chaos for a quantum many body theory with non-local
interaction.

Chaotic behavior in a quantum system with large number of degrees of freedoms
can be quantified as the rate with which quantum information spreads across
the entire system. Moreover, in many body systems which are governed by
complicated hamiltonian, information encoded within any small subsystem
scrambles among the degrees of freedom of the whole system such that it cannot
be recovered by doing local measurement on that particular subsystem. Let the
system under consideration is defined initially by the density matrix
$\rho_{0}$ which evolves in time according to the usual time evaluation rules
in Heisenberg picture, $\rho(t)=e^{-iHt}\rho_{0}e^{iHt}$, $H$ being the
hamiltonian of the system. If the system is partitioned into two parts, $A$
and $B$ then any local measurement on subsystem $A$ can be performed with the
reduced density matrix $\rho_{A}=tr_{B}(\rho(t))$. As time evolves the
subsystem $A$ starts loosing information and it is no longer defined by the
reduced density matrix $\rho_{A}$ but instead by the thermal ensemble density
matrix $\rho_{\beta}$ at some inverse temperature $\beta$ and the information
about the initial state of $A$ is no longer available on local measurements.
Typically quantum chaos is characterized by the thermal average of the
following double commutator for two generic operator $W$ and $V$ as larkin ;
Shenker:2013pqa ,

$C(t,x)=-<[W(t,x),V(0)]^{2}>_{\beta}.$ (1)

In the above equation the double commutator can be simplified only to observe
that it is proportional to the out of time ordered product of operators $W$
and $V$, the OTOC. In quantum many body systems, the chaotic behavior is
captured by the exponential growth of the OTOC, $C(t,x)$ in the above equation
given as, Roberts:2014isa ; Shenker:2014cwa ; Kitaev:2017awl ;
Polchinski:2015cea ; Polchinski:2016xgd ; Maldacena:2016hyu ;
Maldacena:2016upp ; Jensen:2016pah ; Gu:2016oyy ; Patel:2016wdy ;
Alishahiha:2016cjk

$C(t,x)=\frac{1}{N^{2}}e^{\lambda_{L}\left(t-\frac{|x|}{v_{b}}\right)},$ (2)

where the overall growth rate is determined by the Lyapunov exponent
$\lambda_{L}$. As shown in Maldacena:2015waa , that there exists an upper
bound to the Lyapunov exponent given in natural unit as
$\lambda_{L}\leq\frac{2\pi}{\beta}$ which saturates for maximally chaotic
systems. More specifically, for a large class of holographic theories with
black hole in Einstein’s gravity this bound is observed to be saturated
Grozdanov:2017ajz ; Blake:2018leo ; Grozdanov:2018kkt  leading to the
conclusion that black holes are the fastest scramblers of quantum information
Sekino:2008he ,Lashkari:2011yi . In equation (1), the commutator measures the
correlation between a small perturbation $V$ at some early time and some other
operator $W$ inserted at position $x$ and time $t$. The growth of this early
perturbation is characterized by the butterfly velocity $v_{b}$ which defines
an effective light cone such that the operator $W$ if inserted outside of it
will not be effected by the early perturbation, $C(t,x)\approx 0$. However
inside the light cone the commutator is finite and as the initial perturbation
scrambles with time into the degrees of freedom of the entire system, $C(t,x)$
also grows until some time scale $t_{\ast}$, known as the scrambling time at
which $C(t,x)$ becomes $\mathcal{O}(1)$ Hayden:2007cs .

There is another important interpretation of butterfly velocity in the context
of AdS/CFT correspondence. It determines the causal structure of the bulk
geometry Qi:2017ttv  which unlike the microcausality in CFT, is not that
straight forward, especially near the black hole horizon. Given a region $A$
on the boundary, it’s corresponding entanglement wedge is defined as the bulk
region which is bounded by the region $A$ itself and the Ryu-Takayanagi (RT)
surface that is homologous to $A$. Now consider a particle inside the bulk
which is entangled with some reference state. With evolving time as this
particle falls deep inside the bulk, its entanglement with the reference state
can only be recovered as long as it stays inside the entanglement wedge. So
the region $A$ must grow with a rate such that the corresponding RT surface
also penetrate sufficiently deep into the bulk to be able to confine the
particle inside the entanglement wedge. The growth rate of $A$ on the boundary
is exactly equal to the butterfly velocity $v_{b}$. The fact that it is not
possible for the region $A$ to grow any faster than the butterfly velocity
$v_{b}$, sets a restriction for the particle speed in the bulk. This is how
the bulk causality can be interpreted from the knowledge of butterfly velocity
on the boundary. In Mezei:2016wfz , butterfly velocity was computed
holographically for asymptotically AdS-BH spacetime in general dimension to
get,

$v_{b}=\sqrt{\frac{d+1}{2d}},$ (3)

where $d$ is the spatial dimension of the boundary field theory. Usually in
local quantum theory the butterfly velocity is bounded by the speed of light.
However, in the presence of nonlocal interactions this bound can be violated.
As an example in Fischler:2018kwt , the authors observed that in
$\mathcal{N}=4$ non-commutative SYM theory, $v_{b}$ happens to depend on the
non-commutative parameter $\theta$ and for sufficiently large $\theta$ it can
even exceed the speed of light. This violation is a result of the explicit
breakdown of Lorentz invariance due to the non-commutative parameter.

The connection between quantum chaos and information scrambling can be better
understood in a dynamical setting as provided by the process of thermalization
where the expectation value of any simple observable decays with time to the
thermal value. In other words, as a system thermalize, it looses the memory of
initial state resulting in the generation of effective entropy for
entanglement. There exists a number of interesting papers on the
characterization of entanglement growth in free field theory Calabrese:2005in
; Calabrese:2007rg ; Cotler:2016acd . In the presence of strong coupling, the
simple picture of entanglement spreading by the flow of non-interacting
particles in a free theory is no longer applicable. To deal with strong
coupling the authors in Liu:2013iza ; Liu:2013qca ; MohammadiMozaffar:2018vmk
considered a set up which consists of injecting at $t=0$, an uniform energy
density (global quench) for a negligible period of time and then letting the
resulting excited state thermalize. It is observed that in $(1+1)$ dimensional
quantum system as well as in higher dimensional holographic system the
entanglement of any subsystem $\Sigma$ grows linearly in time,

$\frac{dS_{\Sigma}(t)}{dt}=v_{E}s_{th}A_{\Sigma},$ (4)

with $s_{th}$ is the thermal entropy and $A_{\Sigma}$ is the area of the
entangling subsystem $\Sigma$. Physically this growth of entanglement can be
realized in terms of ‘entanglement tsunami’ wave propagating inwards from the
boundary of $\Sigma$ with velocity $v_{E}$ such that only the region which is
already covered by the wave is entangled with the complimentary region
$\Sigma^{c}$. In Calabrese:2005in , $v_{E}$ was obtained for $(1+1)$
dimensional system to be equal to $1$ in natural unit. For higher dimensional
holographic system with $d\geq 3$, the authors in Hartman:2013qma ;
Liu:2013iza ; Liu:2013qca  calculated the following general result for
$v_{E}$,

$v^{holo}_{E}=\frac{\left(\eta-1\right)^{\frac{\eta-1}{2}}}{\eta^{\eta/2}},~{}~{}~{}~{}~{}~{}\eta=\frac{2d-2}{d}.$
(5)

Further, in Casini:2015zua , it was shown that for relativistic theories in
any dimension, $v_{E}$ cannot exceed the velocity of light, $v_{E}\leq 1$.
Finally, the authors in Mezei:2016wfz  managed to came up with the proof that
for any unitary quantum theory, entanglement velocity must be bounded by the
corresponding butterfly velocity,

$v_{E}\leq v_{b}.$ (6)

Later, in Mezei:2016zxg , the same inequality was observed to hold for
holographic theories as well.

In this paper we will concentrate on the holographic calculation of
entanglement velocity by measuring the disruption of mutual information
between two completely disjoint quantum theories coupled together forming a
thermofield double state (TFD). For schematic conveniences, let us call the
two copies of the quantum theories as $QFT_{L}$ and $QFT_{R}$, with the suffix
denoting ‘Left’ and ‘Right’ respectively. A TFD state is defined on the tensor
product of the Hilbert spaces of $QFT_{L}$ and $QFT_{R}$ such that given the
energy eigenstates $|E_{i}>_{L}$ and $|E_{i}>_{R}$ of the two QFTs, it is the
unique pure entangled state given as,

$|TFD>=\frac{1}{Z^{1/2}}\sum_{i}e^{-\frac{\beta}{2}E_{i}}|E_{i}>_{L}|E_{i}>_{R}.$
(7)

---
Figure 1: Kruskal diagram for two sided black hole geometry

It is important to note that each of the two states on the left and right side
are exactly thermal and the thermality in each copy arises due to the
entanglement with the other one. In the language of AdS/CFT correspondence
Maldacena:1997re , thermofield double state is dual to the eternal black hole
geometry which can be realized in Penrose diagram (see figure-1) as a maximal
analytic extension of the usual one sided black hole geometry Maldacena:2001kr
. Here the interior of the two black holes are connected through the wormhole
geometry that indicates the entanglement between the two black holes. Mutual
information $I(A,B)$ between two subsystems $A\in QFT_{L}$ and $B\in QFT_{R}$
is defined as,

$I(A,B)=S(A)+S(B)-S(A\cup B),$ (8)

where $S(A\cup B)$ is the entanglement entropy of the union of $A$ and $B$.
Now initially there is a large amount of entanglement between the two
subregions as evident form the existing upper bound on the correlation between
two operator $\mathcal{O}_{A}$ and $\mathcal{O}_{B}$ defined on $A$ and $B$
respectively given as Wolf:2007tdq ,

$I(A,B)\geq\frac{\left(<\mathcal{O}_{A}\mathcal{O}_{B}>-<\mathcal{O}_{A}><\mathcal{O}_{B}>\right)^{2}}{2<\mathcal{O}_{A}^{2}><\mathcal{O}_{B}^{2}>}$
(9)

In order to characterize the spread of entanglement we first need to disrupt
the TDF structure by applying a small perturbation $W$ on the left side (say)
at some early time $t_{i}$. In the bulk, this corresponds to some localized
energy density injected near the boundary in the distant past. Due to the
strong gravitational force, the excitation will accelerate with time and it’s
energy gets blue shifted as it reaches near the black hole horizon. This
results in a shockwave modified geometry for the bulk spacetime. It is
important to note that only the left-right correlation is destroyed in this
case making the mutual information $I(A,B)$ to vanish for a sufficiently early
time perturbation. As a result in equation (8), only $S(A\cup B)$ shows
increasing behavior with time exactly the same way as in equation (4). That is
the growth rate is controlled by the entanglement velocity $v_{E}$. The
entanglement entropy $S(A)$ and $S(B)$ being unchanged, results $I(A,B)$ to
vanish consequently.

As already mentioned the primary goal of this study is to investigate the
effect of nonlocality to the spread of quantum information. In the context of
black hole information paradox, quantum nonlocality might possibly be the
fundamental ingredient towards the resolution of the paradox. In
Giddings:2006sj , it was argued that in a situation of strong gravitational
interaction the dynamics is governed by the nonlocal degrees of freedom. These
observations/findigs motivate us to study the dynamics of nonlocal information
spreading in a particular quantum theory with inherent nonlocality, the little
string theory DK . Little string theory arises in the decoupling limit
(vanishing string coupling $g_{s}$) of magnetically charged $N\gg 1$,
NS$5$-branes in type-II string theory, $N$ being the number of five branes. In
this case the dynamics of the NS5-branes decouples from the bulk even with
finite string length $l_{s}=\sqrt{\alpha^{\prime}}$. The resulting theory on
five branes is known as the little string theory which is a nonlocal theory
without gravity, it exhibits T-duality upon compactification. Unlike the usual
local QFT, the fundamental object in this theory are one dimensional strings.
The impurity in the form of NS$5$-branes breaks the original $SO(9,1)$ Lorentz
symmetry, however from the worldvolume perspective of the NS$5$-branes, it
still possesses the $SO(5,1)$ Lorentz invariance. Another characteristics of
little string theory is the existence of Hagedorn density of states at
temperature $T_{h}$, known as the Hagedorn temperature. Here the inverse of
the Hagedorn temperature $\beta_{h}$ serves as the nonlocal scale of the
theory Chakraborty:2020fpt . The nonlocal effects are manifest at length scale
less than the inverse Hagedorn temperature $\beta_{h}$ while above $\beta_{h}$
one recovers the usual behavior as observed for the local theory.

Nonlocal quantum theories often shows interesting behavior for physically
measurable quantities at certain length scales. As an example, the holographic
computation of entanglement entropy (using the Ryu-Takayanagi prescription
Ryu:2006bv ) of certain nonlocal theories with appropriate gravity dual shows
volume law behavior for the leading UV diverging term instead of the usual
area law below some critical length scale of the theory Barbon:2008ut ;
Karczmarek:2013xxa ; MohammadiMozaffar:2017nri . A similar conclusion was also
made by the authors in Fischler:2013gsa  regarding the computations of
entanglement entropy and holographic mutual information in large $N$ strongly
coupled noncommutative gauge theory which possesses an inherent non locality
due to the natural UV/IR mixing that the theory exhibits. Regarding the
scrambling of quantum information it is observed that the nonlocal theories
can scramble information much faster compare to the local theories such that
the thermalisation occurs at smaller time scale in theories with nonlocal
interactions Lashkari:2013iga ; Edalati:2012jj . The same conclusion was also
made in Fischler:2018kwt  regarding the information scrambling in
noncommutative geometry by explicitly computing the butterfly velocity. It was
shown that the butterfly velocity keeps on increasing with the noncommutative
parameter. In this paper we also observe the increasing behavior of butterfly
velocity at length scale less than the inverse Hagedorn temperature. However
unlike the case in noncommutative geometry, in this case the butterfly
velocity does not violate the bound imposed by the speed of light. This is
because although the theory of little string is nonlocal, it still possess the
lorentz symmetry as already discussed.

So far in our discussion we talked about the diagnosis of quantum chaos from
the study of out of time ordered correlator. However calculation of OTOC is
sometimes hard, especially when one tries to evaluate at finite temperature.
It is recently observed that the chaotic nature of a many body quantum system
has a direct manifestation in the energy density retarded Green’s function
Blake:2017ris ; Blake:2018leo . In other words the exponential growth of the
OTOC (2) can be realized from the structure two point correlation function
involving the temporal component of the stress energy tensor on the boundary.
According to this phenomenon which is known by the name ‘pole skipping’,
quasinormal frequency ($\omega$) and momentum ($k$) for collective excitations
upon analytic continuation can take special complex values such that the
residue of retarded Green’s function in momentum space vanishes exactly. That
is, at those special values of $\omega$ and $k$, line of poles and line of
zeroes of retarded Green’s function passes through each other and hence it is
not uniquely defined. This phenomenon was first observed in a numerical
computations for a particular holographic system Grozdanov:2017ajz  and then
derived in Blake:2017ris  for an effective field theory of quantum chaos. The
gravitational origin of this phenomenon was nicely explained in Blake:2018leo
. It was observed that at special complex values of $\omega$ and $k$, there
exists two linearly independent ingoing solution for any bulk field equation
of motion near the horizon so that one can define any arbitrary linear
combinations of them to construct another possible solution. Hence the
retarded Green’s function which is determined by the unique choice of the
ingoing solution, is not well defined due to the arbitrariness of the solution
at the horizon. A series of computations has been done recently regarding the
explicit computation of pole skipping points for different theories including
the holographic ones. Blake:2019otz ; Grozdanov:2019uhi ; Natsuume:2019xcy ;
Ceplak:2019ymw ; Natsuume:2019vcv ; Wu:2019esr ; Ahn:2020bks ;
Grozdanov:2020koi ; Ahn:2020baf ; Kim:2020url ; Natsuume:2020snz ;
Abbasi:2020xli ; Ceplak:2021efc ; Kim:2021hqy ; Choi:2020tdj ; Haehl:2018izb ;
Jensen:2019cmr ; Das:2019tga ; Haehl:2019eae ; Ramirez:2020qer ;
Grozdanov:2018kkt ; Natsuume:2019sfp ; Li:2019bgc ; Abbasi:2019rhy ;
Abbasi:2020ykq ; Jansen:2020hfd ; Sil:2020jhr ; Yuan:2020fvv ; Jeong:2021zhz ;
Ahn:2019rnq ; Blake:2021hjj ; Liu:2020yaf . In this paper we intend to carry
on similar calculation for the holographic dual description of little string
theory. We will be particularly interested in the pole skipping points
occurring on upper half of the complex plane which are directly related to the
parameters of chaos.

The rest of the paper is organized as follows, in section-2 we briefly discuss
the holographic dual of little string theory and the corresponding nonlocal
scale. In section-3 and 4, we compute the holographic entanglement entropy and
the entanglement wedge cross section to show directly the evidence of
nonlocality at length scale smaller than some critical one. Section-5 is
entirely based on the computation of butterfly velocity using the
gravitational shockwave analysis in a typical entangled state of two copies of
the theory, known as the thermofield double state. In section-6, we study the
disruption of left-right entanglement in the same thermofield double set up by
computing the two sided holographic mutual information. In section-7, we
evaluate the parameters of quantum chaos, namely the Lyapunov exponent and the
butterfly velocity using the phenomenon of pole skipping. Finally, we conclude
in section-8.

##  2 Holographic description of little string theory

There exists two different classes of non-gravitational theories in higher
dimension that results in some limit of type-II string theory. The usual local
field theories are one such class that arises in the low energy limit defined
by $E\ll m_{s}$, $m_{s}=\frac{1}{\sqrt{\alpha^{\prime}}}$ being the string
scale. On the other hand, there are consistent theories obtained in the limit
of vanishing string coupling, $g_{s}\rightarrow 0$, but at finite string scale
$m_{s}$. This constant string scale is important for the dynamics of these
theories, known as the ‘little string theory’. It is a nonlocal theory defined
on the worldvolume of $N\gg 1$ parallel NS$5$ branes after one considers the
decoupling limit as mentioned above. In the vanishing string coupling limit
however the coupling of the field theory on the NS$5$ branes remains finite
and can be realized by considering the low energy limit Kutasov . In string
frame the supergravity solution can be described in terms of the following ten
dimensional metric, the dilaton and a three form flux corresponding to the NS
$B$ field Horowitz:1991cd ; Maldacena:1997cg ; Parnachev:2005hh ; Chen:2013nma

$\begin{split}ds_{S}^{2}&=-f(r)dt^{2}+dx_{5}^{2}+A(r)\Big{(}\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{3}^{2}\Big{)},~{}~{}\text{where}\\\
A(r)&=1+\frac{N\alpha^{\prime}}{r^{2}},~{}~{}f(r)=1-\frac{r_{H}^{2}}{r^{2}},~{}~{}e^{2\Phi}=gs^{2}A(r),\\\
H_{3}&=2L\sqrt{N\alpha^{\prime}(r_{H}^{2}+N\alpha^{\prime})}V_{3}\end{split}$
(10)

where $V_{3}$ is volume of the compact three sphere and $r_{H}$ is the black
hole horizon. At energy scale $E\sim m_{s}$, the string scale, little string
theory has a density of state which is larger compared to that of the standard
local QFT known as the Hagedorn density of states. It is proportional to
$e^{\beta_{h}E}$, $E$ being the energy density and $\beta_{h}$ as the inverse
Hagedorn temperature. Further, we define $L=\sqrt{N\alpha^{\prime}}$ such that
the inverse Hagedorn temperature is given as,

$\beta_{h}=2\pi\sqrt{N\alpha^{\prime}}=2\pi L.$ (11)

This inverse Hagedorn temperature scale determines the nonlocality of little
string theory, such that at any scale lower than that the nonlocal phenomenon
can be explicitly observed. In the rest of the paper we consider large value
of $L$ (as $\beta_{h}$ is proportional to $L$) to show the nonlocal effects on
different measures of information spread and also quantum chaos. In other
words at separation smaller than $\beta_{h}$, there exist no local operators
but they actually smeared over finite distance scale which is at least
$\sim\beta_{h}$. The most interesting thing about the little string theory is
that it possesses several characteristics of a critical string theory but is
devoid of the nontrivial features of gravity. Using the above dual
gravitational description we will highlight some of the interesting properties
of little string theory relevant to the ongoing research in quantum
information theory.

##  3 Holographic Entanglement Entropy

The Bekenstein-Hawking formula for the entropy of a black hole
Bekenstein:1973ur  relates the entropy to some geometry of the black hole
geometry, it scales as the area of the black hole horizon. The holographic
prescription by Ryu-Takayanagi Ryu:2006bv  is a kind of generalisation of the
Bekenstein-Hawking formula which states that the entropy of any arbitrary
subsystem of a quantum system also has a geometrical interpretation, it equals
the area of some minimal hypersurface embedded into the dual spacetime
background. For a simple bipartite system, the entanglement entropy measures
the strength of correlation between the degrees of freedom of any subsystem
$\mathcal{A}$ with the rest of the system. Given a pure state $|\psi>$ of the
whole system, the entanglement entropy of any subsystem $\mathcal{A}$ is
defined as the Von-Neumann entropy,

$S(\mathcal{A})=-Tr\left(\rho_{\mathcal{A}}\log{\rho_{\mathcal{A}}}\right),$
(12)

where, $\rho_{\mathcal{A}}$ is the reduced density matrix for $\mathcal{A}$.
In this section we will use the above mentioned Ryu-Takayanagi conjecture to
compute the entanglement entropy of an infinitely long strip like entangling
surface extending along the $(x_{2},x_{3},x_{4},x_{5})$ directions but with
finite width $l$ along $x_{1}$ such that, $l/2\leq x_{1}\leq l/2$, $-R/2\leq
x_{2},...,x_{5}\leq R/2$ with $R\rightarrow\infty$. Due to the varying dilaton
profile in this case we will use the generalized version of the formula for
the entanglement entropy of a subregion $A$ as given in Ryu:2006ef ,

$S(A)=\frac{\textrm{Area}(A)}{4G^{10}_{N}}=\frac{1}{4G^{10}_{N}}\int
dx^{8}e^{-2\phi}\sqrt{g^{(8)}_{ind}},$ (13)

where, $G^{10}_{N}=8\pi^{6}\alpha^{\prime 4}$ is the $10$-dimensional Newton’s
constant and $g_{ind}$ is the metric induced on the strip which is embedded
into the spacetime geometry following the equation $x_{1}=x_{1}(r)$ with time
$t$ remaining constant throughout. Using the string frame metric (10), one can
extremize the area functional in (13) with respect to the embedding function
$x_{1}(r)$ to get the equation for the hypersurface extended in the bulk as,

$\frac{dx_{1}}{dr}=\sqrt{\frac{A(r)}{f(r)}}\frac{1}{\sqrt{\frac{r^{6}A(r)}{r_{m}^{6}A(r_{m})}-1}},$
(14)

where $r_{m}$ denotes the turning point for the extremal surface,
$\left(dx_{1}/dr\right)|_{r_{m}}\rightarrow\infty$. We will also define a UV
cut-off at $r=r_{\delta}$ to regularize the area functional defined as,
$x_{1}(r_{\delta})=\pm l/2$ which leads to the following expression for the
strip width $l$ as,

$l=2\int_{r_{m}}^{r_{\delta}}dr~{}\sqrt{\frac{A(r)}{f(r)}}\frac{1}{\sqrt{\frac{r^{6}A(r)}{r_{m}^{6}A(r_{m})}-1}}.$
(15)

The final expression for the area of the subregion $A$ is given as,

$\mathcal{A}=\frac{2\Omega_{3}R^{4}}{g_{s}^{2}}\int_{r_{m}}^{r_{\delta}}dr~{}\frac{A(r)}{\sqrt{f(r)}}\frac{r^{3}}{\sqrt{1-\frac{r_{m}^{6}A(r_{m})}{r^{6}A(r)}}}.$
(16)

---
Figure 2: Figure showing the embedding of the hypersurface as a function the
strip width $l$ for different values of the horizon radius $r_{H}$ with fixed
$L$. Here we have fixed $r_{\delta}$=45.

Before going into the explicit computation of $S(\mathcal{A})$, let us first
present a detailed discussion about the area functional as follows from the
embedding of the extremal bulk hypersurface as a function of the strip width.
As mentioned before, for a theory which exhibit nonlocal effects, there exists
a critical value for the strip width, $l_{c}$ such that the usual area law
behavior for the entanglement entropy is not obeyed and instead it follows the
volume law when the width is less than $l_{c}$. In the following we will
calculate an expression for this critical width which turns out to be
proportional to the UV cut-off scale $r_{\delta}$. So if we set $r_{\delta}$
to infinity then for every finite width of the strip we will find a volume law
behavior. However when we keep the UV cut-off scale at large but finite value,
a phase transition regarding the behavior of entanglement entropy can be
observed as the width of the strip is increases from a value less than $l_{c}$
to the one greater than the same Klebanov:2007ws . In particular, the
entanglement entropy will show a transition from the volume law to the usual
area law behavior.

##### Behavior at UV scale:

Let us consider the plot as given in figure-2 . We observe that at small
values of $l$, that is for very narrow strip, there exist only one smooth
extremal hypersurface which lies very close to the UV cut-off scale
$r_{\delta}$. Also, in this smaller width region all the three hypersurface
corresponding to different values of $r_{H}/L$ has the same form as shown in
figure-2. In order to get an analytic expression for the hypersurface at small
width one can use equation (14) to obtain the following relation between $r$
and $x_{1}$,

$r(x_{1})-r_{m}=\frac{\left(r_{m}^{2}-r_{H}^{2}\right)\left(2L^{2}+3r_{m}^{2}\right)}{8r_{m}\left(L^{2}+r_{m}^{2}\right)^{2}}x_{1}^{2}+\mathcal{O}\left(x_{1}^{4}\right).$
(17)

Using the boundary condition, $x_{1}(r_{\delta})=\pm l/2$ in the above
equation, one can plot the variation of the extremal surface with strip width
for small values of $l$. In figure-3, we have shown the relationship between
$r_{m}$ and $l$ using both the numerical result and the analytic expression
(17) as obtained for small strip width.

---
Figure 3: Figure showing the penetration of the extremal surface in the bulk
as a function of the strip width. The numerical result is denoted by the solid
brown curve while the analytic result is indicated by the orange dashed line.
Note that the analytic result matches nicely with the numerical one for lower
values of the strip width $l$. In this plot we have fixed numerical value the
ratio $r_{H}/L$ to $0.1$.

Substituting $r(x_{1})$ from equation (17) into the area functional in
equation (16), we obtain the approximate analytic expression for the area at
small $l$ region to be,

$\mathcal{A}=\frac{\Omega_{3}R^{4}}{g_{s}^{2}}\biggl{(}lr_{\delta}^{3}+\frac{L^{2}}{2}lr_{\delta}+\frac{15}{128}l^{3}r_{\delta}+\mathcal{O}\left(l^{5}\right)\biggr{)}$
(18)

In a general black hole background that corresponds to an uniform thermal
state in the boundary, the most dominant contribution to the entanglement
entropy (basically the divergent part) comes from the correlation between the
degrees of freedoms which are lying very close to the boundary of the
entangling surface. Hence the most dominant part of entanglement entropy
follows an area law. However there are also finite contributions to the
entanglement entropy that follows a volume law behavior arising due to the
correlation between the degrees of freedom from the entire volume of the
entangling region. However in this case we notice that the leading order term
in $r_{\delta}$ in the above expression for the area depends on the size $l$
of the system, so that even the diverging part of the entanglement entropy
shows an extensive behavior (volume law) which can be linked to the nonlocal
nature of the underlying theory.

##### Behavior at IR scale:

As it follows from figure-2 that there exists three solutions for the extremal
surface at a given strip width around the critical value $l=l_{c}$. Among
these three surfaces, the one at the middle always shows increasing behavior
and hence it is not physical. The surface at the bottom has minimum area
compared to the surface at the top and according to the RT prescription the
bottom surface will be favourable. At even larger strip width there exists
unique solution for the extremal surface and it will stay close to the horizon
and far away from the UV cutoff. As a result the area functional in this scale
is independent of the position of the UV cutoff and shows the usual area law
behavior for the leading divergent part of the entanglement entropy. The area
of the extremal surface in this case is given as,

$\mathcal{A}=\frac{\Omega_{3}R^{4}}{g_{s}^{2}}\biggl{(}r_{\delta}^{4}+\frac{2L^{2}+r_{H}^{2}}{2}r_{\delta}^{2}+\frac{1}{4}\left(4L^{2}r_{H}^{2}+3r_{H}^{4}\right)\log{r_{\delta}}+\mathcal{O}\left(1/r_{\delta}\right)\biggr{)}.$
(19)

The above result for the area is independent of the width $l$ of the infinite
strip. Also with this result the entanglement entropy now follows an area law
behavior as observed for a local theory. However, keeping the width of the
strip fixed if one takes the limit $r_{\delta}\rightarrow\infty$, then the
area again increases extensively similar to equation (18).

The transition from a volume law behavior to an area law for the entanglement
entropy occurs continuously as one moves from UV to IR scale. Hence one can
calculate the critical length scale $l_{c}$ by comparing the two expressions
for the area at high and low energy scales as given in (18) and (19),
respectively and then solving for $l$.

$l_{c}=\frac{r_{\delta}}{2}+\frac{3L^{2}+2r_{H}^{2}}{4r_{\delta}}+\mathcal{O}(1/r_{\delta}^{2}).$
(20)

So we see that the critical length scale is proportional to the UV cutoff
which is a signature of nonlocal behavior of the theory. The contribution to
the entanglement entropy coming from different energy scales differs
nontrivially. Also in standard local theory, the UV contribution to the
entanglement entropy do factorize from that coming from the IR and they just
adds up to the total entanglement entropy. In particular, the UV part of the
entanglement entropy depends on the UV-cutoff $r_{\delta}$ and also it is
local in a sense that only the degrees of freedom very close to the boundary
of the subsystem gets involved to the quantum correlation across the boundary
which results in an area law behavior. Conversely, the IR contribution is not
only independent of the UV cutoff but does not depend on the scheme of the
regularization either such that a good cutoff will only change the physics of
the UV without affecting the IR. However, in a nonlocal theory the above
mentioned factorisation between contributions coming from UV and the IR scales
is no longer a valid assumption. For instance, as we have found, below the
critical length scale the UV contribution of the entanglement entropy depends
on the width $l$ of the infinite strip which is the IR scale of our theory.
Hence the UV part of the entanglement entropy is no longer local and for a
sufficiently narrow strip the degrees of freedom living on the entire strip
now correlates to the one outside and hence it develops a dependance on the IR
scale. This dependance on $l$ also leads to the violation of the area law.
Finally, the critical length $l_{c}$ for the strip width also depends on the
UV cutoff. So if the cutoff is taken at infinity, then a volume law for the
entanglement entropy would follow for all energy scales. All the above results
indicates the UV/IR mixing of the underlying nonlocal theory.

In figure-4 we have shown the variation of the area with respect to the width
$l$ varying from very small value to sufficiently large one. The area
increases very quickly till the critical length scale and after that it slows
down and eventually saturates. The transition from volume law to area law
behavior is clearly realized from the same plot. Also note that at the
critical value of strip width, the entanglement entropy is continuous but it’s
first derivative suffers a discontinuity indicating a first order phase
transition. However as we increase the size of the black hole by considering
larger value of $r_{H}$, there is no such phase transition which is explicitly
shown in the same figure (see the green colored plot).

---
Figure 4: Figure showing the variation of entanglement entropy with strip
length $l$ for three different values of $r_{H}/L$.

##  4 Holographic entanglement wedge cross section

In the last section we analyse the entropy of entanglement for a bipartite
system in a pure state. However, for total correlation in a bipartite mixed
state the appropriate measure as defined in quantum information theory is the
entanglement of purification (EoP) terhal . In the context of AdS/CFT
correspondence, the geometrical quantity which is dual to EoP is conjectured
in Takayanagi:2017knl ; Nguyen:2017yqw  to be the Entanglement wedge cross
section (EWCS). In holographic theories the computation of EWCS can be done in
a straight forward manner by evaluating the area of the minimal hypersurface
$\Gamma_{W}$ (see figure-) inside the entanglement wedge in the bulk
corresponding to the union of two subregions $\mathcal{A}$ and $\mathcal{B}$
on the boundary. In this section we adopt the holographic technique to
calculate EWCS for the same gravitational background dual to the theory of
little string theory. The goal is to see if the nonlocal nature of little
string hteory has any direct manifestation in such measure of correlation in a
mixed state.

Let us start with the technical definition of Eop. Consider two subregions
$\mathcal{A}$ and $\mathcal{B}$ in the form of a infinite strip of equal width
$l$ on two separate the boundary slices at a constant time. In general, the
two subregions will be in mixed states defined by the density matrix
$\rho_{\mathcal{A}}\in\mathcal{H}_{\mathcal{A}}$ and
$\rho_{\mathcal{B}}\in\mathcal{H}_{\mathcal{B}}$ respectively, where
$\mathcal{H}_{\mathcal{A}}$ and $\mathcal{H}_{\mathcal{B}}$ are the
corresponding Hilbert spaces. The total density matrix for the two bipartite
systems
$\rho_{\mathcal{A}\mathcal{B}}\in\mathcal{H}_{\mathcal{A}}\otimes\mathcal{H}_{\mathcal{B}}$
can be obtained from the pure state
$|\psi>\in\mathcal{H}_{\mathcal{A}}\otimes\mathcal{H}_{\mathcal{B}}$ following
the usual definition of density matrix,
$\rho_{\mathcal{A}\mathcal{B}}=Tr_{\overline{\mathcal{A}}~{}\overline{\mathcal{B}}}|\psi><\psi|$.
In the last definition $\overline{\mathcal{A}}$ and $\overline{\mathcal{B}}$
represents the additional degrees of freedoms required for the purification of
$\rho_{\mathcal{A}\mathcal{B}}$ such that the state $|\psi>$ belongs to the
extended Hilbert space
$\mathcal{H}_{\mathcal{A}\overline{\mathcal{A}}}\otimes\mathcal{H}_{\mathcal{B}\overline{\mathcal{B}}}$.
The EoP is defined as the minimum von-Neumann entropy
$S(\mathcal{A}\overline{\mathcal{A}})$, minimized over all possible splitting
of the purification into $\overline{\mathcal{A}}~{}\overline{\mathcal{B}}$,

$E(\rho_{\mathcal{A}\mathcal{B}})=\mathop{min}_{\rho_{\mathcal{A}\mathcal{B}}=Tr_{\overline{\mathcal{A}}~{}\overline{\mathcal{B}}}|{\psi}\rangle\langle{\psi|}}S(\mathcal{A}\overline{\mathcal{A}}).$
(21)

---
Figure 5: A schematic representation of the holographic set up for the
computation of EWCS.

In order to proceed with the holographic calculation of EWCS, we represent two
subregions on the boundary as infinitely long strip with finite width $l$ in a
similar fashion. The two subregions are separated by a distance $D$ as in
figure-5. The RT surface for $2l+D$ and $D$ are denoted by $\gamma_{2l+D}$ and
$\gamma_{D}$ respectively. In a symmetric configuration around $x_{1}=0$, the
EWCS is the minimal area of the hypersurface denoted by $\Gamma_{W}$ with the
two end points coinciding on the RT surface $\gamma_{D}$ and $\gamma_{2l+D}$.
The induced metric on the hypersurface is given as,

$ds_{\Gamma_{W}}^{2}=\sum_{i=2}^{5}dx_{i}^{2}+A(r)\Big{(}\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{3}^{2}\Big{)}.$
(22)

The area of this minimal surface can be computed from the above induced metric
yielding the final result for the EWCS as,

$\begin{split}E_{W}&=\frac{\Omega_{3}R^{4}}{4G_{N}^{(10)}g_{s}^{2}}\int_{r_{m}\left(2l+D\right)}^{r_{m}\left(D\right)}dr\frac{r^{3}A(r)}{\sqrt{f(r)}},\\\
&=\frac{\Omega_{3}R^{4}}{32G_{N}^{(10)}g_{s}^{2}}\biggl{[}r\sqrt{r^{2}-r_{H}^{2}}\left(4L^{2}+2r^{2}+3r_{H}^{2}\right)+r_{H}^{2}\left(4L^{2}+3r_{H}^{2}\right)\log{\left(r+\sqrt{r^{2}-r_{H}^{2}}\right)}\biggr{]}_{r_{m}\left(2l+D\right)}^{r_{m}\left(D\right)}\end{split}$
(23)

Now, as we have seen in the previous section that the RT surface inside the
bulk follow very different embedding depending on weather the width of the
strip $l$ is greater or less than the critical length scale $l_{c}$ (see
figure-2). So we expect the EWCS to show different behavior with increasing
$l$ in regions where $l<l_{c}$ and $l>l_{c}$.

|  |
---|---|---
Figure 6: a)Plot showing the variation of EWCS with strip width in the region
of smaller values of $l<l_{c}$, b) Variation of EWCS with strip width in the
region of larger $l>l_{c}$.

In the left panel of figure-6, we see that the EWCS increases with the strip
width $l$ for length scale which is less that the critical value $l_{c}$. This
can be explain at least geometrically from the nature of the embedding of RT
surface in the bulk geometry. Again referring to figure-2, we see that the RT
surface for small values of $l<l_{c}$ stays very close to the boundary and as
$l$ increases it starts to penetrate deep into the bulk towards the horizon.
Now the RT surface $\gamma_{2l+D}$, corresponding to the region of total width
$2l+D$ on the boundary, moves towards the horizon faster than the other RT
surface $\gamma_{D}$ as the width $l$ is increased. This is clearly evident
from the increasing slope of the minimal surface in figure-2 in the region of
small $l$. Hence the length of the extremal surface $\Gamma_{W}$ increases
with $l$ initially. However, as the width is increased to sufficiently large
value beyond $l_{c}$, the minimal surface $\gamma_{2l+D}$ now reaches enough
close to the horizon and can no longer penetrate anymore. On the other hand,
the surface $\gamma_{D}$ being comparatively smaller in area, can still
increase in size and move towards the horizon. As a result the length of the
extremal surface $\Gamma_{W}$ decreases as shown in the right panel of
figure-6.

Clearly the increase in EWCS in the region of small $l$ is due to the nonlocal
correlation between the two subregions $\mathcal{A}$ and $\mathcal{B}$. In
other words, for small strip widths, the degrees of freedoms from the entire
volume of the two subregions participate in the correlation due to the
nonlocal nature of the interaction and hence we see an increase in the
correlation with $l$. However beyond the critical length scale the nonlocal
correlation ceased and the usual decreasing behavior is observed. As the strip
width is increased beyond a particular value of $l$, the two subregions gets
in a disconnected phase such that the correlation between them drops
discontinuously to zero. 222Here we have considered the infinitely long
parallel strip configurations of width $l$ to calculate the EWCS. In this case
there exists two different phases, namely (i) the connected phase and (ii) the
disconnected phase which basically indicates the degree of correlation between
two subsystems. In connected phase the correlation is finite, on the other
hand in the disconnected phase there is no correlation between the two
subsystems. Now, in the connected phase the area of the minimum cross-section
is finite till some particular value of the strip width $l$, say $l_{con}$.
However as the width is increased beyond $l_{con}$, the entanglement wedge
doesn’t exist anymore and as a result the corresponding minimum cross-section
vanishes. So the reason for the disconnected phase is the sudden non-existence
of the entanglement wedge as the value of $l$ gets bigger than $l_{con}$.
Also, as already mentioned, physically the occurrence of the disconnected
phase indicates zero correlation between the two subregions.

##  5 Study of shockwave geometries

###  5.1 Gravity setup for Unperturbed Geometry

We will start by considering the following $10$-dimensional metric (10)
corresponding to an extended two sided black brane geometry,

$ds^{2}=-G_{tt}dt^{2}+G_{rr}dr^{2}+G_{ij}dx^{i}dx^{j}+G_{lm}d\theta^{l}d\theta^{m},$
(24)

Where, ${i,j=1,2,3,4,5}$ represents the five dimensional flat directions and
${l,m=1,2,3}$ corresponds to the compact three dimensional unit sphere
$S^{3}$. The asymptotic boundary is located at $r=\infty$ and the horizon at
$r=r_{h}$, $r$ being the radial coordinate. Different metric components of the
above equation (24) can be expanded near the horizon and are given as,

$G_{tt}=Z0(r-rh),~{}G_{rr}=\frac{Z1}{r-rh},~{}G_{ij}=G_{lm}=\text{Constant}.$
(25)

The inverse hawking temperature can be calculated from the above data as

$\beta=4\pi\sqrt{Z1/Z0}=1/T_{H}.$ (26)

For the gravitational shockwave analysis of the above geometry it is
convenient to work with the Kruskal coordinate system $(U,V)$ which is defined
below as,

$UV=e^{\frac{4\pi}{\beta}r_{*}},~{}~{}U/V=-e^{-\frac{4\pi}{\beta}t}.$ (27)

where $r_{*}$ is the tortoise coordinate given below as,

$r_{*}=\int\sqrt{\frac{G_{tt}}{G_{rr}}}dr,$ (28)

In terms of the Kruskal coordinate the metric (24) can be rewritten as,

$ds^{2}=2F(U,V)dUdV+G_{ij}(U,V)dx^{i}dx^{j}+G_{lm}(U,V)d\theta^{l}d\theta^{m},$
(29)

where $F(U,V)=\frac{\beta^{2}}{8\pi}\frac{G_{tt}(UV)}{UV}$. Now in terms of
the $(U,V)$ coordinate, the two asymptotic boundaries on the left and the
right side (see figure-1) are situated at $UV=-1$, while at the horizon we
have $UV=0$. Considering the above metric, Einstein’s field equation can be
written as,

$R_{\mu\nu}-\frac{1}{2}\partial_{\mu}\partial_{\nu}\phi-\frac{1}{4}\exp(-\phi)H_{\mu
bc}H_{\nu}^{bc}+\frac{1}{6(d-2)}g_{\mu\nu}\exp(-\phi)H_{3}^{2}=8\pi
G_{N}T_{0,\mu\nu}^{M},$ (30)

where, $T_{0,\mu\nu}^{M}$ is the energy momentum tensor corresponding to the
matter part of the action computed using the unperturbed geometry. It is
presumed to have the following structure,

$\displaystyle T_{0,\mu\nu}^{M}dx^{\mu}dx^{\nu}=$ $\displaystyle
2T_{UV}(U,V)dUdV+T_{UU}(U,V)dU^{2}+T_{VV}(U,V)dV^{2}+T_{ij}(U,V)dx^{i}dx^{j}$
(31) $\displaystyle+T_{lm}(U,V)d\theta^{l}d\theta^{m}.$

###  5.2 Shockwave geometries

The two sided black hole geometry discussed above is holographically dual to
the thermofield double state as already defined in the introduction section.
In this section we will discuss the possible modifications of the metric in
(29) as a result of a localised perturbation on one of the boundaries (say
left) at early time. In the gravity description this early perturbation
corresponds to an energy pulse that originates from the past horizon and
propagates into the future singularity. Due to the strong gravitational
attraction, the energy of this pulse increases exponentially with time. For
sufficiently early perturbation on the boundary the energy pulse produces a
shockwave in the bulk geometry such that the corresponding backreaction
becomes non negligible. This perturbed geometry is referred to as the
shockwave geometry. Analysing the interaction of this shockwave with the black
hole horizon one can extract important quantities like butterfly velocity
$v_{B}$ and Lyapunov exponent $\lambda_{L}$ etc. These quantities can shed
light on the chaotic features of holographic theories.

The shockwave we consider here is localized at $U=0$ and propagates along the
$V$ direction resulting an extension along $V$ and a compression along $U$.
The energy momentum tensor that describe the whole effect is given as,

$T_{UU}^{Shock}=E\delta(U)e^{\frac{2\pi t}{\beta}}e^{ik.x}.$ (32)

Here we will work in the momentum space due to the absence of any well defined
gauge invariant local operators in position space for the underlying
nonlocality of little string theory 333For detailed discussion on this see
Fischler:2018kwt .. To incorporate the backreaction in our calculation we made
the replacement $V\rightarrow V+\alpha$. Here $\alpha$ captures key
information about the profile of the shockwave,

$\alpha=\tilde{\alpha}(t,k)e^{ik.x}.$ (33)

$\tilde{\alpha}(t,k)$ can be obtained as solution of the Einstein’s field
equation. To ensure the fact that only the region with $U>0$ is modified due
to the shockwave, we replace $V$ with $V+\Theta(U)\alpha$, $\Theta(U)$ being
the Heaviside step function. After doing some algebraic manipulation, the
modified metric and energy momentum tensor can be expressed respectively as,

$ds^{2}=2FdUdV+G_{ij}dx^{i}dx^{j}+G_{lm}d\theta^{l}d\theta^{m}-2A\delta(U)dU^{2},$
(34) $\displaystyle T^{M}$ $\displaystyle=2\biggl{(}T_{UV}-\alpha
T_{VV}\delta(U)\biggr{)}dUdV+T_{VV}dV^{2}+\biggl{(}T_{UU}+\alpha^{2}T_{VV}\delta(U)^{2}-2\alpha~{}T_{UV}\delta(U)\biggr{)}dU^{2}$
(35) $\displaystyle+T_{ij}dx^{i}dx^{j}+T_{lm}d\theta^{l}d\theta^{m}.$

Finally, for perturbed geometry we can write Einstein’s equation of motion as,

$R_{\mu\nu}-\frac{1}{2}\partial_{\mu}\partial_{\nu}\phi-\frac{1}{4}\exp(-\phi)H_{\mu
bc}H_{\nu}^{bc}+\frac{1}{6(d-2)}g_{\mu\nu}\exp(-\phi)H_{3}^{2}=8\pi
G^{10}_{N}(T_{\mu\nu}^{M}+T_{\mu\nu}^{shock}).$ (36)

with $T^{M}$ and $T^{shock}$ is given in equation (32) and (35). Now, we need
to solve the UU component of equation (36). Also notice that the components of
the energy momentum tensor and metric should obey the following conditions
Sfetsos:1994xa ; Jahnke:2017iwi

$F_{,V}=G_{ij,V}=T_{VV}^{M}=0~{}~{}\text{at~{}U}=0.$ (37)

We first solve equation (30) for $(T_{UV},T_{VV},T_{UU})$ and then substitute
these back into the UU component of (36) to finally obtain the following
result for $\tilde{\alpha}$ given as,

$\tilde{\alpha}\left(-Fk_{i}k_{j}+\frac{1}{2}G_{ij,UV}\right)G^{ij}=8\pi
G^{10}_{N}Ee^{\frac{2\pi t}{\beta}}.$ (38)

In terms of coordinate $(t,r)$, the above equation takes the following form

$\left(G^{ij}k_{i}k_{j}+\tilde{M}^{2}\right)\tilde{\alpha}=-e^{\frac{2\pi\left(t-t_{*}\right)}{\beta}},$
(39)

where

$\tilde{M}^{2}=\frac{4\pi^{2}}{\beta^{2}}\Bigg{(}\frac{1}{G_{tt}^{\prime}(r_{h})}\Big{(}5\Big{(}\frac{G_{11}^{\prime}(r_{h})}{G_{11}(r_{h})}\Big{)}+3\Big{(}\frac{G_{\theta\theta}^{\prime}(r_{h})}{G_{\theta\theta}(r_{h})}\Big{)}\Big{)}\Bigg{)},$
(40)

and

$t_{*}=\frac{\beta}{2\pi}\log\left(\frac{F(r_{h})}{8\pi G^{10}_{N}E}\right).$
(41)

where, $t_{*}$ is known as the scrambling time which indicates the time scale
for saturation of complexity or growth of operator size. In other words, it
denotes the time scale for the perturbation to spread over all boundary space
time. For diagonal metric $G_{ij}$, we can estimate the parameter
$\tilde{\alpha}$ as,

$\tilde{\alpha}(t,k)\sim\frac{e^{\frac{2\pi(t-t_{*})}{\beta}}}{G^{ii}(r_{h})k_{i}^{2}+\tilde{M}^{2}}.$
(42)

For homogeneous shock, all the components of $k$ are same and we set it to
zero such that the shock parameter $\tilde{\alpha}$ can be expressed as,

$\tilde{\alpha}(t,k)\sim\text{Constant}\times
e^{(\frac{2\pi(t-t_{*})}{\beta})}.$ (43)

We can compare it with the double commutator in (2) and obtain the Lyapunov
exponent $\lambda_{L}$ as,

$\lambda_{L}=\frac{2\pi}{\beta}.$ (44)

Notice that equation (42) has a pole at a particular value of $k$ proportional
to $\tilde{M}$. The expression for $k^{2}$ evaluated at the pole is given in
terms of $\lambda_{L}$ and $v_{b}$ as,

$k^{2}=-\frac{\lambda_{L}^{2}}{v_{b}^{2}}.$ (45)

Using the above formula together with equation (44) we can calculate butterfly
velocity as,

$v_{b}^{2}=\frac{\lambda_{L}^{2}}{\tilde{M}^{2}}=\frac{1}{G_{11}(r_{h})}\Bigg{(}\frac{1}{G_{tt}^{\prime}(r_{H})}\Big{(}5\frac{G_{11}^{\prime}(r_{H})}{G_{11}(r_{h})}+3\frac{G_{\theta\theta}^{\prime}(r_{h})}{G_{\theta\theta}(r_{h})}\Big{)}\Bigg{)}^{-1}.$
(46)

For the little string theory butterfly velocity can be obtained by
substituting the exact metric components from equation (10) into (46) to
obtain the following result,

$v_{b}^{2}=\frac{L^{2}+r_{H}^{2}}{2L^{2}+3r_{H}^{2}}.$ (47)

Figure 7: Variation of butterfly velocity $v_{b}$ with length scale
$\frac{L}{r_{H}}$

In figure-7, we have plotted $v_{b}$ with respect to a dimensionless quantity
$\frac{L}{r_{H}}$. It can be noticed from figure-7, that for increasing values
of $\frac{L}{r_{H}}$, butterfly velocity $v_{b}$ increases sharply and finally
saturates to a value less than the speed of light, $c=1$. The initial increase
of $v_{b}$ with $\frac{L}{r_{H}}$ indicates a faster spread of information due
to the nonlocal property of little string theory. The saturation of $v_{b}$
below the speed of light is due to the Lorentz invariance of the little string
theory.

##  6 Holographic mutual information of two sided geometry

Mutual information plays an important role in defining nonlocal correlation
between noninteracting and non-overlapping subregions. For two sided black
hole geometry, these two subregions are considered at two asymptotic
boundaries of the corresponding Kruskal diagram. The correlation between these
two distant subregions can be realized through the inverse of the geodesic
distance through the wormhole geometry extending from the left boundary to the
right one. Due to the butterfly effect in holographic theories, a small
perturbation in the asymptotic boundary can destroy the left-right entangled
pattern completely as mentioned above. We will be interested in the process of
this disruption, more precisely the rate with which it happens.

###  6.1 Two sided mutual information for unperturbed geometry

We will start with the computation of mutual information in the unperturbed
gravity dual of little string theory using the HRRT prescription of
Hubeny:2007xt . We consider two identical infinitely long strip like
entangling surfaces $A$ and $B$ in the left and right boundaries extended
along the spacial directions $(x_{2},x_{3},x_{4},x_{5})$ such that $-R/2\leq
x_{2,3,4,5}\leq R/2$ with $R\rightarrow\infty$. However, it has a finite width
$l$ along the $x_{1}$ direction. Considering a constant time slice, the
corresponding Ryu-Takayanagi surfaces in the bulk exterior on both sides of
the geometry is defined by the embedding $x_{1}(r)$. This embedding will
generate a surface $\gamma_{A}$ in the left exterior and $\gamma_{B}$ in the
right exterior. Also the corresponding bulk hypersurface for $A\cup B$ is
given as $\gamma_{1}\cup\gamma_{2}$ as shown schematically by the red colored
lines in figure-8

---
Figure 8: Schematic diagram of unperturbed two sided geometry

With this the mutual information $I(A,B)$ is defined as,

$I(A,B)=S(A)+S(B)-S(A\cup B).$ (48)

where $S(A)$, $S(B)$, $S(A\cup B)$ are the entanglement entropy of $A$, $B$
and $A\cup B$ respectively. Using the HRRT formalism they can be calculated
from the minimum area of the hypersurface $\gamma_{A}$, $\gamma_{B}$,and
$\gamma_{1}\cup\gamma_{2}$ respectively. The induced metric can be written as,

$\displaystyle g_{rr}$ $\displaystyle=$
$\displaystyle\left(G_{rr}+G_{11}x^{\prime}(r)^{2}\right),$ $\displaystyle
g_{\theta_{i}\theta_{i}}$ $\displaystyle=$ $\displaystyle
G_{\theta_{i}\theta_{i}}\times\text{metric
of}~{}S^{3},~{}~{}\text{i}=1,2,3,~{}~{}G_{11}=G_{22}=G_{33}=G_{44}=G_{55}.$
(49)

The area functional for $\gamma_{A}$ can be calculated as,

$\displaystyle\mathcal{A}(\gamma_{A})$ $\displaystyle=$ $\displaystyle\int
d^{8}\sigma\sqrt{(detg_{ab})},$ (50) $\displaystyle=$
$\displaystyle\Omega_{3}R^{4}\int
dr~{}r^{3}A^{5/8}\Big{(}\frac{A^{3/4}}{f}+\frac{x^{\prime}(r)^{2}}{A^{1/4}}\Big{)}^{1/2},$
$\displaystyle=$ $\displaystyle\Omega_{3}R^{4}\int
dr\mathcal{F}(x,x^{\prime},r),$

where $\Omega_{3}$ is the volume of three sphere and $R^{4}=\int
dx_{2}dx_{3}dx_{4}dx_{5}$. The integrand in the last line of the above
equation, $\mathcal{F}$, do not depend on $x$ explicitly hence the
corresponding canonical momentum is a conserved quantity which can be
calculated as,

$P=\frac{\partial\mathcal{F}}{\partial
x^{\prime}}=\frac{r^{2}A(r)^{3/8}x^{\prime}}{\sqrt{\frac{A(r)^{3/4}}{f(r)}+\frac{{x^{\prime}}^{2}}{A(r)^{1/4}}}}.$
(51)

At turning point, $r=r_{m}$, $x^{\prime}$ will become infinitely large. Upon
imposing this condition the conserved quantity $P$ is given as,

$P|_{r=r_{m}}=r_{m}^{3}\sqrt{A(r_{m})}.$ (52)

Solving equation (51) for $x^{\prime}$ along with the condition as given in
(52) we obtain,

$x^{\prime}=\sqrt{\frac{A(r)}{f(r)}}\frac{1}{\frac{r^{6}A(r)}{r_{m}^{6}A(r_{m})}-1},$
(53)

Substituting this expression of $x^{\prime}$ back into equation (50) results
in the following on shell area,

$\mathcal{A}(\gamma_{A})=2\Omega_{3}R^{4}\int_{r_{m}}^{r_{\delta}}\frac{r^{3}A(r)}{\sqrt{f(r)}}\frac{1}{\sqrt{1-\frac{r_{m}^{6}A(r_{m})}{r^{6}A(r)}}}dr,$
(54)

where $r_{\delta}$ is the UV cut-off as already defined. Finally, we can write
down the corresponding entanglement entropy $S_{A}$ as,

$S(A)=\frac{\mathcal{A}(\gamma_{A})}{4G^{10}_{N}}=\frac{\Omega_{3}R^{4}}{2G^{10}_{N}}\int_{r_{m}}^{r_{\delta}}\frac{r^{3}A(r)}{\sqrt{f(r)}}\frac{1}{\sqrt{1-\frac{r_{m}^{6}A(r_{m})}{r^{6}A(r)}}}dr.$
(55)

An exact similarly expression follows for the entanglement entropy of strip
$B$. In order to obtain $S(A\cup B)$, we need to calculate the combined area
of the surfaces passing through the interior connecting both sides of the
geometry. There are two such surfaces, one at $x=0$ which is denoted as
$\gamma_{1}$ and the other one is at $x=l$ denoted as $\gamma_{2}$. Using the
symmetric configuration of these surfaces one can express the area of surface
$\gamma_{1}\cup\gamma_{2}$ as,

$\mathcal{A}(\gamma_{1}\cup\gamma_{2})=4\Omega_{3}R^{4}\int_{r_{h}}^{r_{\delta}}\frac{r^{3}A(r)}{\sqrt{f(r)}}dr.$
(56)

Corresponding entanglement entropy can be written as,

$S(A\cup
B)=\frac{\Omega_{3}R^{4}}{G^{10}_{N}}\int_{r_{h}}^{r_{\delta}}\frac{r^{3}A(r)}{\sqrt{f(r)}}dr.$
(57)

Using these results in (48) the final expression of mutual information can be
written as the following integral form,

$I(A,B)=\frac{\Omega_{3}R^{4}}{G^{10}_{N}}\Bigg{(}\int_{r_{m}}^{r_{\delta}}\frac{r^{3}A(r)}{\sqrt{f(r)}}\frac{1}{\sqrt{1-\frac{r_{m}^{6}A(r_{m})}{r^{6}A(r)}}}dr-\int_{r_{h}}^{r_{\delta}}\frac{r^{3}A(r)}{\sqrt{f(r)}}dr\Bigg{)}.$
(58)

On the other hand the strip width $l$ can be expressed as a function of
$r_{m}$ and is given as,

$l=\int dx=\int
x^{\prime}dr=2\int_{r_{m}}^{r_{\delta}}\Bigg{(}\sqrt{\frac{A(r)}{f(r)}}\frac{1}{\frac{r^{6}A(r)}{r_{m}^{6}A(r_{m})}-1}\Bigg{)}dr.$
(59)

In figure-9, we have shown the variation of the mutual information I(A,B) with
respect to the strip width $l$ for two different values of $\frac{r_{h}}{L}$.
It is qualitatively very much similar to the variation of entanglement entropy
with $l$ as discussed in section-3 which is not surprising at all.

Figure 9: Plot showing variation of mutual information I with strip width $l$

###  6.2 Two sided Mutual Information in presence of shock wave

Now, We will discuss the disruption of two sided mutual information in
presence of a shockwave. For the sake of simplicity we will consider the shock
profile to be homogeneous. In that case the shock parameter $\alpha$ can be
written as $\alpha=\text{Constant}*e^{\frac{2\pi t_{0}}{\beta}}$. The mutual
information in this case can be described as,

$I(A,B;\alpha)=S(A)+S(B)-S(A\cup B;\alpha),$ (60)

As the exponentially boosted shockwave modifies the near horizon region of the
black hole geometry quite significantly, only $S(A\cup B)$ will receive the
possible correction with increasing shockwave parameter $\alpha$. Because of
the same reason, $S(A)$ and $S(B)$ will not get modified at all. To get rid of
the UV divergences that appeared in $S_{A\cup B}$, it is convenient to define
the following regularized version of it, which is given as,

$S^{reg}(A\cup B;\alpha)=S(A\cup B;\alpha)-S(A\cup B;\alpha=0).$ (61)

With this regularised entanglement entropy, one can rewrite the definition of
mutual information (60) as,

$I(A,B;\alpha)=I(A,B;\alpha=0)-S^{reg}(A\cup B;\alpha).$ (62)

Here we will consider sufficiently large value for the strip width $l$ so as
to get a finite positive result of the mutual information. In presence of the
shockwave, there is an effective increase of the size of the wormhole at
boundary time $t=0$ which is schematically represented in figure-10.

---
Figure 10: Schematic diagram of perturbed two sided geometry

As already explained, in this case only $S(A\cup B)$ will be modified.
Initially, It is given by the area of the hypersurface
$\gamma_{1}\cup\gamma_{2}\equiv\gamma_{wormhole}$ and one obtains a nonzero
value for the mutual information. However, as the strength of the shockwave
parameter $\alpha$ increases with time, the area of the wormhole surface also
increases eventually decreasing the mutual information. With further increase
of $\alpha$, the area of $\gamma_{wormhole}$ at some point gets bigger than
the combined area $\gamma_{A}\cup\gamma_{B}$. Hence in the definition of
mutual information (62), instead of $\gamma_{wormhole}$, the combined area of
$\gamma_{A}$ and $\gamma_{B}$ will be favourable for the entropy $S_{A\cup B}$
due to the minimum area criteria of HRRT formalism. As a result the mutual
information goes to zero, $I(A,B;\alpha)=0$. To construct the wormhole surface
we choose an appropriate embedding given as,
$x^{m}=(t,0,x_{2},x_{3},x_{4},x_{5},r(t),\theta_{1},\theta_{2},\theta_{3})$.
Using this embedding one can easily obtain the components of the induced
metric as,

$\displaystyle g_{tt}$ $\displaystyle=$ $\displaystyle
G_{tt}+G_{rr}\dot{r}^{2},$ $\displaystyle g_{\theta_{i}\theta_{i}}$
$\displaystyle=$ $\displaystyle G_{\theta_{i}\theta_{i}}\times~{}\text{
metric~{} of~{}}S^{3},$ (63)

The area functional can be expressed as,

$\displaystyle\mathcal{A}(\gamma_{wormhole})$ $\displaystyle=$ $\displaystyle
2\Omega_{3}R^{4}\int
dt~{}r^{3}\sqrt{\frac{A(r)}{f(r)}}\left(-f(r)^{2}+A(r)\dot{r}^{2}\right)^{1/2},$
(64) $\displaystyle=$ $\displaystyle 2\Omega_{3}R^{4}\int
dt\mathcal{F}(r,\dot{r},t).$

As the integrand $\mathcal{F}(r,\dot{r},t)$ in the above equation is invariant
under time translation, the corresponding conserved quantity can be calculated
as,

$\displaystyle C$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{F}}{\partial\dot{r}}\dot{r}-\mathcal{F},$
(65) $\displaystyle=$
$\displaystyle\frac{r^{3}\sqrt{A(r)}f(r)^{3/2}}{\left(-f(r)^{2}+A(r)\dot{r}^{2}\right)^{1/2}}.$

---
Figure 11: Penrose diagram of perturbed geometry with the constant $r=r_{0}$
surface and the horizontal extremal surface as indicated by black and red
lines respectively.

Referring to figure-11, we note that due to the shockwave perturbation, the
left half of the horizontal extremal surface will be divided into three parts
with $r_{0}$ being the contact point of the constant $r$ surface and the same
horizontal extremal surface. The conserved quantity $C$ can be calculated at
$r=r_{0}$ as,

$C(r_{0})=-r_{0}^{3}\sqrt{A(r_{0})}\sqrt{-f(r_{0})}.$ (66)

Next we substitute the above expression of $C(r_{0})$ into equation (65) and
solve for $\dot{r}$ to get,

$\dot{r}=\frac{f(C(r_{0})^{-2}r^{6}f(r)A(r)+1)^{1/2}}{\sqrt{A(r)}}.$ (67)

Putting the above result for $\dot{r}$ in equation (64), we get the on shell
action as,

$\mathcal{A}(\gamma_{wormhole})=2\Omega_{3}R^{4}\int
dr\frac{r^{6}A^{3/2}}{(fAr^{6}C^{-2}+1)^{1/2}}.$ (68)

Also from the expression of $\dot{r}$ as given in (67), one get the following
result for time $t$ running along the wormhole as,

$t=\int\frac{dr}{\dot{r}}=\int
dr\frac{\sqrt{A}}{f\left(C^{-2}fAr^{6}+1\right)^{1/2}}.$ (69)

Taking all the three segments of the left half of the extremal surface in
figure-11 into consideration, the final integral form of $S_{A\cup B}$ is
given as,

$S(A\cup
B;r_{0})=\frac{\Omega_{3}R^{4}}{G^{10}_{N}}\Bigg{(}\int_{r_{H}}^{r_{\delta}}dr\frac{r^{6}A^{3/2}}{(fAr^{6}C^{-2}+1)^{1/2}}+2\int_{r_{0}}^{r_{H}}dr\frac{r^{6}A^{3/2}}{(fAr^{6}C^{-2}+1)^{1/2}}\Bigg{)}.$
(70)

Here the turning point at $r=r_{0}$ also acts as the strength of the shockwave
such that as the shockwave parameter $\alpha$ is increased the turning point
moves further into the black hole interior and the value of $r_{0}$ decreases.
Notice also that in the above equation we expressed the entanglement entropy
$S_{A\cup B}$ as a function of the turning point $r_{0}$. Finally, the
corresponding regularized entropy is given as,

$S^{reg}(A\cup
B;r_{0})=\frac{\Omega_{3}R^{4}}{G^{10}_{N}}\Bigg{(}\int_{r_{H}}^{r_{\delta}}dr\Big{(}\frac{r^{6}A^{3/2}}{(fAr^{6}C^{-2}+1)^{1/2}}-\frac{r^{3}A}{\sqrt{f}}\Big{)}+2\int_{r_{0}}^{r_{H}}dr\frac{r^{6}A^{3/2}}{(fAr^{6}C^{-2}+1)^{1/2}}\Bigg{)}.$
(71)

Following the detailed analysis as presented in Fischler:2018kwt , one can
determine the shockwave parameter $\alpha$ in terms of the turning point
$r_{0}$ which takes the following form,

$\alpha(r_{0})=2\exp(K_{1}(r_{0})+K_{2}(r_{0})+K_{3}(r_{0})),$ (72)

where $K_{1},K_{2}$ and $K_{3}$ are given as follows,

$\displaystyle K_{1}(r_{0})$ $\displaystyle=$
$\displaystyle\frac{4\pi}{\beta}\int_{\overline{r}}^{r_{0}}\frac{\sqrt{A}}{f}dr,$
(73) $\displaystyle K_{2}(r_{0})$ $\displaystyle=$
$\displaystyle\frac{2\pi}{\beta}\int_{r_{H}}^{r_{\delta}}\frac{\sqrt{A}}{f}(1-\frac{1}{\sqrt{C^{-2}fAr^{6}+1}})dr,$
(74) $\displaystyle K_{3}(r_{0})$ $\displaystyle=$
$\displaystyle\frac{4\pi}{\beta}\int_{r_{0}}^{r_{H}}\frac{\sqrt{A}}{f}(1-\frac{1}{\sqrt{C^{-2}fAr^{6}+1}})dr.$
(75)

Using the above we have shown the explicit dependence of $\alpha$ on the
dimensionless ratio $r_{0}/r_{h}$ in figure-12. We observe that the shockwave
parameter $\alpha$ increases with decreasing values of $r_{0}/r_{H}$ and
beyond certain value $\alpha$ sharply diverges. Also as expected, at
$r_{0}=r_{H}$ the shockwave parameter $\alpha$ exactly goes to zero.

Figure 12: Variation of shockwave parameter $\alpha$ with dimensionless ratio
$\frac{r_{0}}{r_{H}}$

Analysing the nature of the integral form of $K_{3}(r_{0})$, we find that the
sharp divergence of $\alpha$ below some critical value of $r_{0}=r_{c}$ is due
to the diverging nature of $K_{3}(r_{0})$ which can be realized from equation
(72). To obtain this critical value, one needs to series expand the integrand
of $K_{3}$ about $r=r_{0}$ and then equate the coefficient of $(r-r_{0})$ in
the resulting expansion to zero Fischler:2018kwt , yielding the following
expression of $r_{c}$,

$r_{c}=\frac{r_{H}\sqrt{1-(\frac{L}{r_{H}})^{2}+\sqrt{1+(\frac{L}{r_{H}})^{2}+(\frac{L}{r_{H}})^{4}}}}{\sqrt{3}}.$
(76)

####  6.2.1 Variation of mutual information with Shock parameter $\alpha$

Two sided mutual information in presence of shockwave is already obtained in
equation (62). It will be convenient to normalize it with the corresponding
result at zero perturbation, that is with $\alpha=0$,

$\frac{I(A,B:\alpha)}{I(A,B:\alpha=0)}=1-\frac{S^{reg}(A\cup
B;r_{0})}{I(A,B:\alpha=0)}.$ (77)

In figure-13 we have plotted the normalized mutual information as given in
(77) with increasing $\alpha$ for three different values of $L$.

Figure 13: Plot shows variation of normalised mutual inforamtion
$I(A,B:\alpha)$ with shockwave parameter $\alpha$ for different $L$

Form the figure we see that mutual information monotonically decreases with
increasing values of $\alpha$ and eventually becomes zero. This is an expected
result as the increase in shockwave parameter destroys the correlation between
the left and the right side of the extended geometry. Apart from this the most
important point of this analysis is the fact that for smaller length scale,
for example the plot with $L=10$ (blue), the rate of decrease of mutual
information with $\alpha$ is relatively slower than the one with larger value,
that is $L=15$ (green). This behavior reflects the fact that the information
scrambling is faster as we probe into the nonlocal regime with larger values
of $L$.

####  6.2.2 Entanglement velocity

In this section we will calculate entanglement velocity $v_{E}$ which
characterizes the spreading of entanglement pattern in chaotic systems after a
global quench Avila:2018sqf ; Hartman:2013qma ; Liu:2013qca ; Liu:2013iza ;
Mezei:2016zxg . To calculate $v_{E}$ we need to study the variation of
$S^{reg}(A\cup B;r_{0})$ (71) with boundary time $t_{0}$ at which the small
perturbation is applied. We observe a linear growth of $S^{reg}(A\cup
B;r_{0})$ with $t_{0}$ or equivalently with $\log{\alpha}$ for homogeneous
shock as shown in figure-14 Avila:2018sqf .

Figure 14: Plot shows variation of normalised entropy $S^{reg}(A\cup B)$ with
$\text{log}~{}\alpha$

Now, one can approximate the relation between $S^{reg}(A\cup B;r_{0})$ and
$t_{0}$ in the region where it is growing linearly as,

$S^{reg}(A\cup
B;r_{0})=\frac{R^{4}r_{c}^{3}}{\tilde{G}^{10}_{N}}\sqrt{-f(r_{c})A(r_{c})}t_{0},$
(78)

where, $\tilde{G}^{10}_{N}=\frac{G^{10}_{N}}{\Omega_{3}}$ is the scaled
Newton’s constant of gravity in $10$-dimension. This approximation is valid
only if $r_{0}$ is chosen in the vicinity of $r_{c}$ where $\alpha$ takes
large values. Differentiating the above equation with respect to $t_{0}$, the
rate of change of $S^{reg}(A\cup B)$ can be obtained as,

$\frac{dS^{reg}(A\cup
B;r_{0})}{dt_{0}}=\frac{R^{4}r_{c}^{3}}{\tilde{G}^{10}_{N}}\sqrt{-f(r_{c})A(r_{c})}.$
(79)

Next, we calculate the thermal entropy using the Bekenstein-Hawking formula
$s_{th}=\frac{\mathcal{A}(\text{horizon})}{4\tilde{G}^{10}_{N}}$
Bekenstein:1973ur . For little string theory it can expressed in the following
form,

$s_{th}=\frac{\sqrt{A(r_{H})}r_{H}^{3}}{4\tilde{G}^{10}_{N}}.$ (80)

Using the above expression of the thermal entropy, one can rewrite equation
(79) as,

$\frac{dS^{reg}(A\cup
B;r_{0})}{dt_{0}}=s_{th}A_{\Sigma}\frac{r_{c}^{3}}{r_{H}^{3}}\sqrt{\frac{-f(r_{c})A(r_{c})}{A(r_{H})}},$
(81)

where $A_{\Sigma}=4R^{4}$ denotes the area of $\Sigma=\partial(A\cup B)$. Now
comparing (81) with equation (4) we can obtain entanglement velocity as

$v_{E}=\frac{r_{c}^{3}}{r_{H}^{3}}\sqrt{\frac{-f(r_{c})A(r_{c})}{A(r_{H})}}.$
(82)

Figure 15: Plot of butterfly velocity $v_{B}$ and entanglement velocity
$v_{E}$ with lengthscale $\frac{L}{r_{H}}$

We have shown the variation of $v_{b}$ and $v_{E}$ with $L/r_{H}$ in
figure-15. From the plot we can conclude that entanglement velocity $v_{E}$
and butterfly velocity $v_{b}$ both increases with increasing $L$. However
$v_{E}$ always remains well below butterfly velocity $v_{b}$ obeying the bound
as discussed in the introduction section.

##  7 Pole skipping

In this section we will explicitly compute the special locations in the
complex $w-q$ plane known as the pole skipping points where the near horizon
solution of bulk field equation of motion cannot be determined uniquely
resulting in non-uniqueness of the corresponding retarded Green’s function.
Depending upon the kind of bulk field perturbations, this phenomenon can occur
both in the upper half and in the lower half complex plane. For example with
the components of metric perturbation in the sound channel which corresponds
to the energy density retarded Green’s function, the pole skipping happens in
the upper half of the complex plane. Also only the upper half pole skipping
points are related directly to the parameters of quantum chaos. In the
following we will study the pole skipping phenomenon at lowest order in the
near horizon expansion occurring on the upper half plane and compute the
Lyapunov exponent and butterfly velocity. Later we will also extend the
analysis for higher order poles in the lower half plane.

###  7.1 Pole skipping in the upper half plane: Butterfly velocity

In this computation we need to do a near horizon analysis of the bulk field
equation of motion. Hence it is convenient to work in the ingoing coordinate
system. The ten dimensional supergravity solution for the stack of NS$5$
branes in terms of the ingoing Eddington–Finkelstein coordinate $v=t+r_{*}$ is
given as,

$ds^{2}=-f(r)A^{-1/4}dv^{2}+2A^{1/4}dvdr+A^{-1/4}\left(\sum_{i=1}^{5}dx_{i}^{2}\right)+r^{2}A^{3/4}d\Omega_{3}^{2}.$
(83)

Linearized perturbation of the metric and the scalar field has the following
form,

$\begin{split}g_{\mu\nu}&=g_{\mu\nu}^{(0)}+h_{\mu\nu},\\\
\Phi&=\Phi_{0}+\phi,\end{split}$ (84)

where $g_{\mu\nu}^{(0)}$ and $\Phi_{0}$ denotes the background values for the
metric and the scalar field with $h_{\mu\nu}$ and $\phi$ as the corresponding
linear perturbations. The fourier transformation for the field fluctuations
are given as,

$\begin{split}h_{\mu\nu}(t,x_{1},r)&=e^{-iwv+iqx_{1}}h_{\mu\nu}(r),\\\
\phi(t,x_{1},r)&=e^{-iwv+iqx_{1}}\phi(r),\end{split}$ (85)

where we have assume the perturbation to travel along the $x_{1}$ direction.
Moreover, based on the rotational symmetry in the space formed by the
coordinates $x_{2},x_{3},x_{4},x_{5}$, one can categorize the metric
perturbations as the following three modes,

  * •

Sound modes:
$h_{vv},h_{vr},h_{rr},h_{vx_{1}},h_{rx_{1}},h_{x_{1}x_{1}},h_{x_{2}x_{2}}=h_{x_{3}x_{3}}=h_{x_{4}x_{4}}=h_{x_{5}x_{5}}$,

  * •

Shear modes: $h_{vx_{i}},h_{rx_{i}},h_{x_{1}x_{2}},~{}~{}i=2,...5$,

  * •

Tensor mode: $h_{x_{i}x_{j}},~{}~{}i\neq j,~{}~{}i,j=2,...5$.

We will start with the computation of butterfly velocity for which we need to
consider the components of the metric perturbations in the sound channel along
with the scalar field which couples to the metric. The linearized Einstein’s
equation is given as,

$\begin{split}&R^{(1)}_{\mu\nu}-\frac{1}{2}\left(\partial_{\mu}\Phi_{0}\partial_{\nu}\phi+\partial_{\mu}\phi\partial_{\nu}\Phi_{0}\right)+\frac{1}{4}e^{-\Phi_{0}}\biggl{\\{}H_{\mu
bc}H_{\nu
b^{\prime}c^{\prime}}\left(g^{(0)cc^{\prime}}h^{bb^{\prime}}+g^{(0)bb^{\prime}}h^{cc^{\prime}}\right)+\phi
H_{\mu bc}H_{\nu}^{bc}\biggr{\\}}\\\
&-\frac{1}{48}g^{(0)\mu\nu}e^{-\Phi_{0}}\biggl{\\{}H_{abc}H_{a^{\prime}b^{\prime}c^{\prime}}\left(g^{(0)aa^{\prime}}g^{(0)bb^{\prime}}h^{cc^{\prime}}+g^{(0)bb^{\prime}}g^{(0)cc^{\prime}}h^{aa^{\prime}}+g^{(0)cc^{\prime}}g^{(0)cc^{\prime}}h^{bb^{\prime}}\right)-\phi
H_{3}^{2}\biggr{\\}}\\\
&+\frac{1}{48}e^{-\Phi_{0}}H_{3}^{2}h_{\mu\nu}=0.\end{split}$ (86)

Also the linearized scalar field equation of motion is given as,

$\begin{split}&g^{(0)\mu\nu}\left(\partial_{\mu}\partial_{\nu}\phi-\Gamma^{(1)\rho}_{\mu\nu}\partial_{\rho}\Phi_{0}-\Gamma^{(0)\rho}_{\mu\nu}\partial_{\rho}\phi\right)+h^{\mu\nu}\left(\partial_{\mu}\partial_{\nu}\Phi_{0}-\Gamma^{(0)\rho}_{\mu\nu}\partial_{\rho}\Phi_{0}\right)-\frac{1}{12}e^{-\Phi_{0}}\phi
H_{3}^{2}\\\
&-\frac{1}{12}e^{-\Phi_{0}}H_{abc}H_{a^{\prime}b^{\prime}c^{\prime}}\left(g^{(0)aa^{\prime}}g^{(0)bb^{\prime}}h^{cc^{\prime}}+g^{(0)bb^{\prime}}g^{(0)cc^{\prime}}h^{aa^{\prime}}+g^{(0)cc^{\prime}}g^{(0)cc^{\prime}}h^{bb^{\prime}}\right)=0,\end{split}$
(87)

where, $\Gamma^{(0)\rho}_{\mu\nu}$ is the affine connection corresponding to
the background value of the metric while $\Gamma^{(1)\rho}_{\mu\nu}$,
$R^{(1)}_{\mu\nu}$ are the linearized fluctuations to the affine connection
and the ricci tensor respectively defined as,

$\begin{split}\Gamma^{(0)\rho}_{\mu\nu}&=\frac{1}{2}g^{(0)\lambda\rho}\left(\partial_{\mu}g^{(0)}_{\lambda\nu}+\partial_{\nu}g^{(0)}_{\lambda\mu}-\partial_{\lambda}g^{(0)}_{\mu\nu}\right),\\\
\Gamma^{(1)\rho}_{\mu\nu}&=\frac{1}{2}\left[g^{(0)\lambda\rho}\left(\partial_{\mu}h_{\lambda\nu}+\partial_{\nu}h_{\lambda\mu}-\partial_{\lambda}h_{\mu\nu}\right)-h^{\lambda\rho}\left(\partial_{\mu}g^{(0)}_{\lambda\nu}+\partial_{\nu}g^{(0)}_{\lambda\mu}-\partial_{\lambda}g^{(0)}_{\mu\nu}\right)\right],\\\
R^{(1)}_{\mu\nu}&=\partial_{\rho}\Gamma^{(1)\rho}_{\mu\nu}-\partial_{\mu}\Gamma^{(1)\rho}_{\rho\nu}+\Gamma^{(0)\rho}_{\rho\lambda}\Gamma^{(1)\lambda}_{\mu\nu}+\Gamma^{(1)\rho}_{\rho\lambda}\Gamma^{(0)\lambda}_{\mu\nu}-\Gamma^{(0)\rho}_{\mu\lambda}\Gamma^{(1)\lambda}_{\rho\nu}-\Gamma^{(1)\rho}_{\mu\lambda}\Gamma^{(0)\lambda}_{\rho\nu}.\end{split}$
(88)

In this case the non-uniqueness of retarded Green’s function is due to the
fact that the $vv$ component of the linearized Einstein’s equation (86)
becomes trivial at the pole skipping point. Hence in the near horizon limit we
first expand the field perturbations as,

$\begin{split}h_{\mu\nu}&=\eta^{(0)}_{\mu\nu}+\eta^{(1)}_{\mu\nu}\left(r-r_{H}\right)+\mathcal{O}(r-r_{H})^{2},\\\
\phi&=\phi^{(0)}+\phi^{(1)}\left(r-r_{H}\right)+\mathcal{O}(r-r_{H})^{2}.\end{split}$
(89)

Substituting the above expansion in the Einstein’s equation and expanding the
same near the horizon we obtain,

$\begin{split}&h^{(0)}_{vv}\biggl{\\{}L^{2}\left(Q^{2}-2iW\right)+r_{H}^{2}\left(Q^{2}-3iW\right)\biggr{\\}}\\\
&+\left(W-i\right)\left(L^{2}+r_{H}^{2}\right)\biggl{\\{}2Qh_{vx_{1}}^{(0)}+W\left(h_{x_{1}x_{1}}^{(0)}+4h_{x_{2}x_{2}}^{(0)}\right)\biggr{\\}}=0,\end{split}$
(90)

where we define $W=w/2\pi T_{H}$ and $Q=q/2\pi T_{H}$, The above equation is
satisfied for the particular values of $W$ and $Q$ given as,

$W=i,~{}~{}~{}~{}Q=i\sqrt{\frac{2L^{2}+3r_{H}^{2}}{L^{2}+r_{H}^{2}}}.$ (91)

The Lyapunov exponent and the butterfly velocity is given as,

$\lambda_{L}=2\pi
T_{H},~{}~{}~{}~{}v_{B}^{2}=\frac{L^{2}+r_{H}^{2}}{2L^{2}+3r_{H}^{2}}.$ (92)

###  7.2 Pole-skipping at higher order complex frequencies

In the last subsection we discussed the pole skipping point at the lowest
order in the upper half plane and obtained the expression of butterfly
velocity. In this section we will extend the analysis of pole skipping
phenomenon for higher ordered complex frequencies $W_{n}$, with $n=1,2,3,...$.

####  7.2.1 Scalar field fluctuations

Considering linearized fluctuation of the scalar field as given in (85), the
corresponding equation of motion can be obtained as,

$\phi^{\prime\prime}(r)+\mathcal{S}(r)\phi^{\prime}(r)+\mathcal{R}(r)\phi(r)=0,$
(93)

where $\mathcal{S}(r)$ and $\mathcal{R}(r)$ are given as,

$\begin{split}\mathcal{S}(r)&=\frac{1}{r\left(r^{2}-r_{H}^{2}\right)}\biggl{(}-3r^{2}+r_{H}^{2}+2iWr^{2}\sqrt{\frac{L^{2}+r^{2}}{L^{2}+r_{H}^{2}}}\biggr{)},\\\
\mathcal{R}(r)&=\frac{1}{r^{2}-r_{H}^{2}}\Biggl{(}2L^{2}\frac{L^{2}+r_{H}^{2}}{\left(L^{2}+r^{2}\right)^{2}}+Q^{2}\left(\frac{L^{2}+r^{2}}{L^{2}+r_{H}^{2}}\right)+iW\frac{\left(2L^{2}+3r^{2}\right)}{\sqrt{L^{2}+r^{2}}\sqrt{L^{2}+r_{H}^{2}}}\Biggr{)}.\end{split}$
(94)

To determine the pole skipping points at different orders one needs to
consider the near horizon expansion for the solution $\phi(r)$ of the above
equation as,

$\phi(r)=\phi_{0}+\left(r-r_{H}\right)\phi_{1}+\left(r-r_{H}\right)^{2}\phi_{2}+\left(r-r_{H}\right)^{3}\phi_{3}+...$
(95)

Substituting the above into equation (93), and equating the coefficients of
$\left(r-r_{H}\right)^{n}$ to zero for $n=0,1,2,3,..$ one gets the following
system of linear equations,

$\begin{split}c_{00}\phi_{0}+c_{01}\phi_{1}&=0,\\\
c_{10}\phi_{0}+c_{11}\phi_{1}+c_{12}\phi_{2}&=0,\\\
c_{20}\phi_{0}+c_{21}\phi_{1}+c_{22}z_{2}+c_{23}\phi_{3}&=0,\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\vdots\end{split}$ (96)

where the coefficients $c_{ij}$’s can be arranged in a square matrix of a
given order as,

$C=\begin{pmatrix}c_{00}&&c_{01}&&0&&0&&\cdots\\\
c_{10}&&c_{11}&&c_{12}&&0&&\cdots\\\ c_{20}&&c_{21}&&c_{22}&&c_{23}&&\cdots\\\
\vdots&&\vdots&&\vdots&&\vdots&&\cdots\\\ ,\end{pmatrix},$ (97)

such that the pole skipping points are obtained by solving simultaneously the
following two equations,

$c_{n-1~{}n}=0,~{}~{}~{}~{}~{}~{}\det{C}=0.$ (98)

In this case with the large $L$ limit, the first few pole skipping points are
given as,

$\begin{split}W_{1}&=-i~{}~{}~{}~{}~{}~{}Q_{1}^{2}=-2+2\frac{r_{H}^{2}}{L^{2}}+\mathcal{O}\biggl{(}\frac{r_{H}^{4}}{L^{4}}\biggr{)}\\\
W_{2}&=-2i~{}~{}~{}~{}~{}Q_{2}^{2}=\left\\{-4-i2\sqrt{6}\frac{r_{H}}{L},-4+i2\sqrt{6}\frac{r_{H}}{L}\right\\}+\mathcal{O}\biggl{(}\frac{r_{H}^{2}}{L^{2}}\biggr{)}\\\
W_{3}&=-3i~{}~{}~{}~{}~{}Q_{3}^{2}=\left\\{-6-\left(12+i6\sqrt{3}\right)\frac{r_{H}^{2}}{L^{2}},-6-\left(12-i6\sqrt{3}\right)\frac{r_{H}^{2}}{L^{2}},-10+22\frac{r_{H}^{2}}{L^{2}}\right\\}+\mathcal{O}\biggl{(}\frac{r_{H}^{4}}{L^{4}}\biggr{)}\end{split}$
(99)

####  7.2.2 Shear mode of metric fluctuations

The gauge invariant variable for the shear modes of metric perturbations is
given as,

$Z_{2}(u)=QH_{vx_{2}}(u)+WH_{x_{1}x_{2}}(u),$ (100)

where we have considered the redial gauge such that $h_{\mu u}=0$ for any
$\mu$. In the above equation the field fluctuations are defined as
$H_{x_{1}x_{2}}=h_{x_{1}x_{2}}/g_{x_{1}x_{1}}$ and
$H_{vx_{2}}=h_{vx_{2}}/g_{x_{2}x_{2}}$. The Einstein’s equation of motion for
the shear modes of field perturbation is obtained in terms of the gauge
invariant variable $Z_{2}$ as,

$Z_{2}^{\prime\prime}(u)-\frac{W^{2}}{\left(1-u\right)\left(W^{2}-(1-u)Q^{2}\right)}Z_{2}^{\prime}(u)+\frac{\left(r_{H}^{2}+uL^{2}\right)\left(W^{2}-(1-u)Q^{2}\right)}{4u^{3}\left(1-u\right)^{2}\left(r_{H}^{2}+L^{2}\right)}Z_{2}(u).$
(101)

Using the method as discussed previously we obtain the pole skipping points
for the first few orders in the limit $\left(r_{H}/L\right)\rightarrow 0$ as,

$\begin{split}W_{1}&=-i~{}~{}~{}~{}~{}~{}Q_{1}^{2}=2-2\frac{r_{H}^{2}}{L^{2}}+\mathcal{O}\biggl{(}\frac{r_{H}^{4}}{L^{4}}\biggr{)}\\\
W_{2}&=-2i~{}~{}~{}~{}~{}Q_{2}^{2}=\left\\{4-\frac{r_{H}^{2}}{L^{2}},-4-5\frac{r_{H}^{2}}{L^{2}}\right\\}+\mathcal{O}\biggl{(}\frac{r_{H}^{4}}{L^{4}}\biggr{)}\\\
W_{3}&=-3i~{}~{}~{}~{}~{}Q_{3}^{2}=\left\\{6+3\frac{r_{H}^{2}}{L^{2}},-6-3\frac{r_{H}^{2}}{L^{2}},-10-5\frac{r_{H}^{2}}{L^{2}}\right\\}+\mathcal{O}\biggl{(}\frac{r_{H}^{4}}{L^{4}}\biggr{)}\end{split}$
(102)

##  8 Conclusion

In this paper we have used different well known holographic techniques to
study quantum information scrambling and quantum chaos in a nonlocal theory
known as the little string theory. The fundamental degrees of freedom of this
theory are strings with finite length $l_{s}$. The nonlocal effects appear
only at length scale lower than the inverse Hagedorn temperature $\beta_{h}$
which is proportional to the square root of the string length. The main
conclusion of this analysis is that the nonlocality accelerates the scrambling
of quantum information. In other words the theory becomes more and more
chaotic as one probe into length scale less than the inverse Hagedorn
temperature. This can be explicitly realized from the variation of butterfly
velocity $v_{b}$ with $\beta_{h}=2\pi L$ as already discussed. Butterfly
velocity acts as an effective bound at low energy on the spread of quantum
information. An increasing behavior of $v_{b}$ with the nonlocal scale
basically widen the effective butterfly cone resulting in faster growth of
local operators. However the butterfly velocity is observed to saturate to a
value less than the speed of light due to the Lorentz invariance.

For the holographic computation we work in the usual thermofield double set up
whose dual description is given in terms of the extended two sided black hole
geometry. A small perturbation on the boundary correspond to an energy quanta
in the bulk which gets exponentially boosted with time and produces a
shockwave. This highly energetic quanta can disrupt the entanglement structure
between the left and the right side of the geometry which is actually the
butterfly effect for this holographic set up. This disruption of entangled
structure is captured by the decrease of holographic two sided mutual
information with increasing shockwave parameter. In this case we also observe
that the rate of disruption of mutual information is faster as the nonlocal
scale is increased.

Regarding the chaotic behavior, we considered the recently observed phenomenon
called pole skipping that provides an efficient and easier way of computing
Lyapunov exponent and also butterfly velocity which characterizes quantum
chaos in many body system. Analysing linearized Einstein’s equation involving
different bulk field perturbations near the black hole horizon singles out
special quasinormal frequency and momentum in the complex plane which can be
connected directly to the chaos parameters due to the phenomenon of pole
skipping. In this paper we have computed the pole skipping points in the upper
half plane to obtain the butterfly velocity which exactly matches with the
result as obtained through the gravitational shockwave analysis in section-5.

In future, it will be interesting to repeat the same exercise for the gravity
dual of $\mathcal{N}=4$ SYM theory which is deformed by the presence of
external heavy quarks Chakrabortty:2011sp ; Chakrabortty:2020ptb ;
Chakrabortty:2022kvq . Further in Boruch:2020wbe , the author has studied the
dynamics of entanglement wedge cross section in shockwave geometry for three
dimensional AdS geometry. A generalization of this result in higher
dimensional black hole background might be an useful future direction.

## Acknowledgements

We would like thank Chandrasekhar Bhamidipati for helpful discussion and
comments on the final draft of this paper. KS would like to thank Shankhadeep
Chakrabortty, Hironori Hoshino and especially Sanjay Pant for useful regular
discussions. KS is supported by the institute post doctoral fellowship from
IIT Bhubaneswar.

## References

  * (1) B. Swingle, “Quantum Information Scrambling: Boulder Lectures,” delivered at the Quantum Information Boulder Summer School in Boulder, Colorado, July 2018.
  * (2) E. H. Lieb and D. W. Robinson, “The finite group velocity of quantum spin systems,” Commun. Math. Phys. 28 (1972), 251-257 doi:10.1007/BF01645779
  * (3) S. H. Shenker and D. Stanford, “Black holes and the butterfly effect,” JHEP 03 (2014), 067 doi:10.1007/JHEP03(2014)067 [arXiv:1306.0622 [hep-th]].
  * (4) D. A. Roberts and B. Swingle, “Lieb-Robinson Bound and the Butterfly Effect in Quantum Field Theories,” Phys. Rev. Lett. 117 (2016) no.9, 091602 doi:10.1103/PhysRevLett.117.091602 [arXiv:1603.09298 [hep-th]].
  * (5) A. Larkin and Y. Ovchinnikov, JETP 28,6, 1200 (1969).
  * (6) D. A. Roberts, D. Stanford and L. Susskind, “Localized shocks,” JHEP 03 (2015), 051 doi:10.1007/JHEP03(2015)051 [arXiv:1409.8180 [hep-th]].
  * (7) S. H. Shenker and D. Stanford, “Stringy effects in scrambling,” JHEP 05 (2015), 132 doi:10.1007/JHEP05(2015)132 [arXiv:1412.6087 [hep-th]].
  * (8) A. Kitaev and S. J. Suh, “The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual,” JHEP 05 (2018), 183 doi:10.1007/JHEP05(2018)183 [arXiv:1711.08467 [hep-th]].
  * (9) J. Polchinski, “Chaos in the black hole S-matrix,” [arXiv:1505.08108 [hep-th]].
  * (10) J. Polchinski and V. Rosenhaus, “The Spectrum in the Sachdev-Ye-Kitaev Model,” JHEP 04 (2016), 001 doi:10.1007/JHEP04(2016)001 [arXiv:1601.06768 [hep-th]].
  * (11) J. Maldacena and D. Stanford, “Remarks on the Sachdev-Ye-Kitaev model,” Phys. Rev. D 94 (2016) no.10, 106002 doi:10.1103/PhysRevD.94.106002 [arXiv:1604.07818 [hep-th]].
  * (12) J. Maldacena, D. Stanford and Z. Yang, “Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space,” PTEP 2016 (2016) no.12, 12C104 doi:10.1093/ptep/ptw124 [arXiv:1606.01857 [hep-th]].
  * (13) K. Jensen, “Chaos in AdS2 Holography,” Phys. Rev. Lett. 117 (2016) no.11, 111601 doi:10.1103/PhysRevLett.117.111601 [arXiv:1605.06098 [hep-th]].
  * (14) Y. Gu, X. L. Qi and D. Stanford, “Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models,” JHEP 05 (2017), 125 doi:10.1007/JHEP05(2017)125 [arXiv:1609.07832 [hep-th]].
  * (15) A. A. Patel and S. Sachdev, “Quantum chaos on a critical Fermi surface,” Proc. Nat. Acad. Sci. 114 (2017), 1844-1849 doi:10.1073/pnas.1618185114 [arXiv:1611.00003 [cond-mat.str-el]].
  * (16) M. Alishahiha, A. Davody, A. Naseh and S. F. Taghavi, “On Butterfly effect in Higher Derivative Gravities,” JHEP 11 (2016), 032 doi:10.1007/JHEP11(2016)032 [arXiv:1610.02890 [hep-th]].
  * (17) J. Maldacena, S. H. Shenker and D. Stanford, “A bound on chaos,” JHEP 08 (2016), 106 doi:10.1007/JHEP08(2016)106 [arXiv:1503.01409 [hep-th]].
  * (18) S. Grozdanov, K. Schalm and V. Scopelliti, “Black hole scrambling from hydrodynamics,” Phys. Rev. Lett. 120 (2018) no.23, 231601 doi:10.1103/PhysRevLett.120.231601 [arXiv:1710.00921 [hep-th]].
  * (19) M. Blake, R. A. Davison, S. Grozdanov and H. Liu, “Many-body chaos and energy dynamics in holography,” JHEP 10 (2018), 035 doi:10.1007/JHEP10(2018)035 [arXiv:1809.01169 [hep-th]].
  * (20) M. Blake, H. Lee and H. Liu, “A quantum hydrodynamical description for scrambling and many-body chaos,” JHEP 10 (2018), 127 doi:10.1007/JHEP10(2018)127 [arXiv:1801.00010 [hep-th]].
  * (21) S. Grozdanov, “On the connection between hydrodynamics and quantum chaos in holographic theories with stringy corrections,” JHEP 01 (2019), 048 doi:10.1007/JHEP01(2019)048 [arXiv:1811.09641 [hep-th]].
  * (22) Y. Sekino and L. Susskind, “Fast Scramblers,” JHEP 10 (2008), 065 doi:10.1088/1126-6708/2008/10/065 [arXiv:0808.2096 [hep-th]].
  * (23) N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, “Towards the Fast Scrambling Conjecture,” JHEP 04 (2013), 022 doi:10.1007/JHEP04(2013)022 [arXiv:1111.6580 [hep-th]].
  * (24) P. Hayden and J. Preskill, “Black holes as mirrors: Quantum information in random subsystems,” JHEP 09 (2007), 120 doi:10.1088/1126-6708/2007/09/120 [arXiv:0708.4025 [hep-th]].
  * (25) X. L. Qi and Z. Yang, “Butterfly velocity and bulk causal structure,” [arXiv:1705.01728 [hep-th]].
  * (26) M. Mezei and D. Stanford, “On entanglement spreading in chaotic systems,” JHEP 05 (2017), 065 doi:10.1007/JHEP05(2017)065 [arXiv:1608.05101 [hep-th]].
  * (27) W. Fischler, V. Jahnke and J. F. Pedraza, “Chaos and entanglement spreading in a non-commutative gauge theory,” JHEP 11, 072 (2018) [erratum: JHEP 02, 149 (2021)] doi:10.1007/JHEP11(2018)072 [arXiv:1808.10050 [hep-th]].
  * (28) K. Sfetsos, “On gravitational shock waves in curved space-times,” Nucl. Phys. B 436 (1995), 721-745 doi:10.1016/0550-3213(94)00573-W [arXiv:hep-th/9408169 [hep-th]].
  * (29) V. Jahnke, “Delocalizing entanglement of anisotropic black branes,” JHEP 01, 102 (2018) doi:10.1007/JHEP01(2018)102 [arXiv:1708.07243 [hep-th]].
  * (30) P. Calabrese and J. L. Cardy, “Evolution of entanglement entropy in one-dimensional systems,” J. Stat. Mech. 0504 (2005), P04010 doi:10.1088/1742-5468/2005/04/P04010 [arXiv:cond-mat/0503393 [cond-mat]].
  * (31) P. Calabrese and J. Cardy, “Quantum Quenches in Extended Systems,” J. Stat. Mech. 0706 (2007), P06008 doi:10.1088/1742-5468/2007/06/P06008 [arXiv:0704.1880 [cond-mat.stat-mech]].
  * (32) J. S. Cotler, M. P. Hertzberg, M. Mezei and M. T. Mueller, “Entanglement Growth after a Global Quench in Free Scalar Field Theory,” JHEP 11 (2016), 166 doi:10.1007/JHEP11(2016)166 [arXiv:1609.00872 [hep-th]].
  * (33) H. Liu and S. J. Suh, “Entanglement Tsunami: Universal Scaling in Holographic Thermalization,” Phys. Rev. Lett. 112 (2014), 011601 doi:10.1103/PhysRevLett.112.011601 [arXiv:1305.7244 [hep-th]].
  * (34) H. Liu and S. J. Suh, “Entanglement growth during thermalization in holographic systems,” Phys. Rev. D 89 (2014) no.6, 066012 doi:10.1103/PhysRevD.89.066012 [arXiv:1311.1200 [hep-th]].
  * (35) M. R. Mohammadi Mozaffar and A. Mollabashi, “Entanglement Evolution in Lifshitz-type Scalar Theories,” JHEP 01 (2019), 137 doi:10.1007/JHEP01(2019)137 [arXiv:1811.11470 [hep-th]].
  * (36) T. Hartman and J. Maldacena, “Time Evolution of Entanglement Entropy from Black Hole Interiors,” JHEP 05 (2013), 014 doi:10.1007/JHEP05(2013)014 [arXiv:1303.1080 [hep-th]].
  * (37) H. Casini, H. Liu and M. Mezei, “Spread of entanglement and causality,” JHEP 07 (2016), 077 doi:10.1007/JHEP07(2016)077 [arXiv:1509.05044 [hep-th]].
  * (38) M. Mezei, “On entanglement spreading from holography,” JHEP 05 (2017), 064 doi:10.1007/JHEP05(2017)064 [arXiv:1612.00082 [hep-th]].
  * (39) J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998), 231-252 doi:10.1023/A:1026654312961 [arXiv:hep-th/9711200 [hep-th]].
  * (40) J. M. Maldacena, “Eternal black holes in anti-de Sitter,” JHEP 04 (2003), 021 doi:10.1088/1126-6708/2003/04/021 [arXiv:hep-th/0106112 [hep-th]].
  * (41) M. M. Wolf, F. Verstraete, M. B. Hastings and J. I. Cirac, “Area Laws in Quantum Systems: Mutual Information and Correlations,” Phys. Rev. Lett. 100 (2008) no.7, 070502 doi:10.1103/PhysRevLett.100.070502 [arXiv:0704.3906 [quant-ph]].
  * (42) S. B. Giddings, “Black hole information, unitarity, and nonlocality,” Phys. Rev. D 74 (2006), 106005 doi:10.1103/PhysRevD.74.106005 [arXiv:hep-th/0605196 [hep-th]].
  * (43) D. Kutasov, “Introduction to little string theory,” ICTP Lect. Notes Ser. 7 (2002) 165
  * (44) S. Chakraborty, G. Katoch and S. R. Roy, “Holographic complexity of LST and single trace $T\overline{T}$,” JHEP 03 (2021), 275 doi:10.1007/JHEP03(2021)275 [arXiv:2012.11644 [hep-th]].
  * (45) S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96, 181602 (2006) doi:10.1103/PhysRevLett.96.181602 [arXiv:hep-th/0603001 [hep-th]].
  * (46) J. L. Karczmarek and C. Rabideau, “Holographic entanglement entropy in nonlocal theories,” JHEP 10 (2013), 078 doi:10.1007/JHEP10(2013)078 [arXiv:1307.3517 [hep-th]].
  * (47) J. L. F. Barbon and C. A. Fuertes, “Holographic entanglement entropy probes (non)locality,” JHEP 04, 096 (2008) doi:10.1088/1126-6708/2008/04/096 [arXiv:0803.1928 [hep-th]].
  * (48) M. R. Mohammadi Mozaffar and A. Mollabashi, “Entanglement in Lifshitz-type Quantum Field Theories,” JHEP 07 (2017), 120 doi:10.1007/JHEP07(2017)120 [arXiv:1705.00483 [hep-th]].
  * (49) W. Fischler, A. Kundu and S. Kundu, “Holographic Entanglement in a Noncommutative Gauge Theory,” JHEP 01, 137 (2014) doi:10.1007/JHEP01(2014)137 [arXiv:1307.2932 [hep-th]].
  * (50) N. Lashkari, “Equilibration of Small and Large Subsystems in Field Theories and Matrix Models,” Commun. Math. Phys. 333 (2015) no.3, 1199-1224 doi:10.1007/s00220-014-2263-y [arXiv:1304.6416 [hep-th]].
  * (51) M. Edalati, W. Fischler, J. F. Pedraza and W. Tangarife Garcia, “Fast Scramblers and Non-commutative Gauge Theories,” JHEP 07 (2012), 043 doi:10.1007/JHEP07(2012)043 [arXiv:1204.5748 [hep-th]].
  * (52) M. Blake, R. A. Davison and D. Vegh, “Horizon constraints on holographic Green’s functions,” JHEP 01, 077 (2020) doi:10.1007/JHEP01(2020)077 [arXiv:1904.12883 [hep-th]].
  * (53) S. Grozdanov, P. K. Kovtun, A. O. Starinets and P. Tadić, “The complex life of hydrodynamic modes,” JHEP 11, 097 (2019) doi:10.1007/JHEP11(2019)097 [arXiv:1904.12862 [hep-th]].
  * (54) M. Natsuume and T. Okamura, “Nonuniqueness of Green’s functions at special points,” JHEP 12, 139 (2019) doi:10.1007/JHEP12(2019)139 [arXiv:1905.12015 [hep-th]].
  * (55) N. Ceplak, K. Ramdial and D. Vegh, “Fermionic pole-skipping in holography,” JHEP 07, 203 (2020) doi:10.1007/JHEP07(2020)203 [arXiv:1910.02975 [hep-th]].
  * (56) M. Natsuume and T. Okamura, “Pole-skipping with finite-coupling corrections,” Phys. Rev. D 100, no.12, 126012 (2019) doi:10.1103/PhysRevD.100.126012 [arXiv:1909.09168 [hep-th]].
  * (57) X. Wu, “Higher curvature corrections to pole-skipping,” JHEP 12, 140 (2019) doi:10.1007/JHEP12(2019)140 [arXiv:1909.10223 [hep-th]].
  * (58) Y. Ahn, V. Jahnke, H. S. Jeong, K. Y. Kim, K. S. Lee and M. Nishida, “Pole-skipping of scalar and vector fields in hyperbolic space: conformal blocks and holography,” JHEP 09, 111 (2020) doi:10.1007/JHEP09(2020)111 [arXiv:2006.00974 [hep-th]].
  * (59) S. Grozdanov, “Bounds on transport from univalence and pole-skipping,” Phys. Rev. Lett. 126, no.5, 051601 (2021) doi:10.1103/PhysRevLett.126.051601 [arXiv:2008.00888 [hep-th]].
  * (60) Y. Ahn, V. Jahnke, H. S. Jeong, K. S. Lee, M. Nishida and K. Y. Kim, “Classifying pole-skipping points,” JHEP 03, 175 (2021) doi:10.1007/JHEP03(2021)175 [arXiv:2010.16166 [hep-th]].
  * (61) K. Y. Kim, K. S. Lee and M. Nishida, “Holographic scalar and vector exchange in OTOCs and pole-skipping phenomena,” JHEP 04, 092 (2021) [erratum: JHEP 04, 229 (2021)] doi:10.1007/JHEP04(2021)092 [arXiv:2011.13716 [hep-th]].
  * (62) M. Natsuume and T. Okamura, “Pole-skipping and zero temperature,” Phys. Rev. D 103, no.6, 066017 (2021) doi:10.1103/PhysRevD.103.066017 [arXiv:2011.10093 [hep-th]].
  * (63) N. Abbasi and M. Kaminski, “Constraints on quasinormal modes and bounds for critical points from pole-skipping,” JHEP 03, 265 (2021) doi:10.1007/JHEP03(2021)265 [arXiv:2012.15820 [hep-th]].
  * (64) N. Ceplak and D. Vegh, “Pole-skipping and Rarita-Schwinger fields,” Phys. Rev. D 103, no.10, 106009 (2021) doi:10.1103/PhysRevD.103.106009 [arXiv:2101.01490 [hep-th]].
  * (65) K. Y. Kim, K. S. Lee and M. Nishida, “Regge conformal blocks from the Rindler-AdS black hole and the pole-skipping phenomena,” JHEP 11, 020 (2021) doi:10.1007/JHEP11(2021)020 [arXiv:2105.07778 [hep-th]].
  * (66) C. Choi, M. Mezei and G. Sárosi, “Pole skipping away from maximal chaos,” doi:10.1007/JHEP02(2021)207 [arXiv:2010.08558 [hep-th]].
  * (67) F. M. Haehl and M. Rozali, “Effective Field Theory for Chaotic CFTs,” JHEP 10, 118 (2018) doi:10.1007/JHEP10(2018)118 [arXiv:1808.02898 [hep-th]].
  * (68) K. Jensen, “Scrambling in nearly thermalized states at large central charge,” [arXiv:1906.05852 [hep-th]].
  * (69) S. Das, B. Ezhuthachan and A. Kundu, “Real time dynamics from low point correlators in 2d BCFT,” JHEP 12, 141 (2019) doi:10.1007/JHEP12(2019)141 [arXiv:1907.08763 [hep-th]].
  * (70) F. M. Haehl, W. Reeves and M. Rozali, “Reparametrization modes, shadow operators, and quantum chaos in higher-dimensional CFTs,” JHEP 11, 102 (2019) doi:10.1007/JHEP11(2019)102 [arXiv:1909.05847 [hep-th]].
  * (71) D. M. Ramirez, “Chaos and pole skipping in CFT2,” JHEP 12, 006 (2021) doi:10.1007/JHEP12(2021)006 [arXiv:2009.00500 [hep-th]].
  * (72) M. Natsuume and T. Okamura, “Holographic chaos, pole-skipping, and regularity,” PTEP 2020, no.1, 013B07 (2020) doi:10.1093/ptep/ptz155 [arXiv:1905.12014 [hep-th]].
  * (73) W. Li, S. Lin and J. Mei, “Thermal diffusion and quantum chaos in neutral magnetized plasma,” Phys. Rev. D 100, no.4, 046012 (2019) doi:10.1103/PhysRevD.100.046012 [arXiv:1905.07684 [hep-th]].
  * (74) N. Abbasi and J. Tabatabaei, “Quantum chaos, pole-skipping and hydrodynamics in a holographic system with chiral anomaly,” JHEP 03, 050 (2020) doi:10.1007/JHEP03(2020)050 [arXiv:1910.13696 [hep-th]].
  * (75) N. Abbasi and S. Tahery, “Complexified quasinormal modes and the pole-skipping in a holographic system at finite chemical potential,” JHEP 10, 076 (2020) doi:10.1007/JHEP10(2020)076 [arXiv:2007.10024 [hep-th]].
  * (76) A. Jansen and C. Pantelidou, “Quasinormal modes in charged fluids at complex momentum,” JHEP 10, 121 (2020) doi:10.1007/JHEP10(2020)121 [arXiv:2007.14418 [hep-th]].
  * (77) K. Sil, “Pole skipping and chaos in anisotropic plasma: a holographic study,” JHEP 03, 232 (2021) doi:10.1007/JHEP03(2021)232 [arXiv:2012.07710 [hep-th]].
  * (78) H. Yuan and X. H. Ge, “Pole-skipping and hydrodynamic analysis in Lifshitz, AdS2 and Rindler geometries,” JHEP 06, 165 (2021) doi:10.1007/JHEP06(2021)165 [arXiv:2012.15396 [hep-th]].
  * (79) H. S. Jeong, K. Y. Kim and Y. W. Sun, “Bound of diffusion constants from pole-skipping points: spontaneous symmetry breaking and magnetic field,” JHEP 07, 105 (2021) doi:10.1007/JHEP07(2021)105 [arXiv:2104.13084 [hep-th]].
  * (80) Y. Ahn, V. Jahnke, H. S. Jeong and K. Y. Kim, “Scrambling in Hyperbolic Black Holes: shock waves and pole-skipping,” JHEP 10, 257 (2019) doi:10.1007/JHEP10(2019)257 [arXiv:1907.08030 [hep-th]].
  * (81) M. Blake and R. A. Davison, “Chaos and pole-skipping in rotating black holes,” JHEP 01, 013 (2022) doi:10.1007/JHEP01(2022)013 [arXiv:2111.11093 [hep-th]].
  * (82) Y. Liu and A. Raju, “Quantum Chaos in Topologically Massive Gravity,” JHEP 12 (2020), 027 doi:10.1007/JHEP12(2020)027 [arXiv:2005.08508 [hep-th]].
  * (83) D. Kutasov “Introduction to little string theory”, ICTP Lect. Notes Ser. 7 165.
  * (84) G. T. Horowitz and A. Strominger, “Black strings and P-branes,” Nucl. Phys. B 360, 197-209 (1991) doi:10.1016/0550-3213(91)90440-9
  * (85) A. Parnachev and A. Starinets, “The Silence of the little strings,” JHEP 10, 027 (2005) doi:10.1088/1126-6708/2005/10/027 [arXiv:hep-th/0506144 [hep-th]].
  * (86) J. M. Maldacena and A. Strominger, “Semiclassical decay of near extremal five-branes,” JHEP 12, 008 (1997) doi:10.1088/1126-6708/1997/12/008 [arXiv:hep-th/9710014 [hep-th]].
  * (87) F. Chen, K. Dasgupta, J. M. Lapan, J. Seo and R. Tatar, “Gauge/Gravity Duality in Heterotic String Theory,” Phys. Rev. D 88 (2013), 066003 doi:10.1103/PhysRevD.88.066003 [arXiv:1303.4750 [hep-th]].
  * (88) J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7 (1973), 2333-2346 doi:10.1103/PhysRevD.7.2333
  * (89) S. Ryu and T. Takayanagi, “Aspects of Holographic Entanglement Entropy,” JHEP 08 (2006), 045 doi:10.1088/1126-6708/2006/08/045 [arXiv:hep-th/0605073 [hep-th]].
  * (90) V. E. Hubeny, M. Rangamani and T. Takayanagi, “A Covariant holographic entanglement entropy proposal,” JHEP 07 (2007), 062 doi:10.1088/1126-6708/2007/07/062 [arXiv:0705.0016 [hep-th]].
  * (91) I. R. Klebanov, D. Kutasov and A. Murugan, “Entanglement as a probe of confinement,” Nucl. Phys. B 796 (2008), 274-293 doi:10.1016/j.nuclphysb.2007.12.017 [arXiv:0709.2140 [hep-th]].
  * (92) N. Seiberg, ‘New theories in six-dimensions and matrix description of M theory on T**5 and T**5 / Z(2),” Phys. Lett. B 408, 98-104 (1997) doi:10.1016/S0370-2693(97)00805-8 [arXiv:hep-th/9705221 [hep-th]].
  * (93) M. Rangamani, “Little string thermodynamics,” JHEP 06, 042 (2001) doi:10.1088/1126-6708/2001/06/042 [arXiv:hep-th/0104125 [hep-th]].
  * (94) B. M. Terhal, M. Horodecki, D. W. Leung and D. P. DiVincenzo, “The entanglement of purification”, J. Math. Phys. 43 (2002) 4286 [quant-ph/0202044].
  * (95) T. Takayanagi and K. Umemoto, “Entanglement of purification through holographic duality,” Nature Phys. 14 (2018) no.6, 573-577 doi:10.1038/s41567-018-0075-2 [arXiv:1708.09393 [hep-th]].
  * (96) P. Nguyen, T. Devakul, M. G. Halbasch, M. P. Zaletel and B. Swingle, “Entanglement of purification: from spin chains to holography,” JHEP 01 (2018), 098 doi:10.1007/JHEP01(2018)098 [arXiv:1709.07424 [hep-th]].
  * (97) D. Ávila, V. Jahnke and L. Patiño, “Chaos, Diffusivity, and Spreading of Entanglement in Magnetic Branes, and the Strengthening of the Internal Interaction,” JHEP 09, 131 (2018) doi:10.1007/JHEP09(2018)131 [arXiv:1805.05351 [hep-th]].
  * (98) T. Hartman and J. Maldacena, “Time Evolution of Entanglement Entropy from Black Hole Interiors,” JHEP 05, 014 (2013) doi:10.1007/JHEP05(2013)014 [arXiv:1303.1080 [hep-th]].
  * (99) H. Liu and S. J. Suh, “Entanglement growth during thermalization in holographic systems,” Phys. Rev. D 89, no.6, 066012 (2014) doi:10.1103/PhysRevD.89.066012 [arXiv:1311.1200 [hep-th]].
  * (100) H. Liu and S. J. Suh, “Entanglement Tsunami: Universal Scaling in Holographic Thermalization,” Phys. Rev. Lett. 112, 011601 (2014) doi:10.1103/PhysRevLett.112.011601 [arXiv:1305.7244 [hep-th]].
  * (101) M. Mezei, “On entanglement spreading from holography,” JHEP 05, 064 (2017) doi:10.1007/JHEP05(2017)064 [arXiv:1612.00082 [hep-th]].
  * (102) S. Chakrabortty, “Dissipative force on an external quark in heavy quark cloud,” Phys. Lett. B 705 (2011), 244-250 doi:10.1016/j.physletb.2011.09.112 [arXiv:1108.0165 [hep-th]].
  * (103) S. Chakrabortty, S. Pant and K. Sil, “Effect of back reaction on entanglement and subregion volume complexity in strongly coupled plasma,” JHEP 06 (2020), 061 doi:10.1007/JHEP06(2020)061 [arXiv:2004.06991 [hep-th]].
  * (104) S. Chakrabortty, H. Hoshino, S. Pant and K. Sil, “A holographic study of the characteristics of chaos and correlation in the presence of backreaction,” [arXiv:2206.12555 [hep-th]].
  * (105) J. Boruch, “Entanglement wedge cross-section in shock wave geometries,” JHEP 07 (2020), 208 doi:10.1007/JHEP07(2020)208 [arXiv:2006.10625 [hep-th]].