aainstitutetext: Theory Group, Department of Physics, The University of Texas
at Austin, Austin TX, USAbbinstitutetext: Department of Physics and Astronomy,
University College London, London WC1E 6BT, UKccinstitutetext: Martin Fisher
School of Physics, Brandeis University, Waltham MA 02453, USA

# Speeding up the spread of quantum information in chaotic systems

Stefan Eccles a Willy Fischler a Tyler Guglielmo b,c Juan F. Pedraza a and
Sarah Racz<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We explore the effect of introducing mild nonlocality into otherwise local,
chaotic quantum systems, on the rate of information spreading and associated
rates of entanglement generation and operator growth. We consider various
forms of nonlocality, both in 1-dimensional spin chain models and in
holographic gauge theories, comparing the phenomenology of each. Generically,
increasing the level of nonlocality increases the rate of information
spreading, but in lattice models we find instances where these rates are
slightly suppressed.

##  1 Introduction

The dynamics of entanglement generation and the growth of local Heisenberg
operators in chaotic quantum systems have been subjects of intense
investigation in recent years, bringing together the interests of condensed
matter, quantum gravity, and quantum information theory communities
Shenker:2013pqa ; Roberts:2014isa ; Shenker:2014cwa ; Sachdev:2015efa ;
Maldacena:2015waa ; Hosur:2015ylk ; Aleiner:2016eni ; Kaufman_2016 ;
Gu:2016oyy ; Bohrdt:2016vhv ; Zhou_2017 ; Luitz:2017jrn ; Nahum:2017yvy ;
vonKeyserlingk:2017dyr ; deBoer:2017xdk ; Grozdanov:2017ajz ; Leviatan:2017vur
; Hallam:2018mmi ; White_2018 ; Xu:2018xfz ; You:2018ons ; Nahum:2017xiy ;
Heyl_2018 ; Dag:2018oym ; Zhou:2018myl ; Zhou:2019pob ; Sahu:2020xdn ;
Jian:2020qpp ; Zanoci:2020jwl ; Cheng:2021mop . Generation of entanglement is
one of the primary features characterizing the thermalization process for out-
of-equilibrium states in a wide variety of chaotic systems. Operator
spreading, as defined through various operator entanglement measures and
through out-of-time-order correlators (OTOCs), has likewise been used to
diagnose chaos in an equally diverse set of systems. In this work we focus on
the related notion of information spreading, which is linked to both
phenomena.

In any local chaotic quantum system, information that is initially localized
will spread, such that its full recovery requires knowledge of an increasing
number of degrees of freedom around the location of origin. Suppose we wish to
track the spread of information that is initially localized on some test
subsystem $T$. The smallest region/subsystem at any time which can be used to
fully reconstruct the initial state of $T$ constitutes the “information cone”
of $T$. At late times in homogeneous systems, the boundary of this cone grows
linearly at a rate dubbed the information velocity, $v_{I}$ Couch:2019zni . It
was argued in Couch:2019zni  that this velocity is limited both by the rate of
entanglement generation and by the speed of local operator spreading. These
disparate phenomena are therefore both brought to bear on a single physical
question of how rapidly information delocalizes. This question is itself
rather fundamental, inducing a form of causal structure for thermal-scale
operators on the theory which may be more stringent than the ultimate
relativistic bounds, and which is present even in nonrelativistic quantum
systems.

The effect of nonlocality on entanglement growth and on various diagnostics of
chaos has been previously explored in holographic theories Fischler:2018kwt .
Interestingly, introducing nonlocality into the system was found to enhance
scrambling as well as the entanglement generation, eluding previous bounds
proposed in the context of local quantum field theory. Further studies support
this idea, including tantalizing results on the speed up of thermalization,
dissipation and complexification rates Edalati:2012jj ; Fischler:2012ff ;
Couch:2017yil  due to nonlocal effects. In this work, we extend these studies
to ask more in detail how the presence of nonlocality affects the rate of
information spreading. The forms of nonlocality we will consider are “mild,”
in the sense that there is always a finite length scale associated with
nonlocal interactions, and we consider regions larger than this length scale.
Thus, upon coarse-graining over short distances we can always recover standard
local physics. This mild nonlocality is in contrast, for example, to matrix
models, or to lattice system with all-to-all couplings which, in the extreme,
could render any notion of local subsystems and neighborhoods to be
irrelevant.

In this paper we will consider 1-dimensional spin chain systems as well as
holographic gauge theories with various forms of mild nonlocality. Even though
the presence of all-to-all interactions seems ubiquitous and necessary for
fast scrambling Belyansky:2020bia , there are examples of systems that exhibit
this phenomenon regardless of the absence of the former Ikeda:2021wfa . These
are spin chains with a specified combination of some sectors, including a
subset of next-to-nearest neighbor interactions. This form of mild nonlocality
can in fact be realized experimentally. Known examples include materials with
high polarizability which, in the presence of a strong electromagnetic field,
mimic the presence of such couplings due to the alignment of their molecular
dipoles. We believe that the results of this study could thence be important
for emerging quantum technologies such as quantum computation and quantum
communications.

We begin the work with a review of related concepts and techniques in section
2. We pay particular attention to describe the effective description for
information spreading that arises from coarse-graining over short distances,
a.k.a. the membrane or hydrodynamic theory, developed in Jonay:2018yei  and
adapted to holographic settings in Mezei:2018jco . Section 3 asks how
nonlocality affects information spreading in chaotic, 1-dimensional spin chain
systems, and discusses our numerical results for spin chains with next-nearest
neighbor interaction and generalizations. Section 4 then poses the same
questions in holographic systems inspired by the nonlocal spin chains
considered previously. These include non-commutative SYM theory (NCSYM) and
dipole-deformed SYM theory (DDSYM), both of which include fundamental dipole-
type interactions. Even though, NCSYM was already studied in Fischler:2018kwt
, the methods employed there (shock waves) are technically different to those
used in this paper (membrane theory). As expected, the results agree with each
other. This provides evidence that the hydrodynamic description is valid for
systems with mild nonlocality as the ones considered in this paper. We
conclude in section 5 with a comparative discussion of our results and future
directions.

##  2 Review of concepts and methods

Before turning to consider the effects of nonlocality, we here summarize the
various terms, concepts, and techniques employed in the investigation. We
begin by reviewing two other characteristic quantities, the entanglement
velocity $v_{E}$ and the butterfly velocity $v_{B}$, and their relationship to
the information velocity $v_{I}$. We describe our method of numerically
tracking the information cone and computing $v_{I}$ in chaotic 1-D spin
chains. We then provide a review of the membrane theory of entanglement
dynamics Jonay:2018yei ; Mezei:2018jco , and introduce the membrane tension
function $\varepsilon(v)$, which encodes information about the aforementioned
quantities. We review its origin, use, and calculation, both in spin chains
and in holographic contexts.

###  2.1 Three characteristic speeds: $v_{I}$, $v_{E}$, and $v_{B}$

In addition to our primary quantity of interest, the information velocity
$v_{I}$, we will frequently refer to two other velocities characteristic of
local quantum chaotic systems: the entanglement velocity $v_{E}$ and the
butterfly velocity $v_{B}$. We review these here for future reference. The
former, $v_{E}$, is not a velocity in the strict sense, but a quantity that
controls the rate of entanglement growth in thermalizing states
Hartman:2013qma ; Liu:2013iza ; Liu:2013qca . For large regions and early
times (but much larger than local thermal scales) after a homogeneous quench,
the entropy of a region $A$ grows linearly according to

$S\left[A(t)\right]=s_{\text{th}}\leavevmode\nobreak\
v_{E}\leavevmode\nobreak\ |\partial A|t+...\,,$ (1)

where $s_{\text{th}}$ is the thermal entropy density, and $|\partial A|$
denotes the area of the boundary of $A$. This linear growth is explained
heuristically in terms of an emergent “entanglement tsunami”, which is valid
in the hydrodynamic regime.111This picture breaks down for small systems
Kundu:2016cgh ; Lokhande:2017jik , however, here we will focus on regions much
larger than both the thermal length and the scale introduced by nonlocality.
For strip-like regions, this linear growth regime persists until saturation.
The ellipses include an initial entropy density, as well as sub-extensive
contributions which we ignore as subleading. Some dependencies are suppressed
in this expression: both $v_{E}$ and $s_{\text{th}}$ may depend on the energy
and charge density (or equivalently, the inverse temperature and chemical
potential) of the state. Additionally, while $v_{E}$ is usually defined in the
context of entropy growth from an unentangled state (or a vacuum state, with
$S\left[A(t)\right]$ taken to be the vacuum-subtracted entropy), it can more
generally depend on details of the initial state. For instance, in this work
we will consider initial states with a uniform entropy density greater than
zero, but less than the thermal value, referring to $f\equiv
s_{\text{initial}}/s_{\text{thermal}}$ as the initial entanglement fraction of
the state. In this case, $v_{E}$ depends on $f$ as well.

The second velocity, $v_{B}$ Shenker:2013pqa ; Roberts:2014isa , is associated
with the spread of local perturbations, as diagnosed through the growth of
local Heisenberg operators. The thermal expectation value of the squared
commutator of one simple local operator and another, displaced in space and
time, e.g.
$\left<[\mathcal{O}_{1}(x),U^{\dagger}(t)\mathcal{O}_{2}(y)U(t)]^{2}\right>_{\beta}$,
vanishes as $|x-y|\rightarrow\infty$, but becomes non-negligible over a
displacement range that grows with time. The region over which the expectation
value of the square of this commutator is $\mathcal{O}(1)$ serves to define
the butterfly cone. After initial transient behavior, the boundaries of such a
cone expand linearly at a rate $v_{B}$. Such cones map the regions of
influence of perturbations by local operators.222Aside from choosing operators
that do not represent locally conserved charge, the cone is largely
insensitive to the choice of local operators.

In Couch:2019zni  it was argued that the information velocity $v_{I}$ can be
understood through an argument that relates it to both $v_{E}$ and $v_{B}$.
The argument follows a variation of the Hayden and Preskill protocol of
information recovery Hayden:2007cs , and states that the information cone of a
test site $T$ can be identified, at any given time, by finding the largest
region centered on the test site which satisfies two conditions. First, it
must have reached entanglement saturation, and second, it must be within the
butterfly cone of the test site $T$. The former condition indicates that the
active degrees of freedom in the region are maximally entangled with degrees
of freedom in the complement system. The latter serves as an indicator that
the region is sufficiently scrambled.333The precise requirement for
“scrambling” is not rigorously identified. In the quantum information argument
by Hayden and Preskill, the scrambling operation corresponds to a Haar-
averaging over unitaries, though this is presumed excessive. In the variation
of Couch:2019zni , it is assumed that degrees of freedom within the butterfly
cone are sufficiently scrambled (for purposes of the argument), because all
local operators at $T$ can be replaced by truncated operators with support
only within the (growing) butterfly cone, and within this region the chaotic
dynamics is presumed sufficiently close to a “random” unitary. When both
criteria are met by a region $R$, a Hayden-Preskill style protocol for
information recovery implies that the complement region $\bar{R}$, and not
$R$, can recover the initial data of $T$.444Actually, the argument implies
that the complement region $\bar{R}$, plus any small region from within $R$,
$\Delta R$, with Hilbert space dimension at least greater than that of $T$,
can recover the information. In our large region limit we assume $|R|>>|T|$
and treat the difference between the size of $R$ and $R-\Delta R$ as
negligible. Considering larger and larger regions centered on $T$ at a fixed
time until one of the two criteria fails identifies the smallest region
$R_{I}$ that can be used to reconstruct the initial data of $T$. This
discussion assumes a sharp transition between regions of recovery, $R$ and
$\bar{R}$. In reality, the “edge” of the information cone may be smoothed,
however we assume this does not affect the asymptotic scaling of the growth of
$R_{I}$, and therefore $v_{I}$.

The above discussion alone indicates that either the scrambling condition or
the entanglement condition could act as a bottleneck to the spread of
information. However, in all systems considered, the condition of entanglement
saturation always sets the more stringent constraint, at least in the large-
region, late-time limit.555This statement is related to several known bounds
in the literature. In Hartman:2013qma ; Casini:2015zua  it was proven that
$v_{E}\leq c$ in relativistic systems and then by a causality argument that
$t_{\text{sat}}\geq r_{\text{ins}}/c$, where $r_{\text{ins}}$ is the radius of
the largest ball that fits in the region being considered. Then in
Mezei:2016wfz ; Mezei:2016zxg  it was shown that $v_{E}\leq v_{LC}$ and
$t_{\text{sat}}\geq r_{\text{ins}}/v_{LC}$, where $v_{LC}$ is an effective
light cone speed, usually assumed equal to $v_{B}$ (though not always, see
Bohrdt:2016vhv ). Therefore we henceforth take for granted that $v_{I}$ can be
identified by finding the growth of the largest region that has just reached
entanglement saturation. For an arbitrary region $A$, the time to entanglement
saturation is not solely controlled by $v_{E}$, but in cases where linear
growth persists until saturation, in the scaling limit it is simply given by
$t_{\text{sat}}=(1-f)v_{E}^{-1}|A|/|\partial A|$. For strips of width $R$ this
gives $t_{\text{sat}}=R(1-f)/v_{E}(f)$. Together with the statement that
$v_{E}(f)\sim v_{B}(1-f)+...$ near $f=1$ Mezei:2016wfz ; Jonay:2018yei ;
Couch:2019zni , this entails that for strip-shaped regions, $v_{I}$
interpolates between $v_{E}$ at $f=0$ and $v_{B}$ at $f=1$. Finding $v_{I}$
for strips across all values of $f$ thereby also provides information about
the rates controlling entanglement growth and operator spreading in the
system. Computing $v_{I}(f)$ for strip-like regions will therefore be our
primary focus, leaving extensions to other shapes and states to future work.

The fact that $v_{I}$ is simply related to the largest region to reach
entanglement saturation also allows us to relate $v_{I}$ directly to the
membrane tension function, a quantity described in the section 2.3. In spin
chain systems, we can only roughly approximate the membrane tension function
due to computational limitations and finite size effects. Hence, we are better
served by computing $v_{I}$ directly through a method explained in section
2.2. Holographic systems, on the other hand, are particularly amenable to the
membrane method. In these systems we compute $v_{I}$ by relating it to the
membrane tension function, as explained in section 2.3.2.

###  2.2 Tracking the information cone in 1-D spin chains

For the purpose of identifying the information cone in a local quantum chaotic
system666Controlled information transport in non-chaotic spin chains is
another interesting vein of research. See, for example, 2007PhRvL..99y0506C
and references therein., we imagine that the degrees of freedom on a local
test site, $T$, are initially maximally entangled with a reference system $R$
of equal or greater Hilbert space dimension. The reference $R$ is thereafter
completely decoupled from the system dynamics. Initially, $R$ has maximal
mutual information with the test degrees of freedom, but as the system
evolves, the mutual information between $R$ and $T$ decreases as data from $T$
scrambles into the rest of the system. The information cone can be identified
by tracking the smallest region around $T$ that retains maximal mutual
information with the reference system. Any other initial state at $T$ (besides
the maximally mixed state used in this procedure) could, in principle, be
inferred from data within this region Barnum:1998da .

For a 1-dimensional chaotic spin chain, we instantiate a version of this setup
as follows (this is a slight modification of a procedure employed in
Couch:2019zni ): We consider $L$-qubit systems with nearest-neighbor
$\sigma_{z}\sigma_{z}$ couplings, with longitudinal and transverse magnetic
field terms chosen to render the Hamiltonian chaotic:

$\begin{split}H^{(0)}_{L}=&J_{zz}\sum_{i=1}^{L-1}\sigma_{z}^{(i)}\sigma_{z}^{(i+1)}+h_{x}\sum_{i=1}^{L}\sigma_{x}^{(i)}+h_{z}\sum_{i=1}^{L}\sigma_{z}^{(i)}\\\
\end{split}$ (2)

where $J_{zz}=1,\leavevmode\nobreak\ \leavevmode\nobreak\
h_{x}=1.05,\leavevmode\nobreak\ \leavevmode\nobreak\ h_{z}=0.5$. Variations of
this Hamiltonian have been widely used in the literature to study
thermalization. The superscript on $H^{(0)}_{L}$ merely indicates that this
serves as our “base” Hamiltonian, to which we will later add various nonlocal
couplings. However, the nonlocal couplings are irrelevant for the
demonstrations of this section. We also define the following set of states:

$\begin{split}\ket{\psi_{L}^{f}}=\prod_{i=1}^{L/2}a_{\sigma,\sigma^{\prime}}\ket{\sigma_{i}}\ket{\sigma_{i+L/2}}\end{split}$
(3)

with the matrix $a$ given by

$\begin{split}a=\left(\frac{1}{2}I+\xi\leavevmode\nobreak\
\sigma_{y}\right)^{1/2}.\end{split}$ (4)

These states allow us to initialize a uniform “volume law” entanglement
fraction $f$, such that the entanglement entropy of a subsystem of
$N_{\text{sub}}<L/2$ neighboring qubits is $f*N_{\text{sub}}$. The $f$
dependence is implicitly related through $\xi$ satisfying
$-(1/2+\xi)\log(1/2+\xi)-(1/2-\xi)\log(1/2-\xi)=f\log(2)$. For all $f$, these
states are at the center of the energy spectrum, in that
$\expectationvalue{H_{L}}{\psi}=0$. An additional Bell pair is then appended
to the system. One member of the pair is dynamically coupled to the system
while the other remains decoupled. These are the “test” and “reference” qubits
respectively.

$\begin{split}\ket{\psi_{L+2}(t=0))}=\left(\ket{0}_{R}\ket{0}_{T}+\ket{1}_{R}\ket{1}_{T}\right)\ket{\psi^{f}_{L}}\end{split}$
(5)

The system is then evolved with the $L+1$ qubit Hamiltonian (2), treating the
test qubit as the first in an $L+1$ qubit spin chain. The reference and test
qubits initially have maximal mutual information, but as the information
associated with the test qubit scrambles into the rest of the system, an ever-
growing number of nearby qubits are required to recover the mutual information
with the reference777In practice, we use a cutoff value of one bit of mutual
information. The subsystems under consideration, which all include the test
qubit and a variable number of its neighbors, evolve from maximal mutual
information to near zero mutual information (with the reference) at late times
as long as the number of qubits is under half the system size. A mutual
information of 1 captures the center of this falloff, and we use it to define
the information “wavefront.”. The growth of the smallest subsystem necessary
to achieve this defines the growth of the information cone.

Our Hamiltonian and states have a quasi-translation invariance, voided by the
existence of the boundaries, or by the fact that we are computationally
restricted to finite $L$. We therefore track the cone across just under half
the system size, extracting $v_{I}$ as a linear fit on the cone in this
region. Most of our results are for systems with total $L=26$. In fitting
$v_{I}$, we exclude the initial and final three subsystems of the half-system
to minimize finite-time and edge effects. We implement time evolution with a
Krylov subspace method to exponentiate the Hamiltonian. This code was written
in C, and based on the well known Fortran package, Expokit EXPOKIT , where
modifications were made to improve efficiency and parallelize the computation.
Each time evolution and entropy computation for a particular Hamiltonian was
run on a single node across 44 cores on TACC’s Stampede2 computer. The results
were verified for smaller $L$ using the Dynamite 2017_dynamite  and Qutip
qutip1 ; qutip2  python packages as well as a Mathematica Mathematica
implementation. Figure 1 shows examples of information cones from states
initialized with varying degrees of uniform entanglement, $f$. The information
velocity is plotted as a function of initial entanglement fraction. As
$f\rightarrow 1$, $v_{I}$ is expected to approach $v_{B}$. As a consistency
check, we computed $v_{B}$ independently by tracking the growth of squared
commutators of simple operators, finding values in approximate agreement,
considering finite size effects in both computations.

$v_{I}$

Figure 1: Left: A sample of information cones in the $H^{(0)}_{L=26}$ system
for various values of initial entanglement fraction $f$. Right: Information
velocity as a function of $f$, extracted from the curves in the left panel.
The dotted lines represent the butterfly velocity $v_{B}$, as obtained through
an independent OTOC computation, and the entanglement velocity $v_{E}$,
computed as the linear growth rate of entanglement on half the system.

###  2.3 Membrane theory of entanglement dynamics

An interesting quantity which can serve to simultaneously characterize aspects
of entanglement growth, operator dynamics, and information spreading in local
quantum chaotic systems is the membrane tension function, ${\cal E}(v)$
Jonay:2018yei . The membrane model of entanglement dynamics emerges in a
course-grained, hydrodynamic type approximation of such systems. It computes
the entanglement entropy of a state via an integrated energy density along a
timelike membrane stretching through an auxiliary Minkowski spacetime,
analogous to a minimal cut through a tensor network or quantum circuit
representing unitary time evolution. The membrane method has been applied with
great success to holographic systems Mezei:2018jco ; Mezei:2019zyt ;
Agon:2019qgh , where the membrane is merely a projection of bulk extremal
surfaces, with the bulk dimension integrated out. In this section, we review
the origin of the membrane method in a thermodynamic limit of chaotic quantum
systems. We review a method of approximating ${\cal E}(v)$ in 1-D spin chain
systems which we will later employ in section 3.3. We then discuss its
derivation and use in the context of the AdS/CFT correspondence as well as its
extension to more general gauge/gravity dualities. This section is largely a
review of work appearing in Jonay:2018yei  and Mezei:2018jco , though, we
generalize the techniques to theories with inherent nonlocality.

####  2.3.1 Membrane theory for spin chains

To motivate the membrane method, we restrict ourselves to states in local
chaotic quantum systems exhibiting a “volume law” entanglement pattern, such
that the entanglement entropy of subregion is simply proportional to the
volume of the subregion (plus subleading corrections). For simplicity we also
first consider 1-dimensional systems, though the generalization to higher
dimensions is immediate. In a system of total length $L$, pure states
partitioned by a cut at position $x$, at equilibrium exhibit a bipartite
entanglement entropy of

$S_{\text{eq}}(x,t)=s_{\text{eq}}\min\left(x,L-x\right)$ (6)

where $s_{\text{eq}}$ is the equilibrium entanglement density. The leading,
course-grained dynamics of out-of-equilibrium states can be captured in a
hydrodynamic description, governed by

$\frac{\partial S}{\partial t}=s_{\text{eq}}\Gamma\left(\frac{\partial
S}{\partial x}\right).$ (7)

Here $\Gamma\left(\frac{\partial S}{\partial x}\right)=\Gamma(s)$ is the
entropy production rate. In general this function could depend explicitly on
position or on any locally conserved charge densities, though we consider the
simplest, spatially uniform case. Physical considerations immediately put some
constraints on $\Gamma(s)$. At saturation, entanglement growth vanishes
$\frac{\partial S}{\partial t}=0$, implying that
$\Gamma(s_{\text{eq}})=\Gamma(-s_{\text{eq}})=0$. Considering the growth of
entanglement from a completely unentangled state leads to $\Gamma(0)=v_{E}$.
The entropy production rate is only defined over $s_{\text{eq}}\leq s\leq
s_{\text{eq}}$, and we assume a smooth, symmetric function with
$\Gamma^{\prime\prime}(v)<0$ over this domain.

Next suppose that there is an equivalent representation of the entropy
dynamics, staged on an auxiliary Minkowski spacetime associated with the time
evolution operator, where the entanglement entropy is computed by minimizing
the integrated local “energy” along a timelike membrane stretching from the
initial to final time and obeying certain boundary conditions. The tension
function ${\cal E}(v)$ specifies the energy density along this membrane as a
function of its orientation in the spacetime (its local “velocity”). For the
one-dimensional translation invariant system described above, the entropy at
position $S(x,t)$ would be obtained by the following minimization (see figure
2):

Figure 2: Minimization over membranes brings the curve on the left where a
membrane stretches from $y$ to $x$ with a velocity $v(t)$ assigned as the
angle from the vertical at each point in time to a straight line.

$S(x,t)=\underset{y}{\min}\left(t\leavevmode\nobreak\
s_{\text{eq}}\leavevmode\nobreak\ {\cal
E}\left(\frac{x-y}{t}\right)+S(y,0)\right)$ (8)

Differentiating with respect to time and comparing to equation (7) entails
that ${\cal E}(v)$ can be directly related to $\Gamma(s)$ (and vice versa) by
Legendre transformation:

$\begin{split}\Gamma(s)=\underset{v}{\min}\left({\cal
E}(v)-\frac{vs}{s_{\text{eq}}}\right)\\\ {\cal
E}(v)=\underset{s}{\max}\left(\Gamma(s)+\frac{vs}{s_{\text{eq}}}\right)\\\
\end{split}$ (9)

From the properties of $\Gamma(s)$, we can infer related properties of
$\varepsilon(v)$. In particular, considering $\Gamma(s_{\text{eq}})=0$ leads
to $v_{\text{max}}=s_{\text{eq}}\Gamma^{\prime}(s_{\text{eq}})$. In the tensor
network or circuit picture, the maximum velocity $v_{\text{max}}$ serves as an
effective lightcone velocity, which in known cases is equal to the butterfly
velocity $v_{B}$. Results in the systems we consider are consistent with this
expectation, and we assume it throughout. In conjunction with the other
characteristics of $\Gamma(s)$ and properties of the Legendre transform, in
the range $-v_{B}\leq v\leq v_{B}$ we have

$\displaystyle{\cal E}(0)=v_{E},$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ {\cal E}(v_{B})=v_{B},$ $\displaystyle{\cal
E}^{\prime}(0)=0,$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
{\cal E}^{\prime}(v_{B})=1,$ $\displaystyle{\cal E}(v)\geq|v|,$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ {\cal
E}^{\prime\prime}(v)\geq 0.$ (10)

The definition of $\varepsilon(v)$ can be extended outside this domain, but
this is the region of dynamical interest. It is evident that the membrane
tension function encodes a lot of information characterizing the system,
including not only entanglement dynamics ($v_{E}$), but also operator growth
($v_{B}$) and indirectly, information spreading ($v_{I}$). For this reason it
will be one of the primary quantities which we compute and compare for systems
with variable degrees of nonlocality.

In general it is not possible to compute ${\cal E}(v)$ analytically for a
given Hamiltonian. A numerical approximation technique was introduced in
Jonay:2018yei , which we summarize here and later employ. The membrane tension
function at infinite temperature can be simply related to the operator
entanglement of the time evolution unitary. Operator entanglement is computed
by treating the operator as a state vector in a doubled Hilbert space. For
instance, from the time evolution operator $U(t)$ we obtain:

$\bra{i}U(t)\ket{j}\ket{i}\bra{j}\longrightarrow U_{ij}(t)\ket{j}\ket{i}$

Here, the first and second Hilbert space factors can be thought of as
representing the same degrees of freedom at time $0$ and time $t$,
respectively. An ordinary entanglement entropy can then be computed on various
subsystems of this state. In a 1-D system, tracing out degrees of freedom up
to $x$ on the first factor and up to $y$ on the second factor gives the
entanglement entropy $S_{U}(x,y,t)$, which in the membrane picture corresponds
to a minimal membrane stretching from $x$ to $y$ through an auxiliary
spacetime of height $t$. This leads to a method of approximating the membrane
tension function via operator entanglements of the time evolution unitary:

${\cal
E}_{\text{eff}}\left(t,v\right)\equiv\frac{1}{s_{\text{eq}}t}S_{U}\left(-\frac{vt}{2},\frac{vt}{2},t\right)$

Here, for system of finite length $L$, the origin of the spatial arguments is
placed at the center. At any given $t$, evaluating the right hand side gives
an approximation of the membrane tension function at infinite temperature.
This approximation is limited by both finite size effects and finite time
effects. In an infinite system, taking a large time limit would render finite
time effects negligible, and the approximation would converge to ${\cal E}(v)$
over the range $-v_{B}<v<v_{B}$, and to $|v|$ outside this range. But for a
finite system as $t$ increases, the range of $v$ over which the cut can be
made without exceeding the boundaries (or more stringently, before edge
effects become substantial) becomes more limited. In our spin chain system, we
are computationally limited to $13$ qubits. So in practice, we obtain a series
of curves for a set of $t$ values (or rather, discreet data points such that
$|vt/2|$ correspond to an integer number of spins traced out) and
schematically piece these together to give a qualitative picture of ${\cal
E}(v)$ under different Hamiltonians. This method is employed in section 3.3 as
a qualitative check on our more direct methods of investigating the effect of
variable nonlocality in spin chain systems.

####  2.3.2 Membrane theory for holographic systems

The membrane theory of entanglement dynamics has been successfully generalized
to strongly-coupled large-$N$ theories with holographic duals Mezei:2018jco ;
Mezei:2019zyt . This provides strong evidence that this effective description
can efficiently encapsulate the entanglement dynamics in all chaotic systems.
In Mezei:2018jco  it was first realized that a membrane description naturally
emerges upon studying the late-time and large-distance regime of the RT
prescription, akin to hydrodynamics. This study was later complemented in
Mezei:2019zyt  showing that this theory is robust under a large set of
generalizations, including different quench protocols and finite coupling
corrections holographically dual to higher derivative gravity corrections. In
this section we will first review the results of Mezei:2018jco , valid for
asymptotically AdS spaces in Einstein gravity. We will then generalize this
framework to more general holographic scenarios, including those described by
non-AdS spaces with extra matter fields and fluxes, ubiquitous in top-down
constructions emerging from string or M theory.

#### The case of asymptotically AdS spaces

We start with a generic asymptotically AdSd+1 black brane

$ds^{2}={L^{2}\over z^{2}}\left[-a(z)dv^{2}-{2\over
b(z)}\,dvdz+d\vec{x}_{d-1}^{2}\right]\,,$ (11)

where $v$ is an “infalling” time coordinate. For instance, for the standard
Schwarzschild-AdSd+1 we have

$a(z)=1-\left(\frac{z}{z_{h}}\right)^{d}\,,\qquad b(z)=1\,,$ (12)

with

$T=\frac{1}{\beta}=-\frac{1}{4\pi}\frac{d}{dz}a(z)\Big{|}_{z_{h}}=\frac{d}{4\pi
z_{h}}\,.$ (13)

We follow Mezei:2018jco  but keep working in cartesian coordinates. In this
way, we can later generalize the analysis to non-AdS settings in a more
straightforward way. First, consider the scalings

$v\equiv\Lambda\,\tau\,,\qquad x_{i}\equiv\Lambda\,\xi_{i}\,,\qquad
z\equiv\zeta\,.$ (14)

Defining $\xi\equiv\xi_{1}$,
$\vec{\xi}_{\perp}\equiv\\{\xi_{2},\ldots,\xi_{d-1}\\}$ and parameterizing the
RT surface with functions $\zeta(\tau,\vec{\xi}_{\perp})$,
$\xi(\tau,\vec{\xi}_{\perp})$, one finds that in the limit $\beta/\Lambda\to
0$ the entanglement entropy functional reduces to:

$\displaystyle S_{A}=\frac{L^{d-1}\Lambda^{d-1}}{4G_{N}}\int d\tau
d\vec{\xi}_{\perp}\ {\sqrt{Q}\over\zeta^{d-1}}\,,\qquad
Q\equiv(\partial_{\tau}\xi)^{2}-a(\zeta)\left[1+(\partial_{\perp}\xi)^{2}\right]\,,$
(15)

where
$(\partial_{\perp}\xi)^{2}\equiv(\partial_{\xi_{2}}\xi)^{2}+\cdots+(\partial_{\xi_{d-1}}\xi)^{2}$.
In this limit, the equation of motion for $\zeta$ is algebraic:

${(\partial_{\tau}\xi)^{2}\over 1+(\partial_{\perp}\xi)^{2}}=a(\zeta)-{\zeta
a^{\prime}(\zeta)\over 2(d-1)}\,.$ (16)

We now define:

$v^{2}\equiv{(\partial_{\tau}\xi)^{2}\over
1+(\partial_{\perp}\xi)^{2}}\,,\qquad c(\zeta)\equiv a(\zeta)-{\zeta
a^{\prime}(\zeta)\over 2(d-1)}\,.$ (17)

so that (16) becomes $v^{2}=c(\zeta)$. Plugging back into the action the
solution to this equation, $\zeta=c^{-1}(v^{2})$, we can rewrite the entropy
functional as a membrane functional:

$\displaystyle S_{A}=\frac{L^{d-1}\Lambda^{d-1}}{4G_{N}}\int d\tau
d\vec{\xi}_{\perp}\ \sqrt{1+(\partial_{\perp}\xi)^{2}}\,{\cal
E}\left(v\right)=s_{\text{th}}\Lambda^{d-1}\int d\text{area}\ {{\cal
E}\left(v\right)\over\sqrt{1-v^{2}}}\,,$ (18)

where

$s_{\text{th}}=\frac{L^{d-1}}{4G_{N}\zeta_{h}^{d-1}},$ (19)

is the thermal entropy density of the black brane. In the above we have
identified the standard “area” element in Minkowski space

$d\text{area}=d\tau
d\vec{\xi}_{\perp}\sqrt{1+(\partial_{\perp}\xi)^{2}-(\partial_{\tau}\xi)^{2}}$
(20)

and defined the membrane tension ${\cal E}(v)$ to be

${\cal E}\left(v\right)\equiv\zeta_{h}^{d-1}{\sqrt{-a^{\prime}(\zeta)\over
2(d-1)\zeta^{2d-3}}}\Bigg{|}_{\zeta=c^{-1}(v^{2})}\,.$ (21)

Notice that the factor $\zeta_{h}^{d-1}$ is included in (41) to make ${\cal
E}(v)$ dimensionless. For example, for Schwarzschild-AdSd+1 (12) we find that
the equation (16) becomes

$v^{2}=1-\left(\frac{\zeta}{\zeta_{h}}\right)^{d}+{d\over
2(d-1)}\left(\frac{\zeta}{\zeta_{h}}\right)^{d}=1-\frac{(d-2)}{2(d-1)}\left(\frac{\zeta}{\zeta_{h}}\right)^{d}\,,$
(22)

which can be inverted as follows:

$\zeta=c^{-1}(v^{2})=\zeta_{h}\left[\frac{2(d-1)}{(d-2)}(1-v^{2})\right]^{1/d}\,.$
(23)

Plugging this back into (41) we find that

${\cal E}\left(v\right)={v_{E}\over(1-v^{2})^{(d-2)/(2d)}}\,,\qquad
v_{E}={\left(d-2\over d\right)^{(d-2)/(2d)}\over\left(2(d-1)\over
d\right)^{(d-1)/d}}\,.$ (24)

It can be checked that this function has all the desired properties
Mezei:2018jco ; Mezei:2019zyt  (in the dynamically relevant regime, i.e.,
$0\leq v\leq v_{B}$):

${\cal E}(0)=v_{E}\,,\qquad{\cal E}^{\prime}(0)=0\,,\qquad{\cal
E}(v_{B})=v_{B}\,,\qquad{\cal E}^{\prime}(v_{B})=1\,,\qquad{\cal
E}^{\prime}(v)\geq 0\,,\qquad{\cal E}^{\prime\prime}(v)\geq 0\,,$ (25)

with

$v_{B}=\sqrt{\frac{d}{2(d-1)}}\,.$ (26)

One final remark is in order. In general, solving for $v_{B}$ from (25) may be
very difficult. However, it is useful to notice that

$v_{B}^{2}=c(\zeta_{h})\,,$ (27)

so $c^{-1}(v_{B}^{2})=\zeta_{h}$ must yield the horizon. On the other hand,

$0=c(\zeta_{\text{HM}})\,,$ (28)

and $c^{-1}(0)=\zeta_{\text{HM}}$ yields the Hartman-Maldacena surface
Hartman:2013qma .

#### General theory for non-AdS spaces

The gravity duals of the theories we want to study are non-asymptotically AdS
and possibly anisotropic. In particular, their metrics are not of the form
(11). This means we need to generalize the membrane theory starting from a
more general ansatz. For concreteness, we will take the metric to be of the
following form:

$ds^{2}=-g_{vv}(z)dv^{2}-{2g_{vz}(z)}\,dvdz+\sum_{i=1}^{d-1}g_{ii}(z)dx_{i}^{2}+d\Omega_{p}^{2}\,,$
(29)

where $d$ is the number of spacetime dimensions in the boundary and
$d\Omega_{p}^{2}$ is the metric on a $p$-dimensional compact space
$\mathcal{M}_{p}$. Importantly, we will assume that this compact space may
have a non-trivial dependence with respect to the radial coordinate $z$, such
that its volume factorizes as

$\int
d^{p}\Omega_{p}=V_{\Omega}(z)\times\text{Vol}(\tilde{\mathcal{M}}_{p})\,,$
(30)

for a suitable “normalized” $\tilde{\mathcal{M}}_{p}$.

Following the derivation of the previous section, we first consider the
scalings

$v\equiv\Lambda\,\tau\,,\qquad x_{i}\equiv\Lambda\,\xi_{i}\,,\qquad
z\equiv\zeta\,,$ (31)

while keeping the coordinates of the compact space untouched. Defining
$\xi\equiv\xi_{i}$, $\vec{\xi}_{\perp}\equiv\xi_{j}$ $(j\neq i)$ and
parameterizing the RT surface with functions $\zeta(\tau,\vec{\xi}_{\perp})$,
$\xi(\tau,\vec{\xi}_{\perp})$ (assuming no dependence with respect to the
coordinates of the compact space), one finds that in the limit
$\beta/\Lambda\to 0$ the entanglement entropy functional reduces to:

$\displaystyle
S_{A}=\frac{\Lambda^{d-1}\text{Vol}(\tilde{\mathcal{M}}_{p})}{4G_{N}}\int
d\tau d\vec{\xi}_{\perp}\sqrt{P}\sqrt{Q}\,,$ (32)

with

$P\equiv V_{\Omega}^{2}(\zeta)\Pi_{i=1}^{d-1}g_{ii}(\zeta)\,,\qquad
Q\equiv(\partial_{\tau}\xi)^{2}-g_{vv}(\zeta)\left[\frac{1}{g_{ii}(\zeta)}+\sum_{j\neq
i}\frac{(\partial_{\xi_{j}}\xi)^{2}}{g_{jj}(\zeta)}\right]\,.$ (33)

In this limit the equation of motion for $\zeta$ is algebraic; however, it
generally cannot be written in the form

$v^{2}=c(\zeta)\,.$ (34)

This is due to the anisotropy in the metric (29). To move forward we
specialize to the case of infinite strips, so that the embedding functions are
independent of the transverse coordinates, i.e., $\zeta(\tau)$, $\xi(\tau)$
and hence $\partial_{\xi_{j}}\xi=0$. In this case we recover (32) but now with

$P\equiv V_{\Omega}^{2}(\zeta)\Pi_{i=1}^{d-1}g_{ii}(\zeta)\,,\qquad
Q\equiv(\partial_{\tau}\xi)^{2}-\frac{g_{vv}(\zeta)}{g_{ii}(\zeta)}\,.$ (35)

The equation of motion for $\zeta$ is now:

$(\partial_{\tau}\xi)^{2}=\frac{g_{vv}(\zeta)}{g_{ii}(\zeta)}-\frac{P(\zeta)}{P^{\prime}(\zeta)g_{ii}(\zeta)^{2}}\left[g_{vv}(\zeta)g_{ii}^{\prime}(\zeta)-g_{ii}(\zeta)g_{vv}^{\prime}(\zeta)\right]\,,$
(36)

which is of the form (34) with

$v^{2}\equiv(\partial_{\tau}\xi)^{2}\,,\qquad
c(\zeta)\equiv\frac{g_{vv}(\zeta)}{g_{ii}(\zeta)}-\frac{P(\zeta)}{P^{\prime}(\zeta)g_{ii}(\zeta)^{2}}\left[g_{vv}(\zeta)g_{ii}^{\prime}(\zeta)-g_{ii}(\zeta)g_{vv}^{\prime}(\zeta)\right]\,.$
(37)

As a consistency check, we note that for asymptotically AdS, isotropic black
branes we have

$P(\zeta)=\frac{L^{2(d-1)}}{\zeta^{2(d-1)}}\,,\qquad
g_{vv}(\zeta)=\frac{L^{2}}{\zeta^{2}}a(\zeta)\,,\qquad
g_{ii}(\zeta)=\frac{L^{2}}{\zeta^{2}}\,,\qquad
g_{jj}(\zeta)=\frac{L^{2}}{\zeta^{2}}$ (38)

and we recover (17) from (37). Plugging back into the action the solution to
this equation, $\zeta=c^{-1}(v^{2})$, we can rewrite the entropy functional as
a membrane functional:

$\displaystyle
S_{A}=\frac{\Lambda^{d-1}\text{Vol}(\tilde{\mathcal{M}})}{4G_{N}}\int d\tau
d\vec{\xi}_{\perp}\,{\cal\tilde{E}}\left(v\right)=\frac{\Lambda^{d-1}\text{Vol}(\tilde{\mathcal{M}})}{4G_{N}}\int
d\text{area}\ {{\cal\tilde{E}}\left(v\right)\over\sqrt{1-v^{2}}}\,,$ (39)

where

$d\text{area}=d\tau d\vec{\xi}_{\perp}\sqrt{1-(\partial_{\tau}\xi)^{2}}$ (40)

and

${\cal\tilde{E}}\left(v\right)\equiv{\sqrt{-\frac{P(\zeta)^{2}}{P^{\prime}(\zeta)g_{ii}(\zeta)^{2}}\left[g_{vv}(\zeta)g_{ii}^{\prime}(\zeta)-g_{ii}(\zeta)g_{vv}^{\prime}(\zeta)\right]}}\Bigg{|}_{\zeta=c^{-1}(v^{2})}\,.$
(41)

We note that ${\cal\tilde{E}}\left(v\right)$ still needs to be normalized. In
order to do so, we identify the entropy density as:

$s_{\text{th}}=\frac{\text{Vol}(\tilde{\mathcal{M}})}{4G_{N}}\sqrt{V_{\Omega}^{2}(\zeta_{h})\Pi_{i=1}^{d-1}g_{ii}(\zeta_{h})}=\frac{\text{Vol}(\tilde{\mathcal{M}})}{4G_{N}}\sqrt{P(\zeta_{h})}\,.$
(42)

Hence, we find that

$\displaystyle S_{A}=s_{\text{th}}\Lambda^{d-1}\int d\text{area}\ {{\cal
E}\left(v\right)\over\sqrt{1-v^{2}}}\,,$ (43)

with

${\cal
E}\left(v\right)\equiv{\sqrt{-\frac{P(\zeta)^{2}}{P(\zeta_{h})P^{\prime}(\zeta)g_{ii}(\zeta)^{2}}\left[g_{vv}(\zeta)g_{ii}^{\prime}(\zeta)-g_{ii}(\zeta)g_{vv}^{\prime}(\zeta)\right]}}\Bigg{|}_{\zeta=c^{-1}(v^{2})}\,.$
(44)

#### Information velocity from the membrane theory

We can include the effects of the entanglement fraction $f$ Couch:2019zni  in
the membrane theory in a manner developed by Mark Mezei, who obtained the
results of this section in unpublished work Mezei:note . The inclusion of $f$
amounts to the introduction of a penalty factor for the lower end of the
membrane:

$S_{A}(A(\tau))=s_{\text{th}}\left(\int d\text{area}\ {{\cal
E}\left(v\right)\over\sqrt{1-v^{2}}}+f\text{vol}(A^{\prime}(0))\right)\,.$
(45)

Thus, the membrane is anchored on $\partial A(\tau)$ and $\partial
A^{\prime}(0)$ on the upper and lower boundaries respectively. Since the
latter is just a boundary term, it does not affect the equations of motion.
For the case of strips of width $2R$ one gets membranes with constant $v$, and
(45) becomes

$S_{A}(A(\tau))=s_{\text{th}}\text{area}(\partial
A)(\mathcal{E}(v)\tau+f(R-v\tau))\,$ (46)

whose minimum in $v$ is at

$f=\mathcal{E}^{\prime}(v)\,,$ (47)

independently of $\tau$. The entropy saturates when
$S_{A}(A(\tau_{\text{sat}}))=s_{\text{th}}\text{area}(\partial A)R$, which
gives

$v_{I}=\frac{R}{\tau_{\text{sat}}}=\frac{\mathcal{E}(v)-\mathcal{E}^{\prime}(v)v}{1-f}\bigg{|}_{\mathcal{E}^{\prime}(v)=f}=\frac{\Gamma(f)}{1-f}\,,$
(48)

where $\Gamma(f)$ is defined as the Legendre transform of $\mathcal{E}(v)$. It
can be checked that $v_{I}(f=0)=v_{E}$ and $v_{I}(f=1)=v_{B}$.

##  3 Information propagation in spin chains with variable nonlocality

We now turn to the question of how the rate of information spreading in
chaotic quantum systems is affected by the presence of nonlocality in various
forms. We begin with the base Hamiltonian $H_{L}^{(0)}$ of equation (2). By
adding next-nearest-neighbor terms (or next-next-nearest-neighbor terms, etc…)
with variable couplings, we introduce a form of mild nonlocality into the
Hamiltonian and investigate the affect of these terms on information
spreading.888Spin chains with next-nearest neighbor interactions or related
couplings have been studied in many other contexts, see for example
PhysRevB.48.3240 ; Gu_2004 ; Liu_2006ent ; Kwek_2009 ; 2009PhRvB.79a4439O .
Entanglement growth in integral models with a form of nonlocality induced by a
Lifschitz scaling has been studied in Mozaffar:2018vmk ; Mozaffar:2021nex .
More precisely, our total Hamiltonian will be

$\begin{split}H_{L}=H_{L}^{(0)}+H_{L}^{(\text{nonlocal})}\end{split}$ (49)

where

$\begin{split}H_{L}^{(\text{nonlocal})}=&J_{zoz}\sum_{i=1}^{L-2}\sigma_{z}^{(i)}\sigma_{z}^{(i+2)}+J_{zooz}\sum_{i=1}^{L-3}\sigma_{z}^{(i)}\sigma_{z}^{(i+3)}+...\\\
&J_{xox}\sum_{i=1}^{L-2}\sigma_{x}^{(i)}\sigma_{x}^{(i+2)}+J_{xoox}\sum_{i=1}^{L-3}\sigma_{x}^{(i)}\sigma_{x}^{(i+3)}+...\leavevmode\nobreak\
.\end{split}$ (50)

All the individual terms in $H_{L}^{(\text{nonlocal})}$ are quadratic, in the
sense that they contain two active sites (i.e. only two Pauli operators per
summed term). The labelling scheme for nonlocal coefficients should be clear:
the number of $o$’s in $J_{xo...ox}$ (for example) indicates the number of
spaces between active $\sigma_{x}$ sites. In most cases, we will set only a
subset of the couplings $J_{xox},J_{zoz},J_{xoox},J_{zooz}...$ to nonzero
values simultaneously, however we do allow them to be on the same order as the
$J_{zz}=1$ coupling in $H_{L}^{(0)}$, so they do not generally constitute
small perturbations to $H^{(0)}$. In fact, in our scans below we choose to
consider nonlocal couplings up to precisely $J_{\text{nonlocal}}=1$ because in
some cases going far beyond this causes the Hamiltonian eigenstate level-
spacing to deviate from the Wigner-Dyson distribution indicative of a chaotic
regime PhysRevLett.110.084101 , where we wish to remain.

Equation (50) obviously does not represent the unique set of nonlocal terms
which could be considered. We choose to work with $\sigma_{z}\sigma_{z}$ terms
and $\sigma_{x}\sigma_{x}$ terms, because unlike $\sigma_{y}\sigma_{y}$, such
terms leave the states (3) at the center of the energy spectrum, regardless of
the values of nonlocal coupling coefficients. These two types of terms also
provide clean examples of nonlocal additions which are, respectively,
commuting and non-commuting with the nearest neighbor piece of $H^{(0)}_{L}$,
which we speculate may lead to qualitatively different behavior.

In the following sections we display some results obtained using the methods
outlined in section 2.2, to show the effects of nonlocal terms on the rate of
information spreading.

###  3.1 Next-nearest neighbor interactions: $J_{xox}$ and $J_{zoz}$

$J_{xox}$$J_{zoz}$$v_{I}$$v_{I}$

Figure 3: Left: Information velocity as a function of initial entanglement
fraction $f$ in $L=26$ systems with different values of $J_{xox}$ nonlocal
coupling (all other nonlocal couplings set to zero). Right: Information
velocity as a function of initial entanglement fraction $f$ in $L=26$ systems
with different values of $J_{zoz}$ (all other nonlocal couplings set to zero).
In both panels, the $x$’s along the $f=1$ line correspond to independent
computations of $v_{B}$, and the $o$’s along $f=0$ correspond to independent
$v_{E}$ computations. See discussion in the main text.

First we consider the case where only one of the nonlocal terms in (50) is
turned on. Figure 3 shows $v_{I}$ over $f$ for the different values of either
$J_{xox}$ (left panel) or $J_{zoz}$ (right panel). All other nonlocal
couplings are set to zero for these scans. In the case of $J_{xox}$, the most
apparent qualitative behavior is perhaps unsurprising: as the amount of
nonlocality is increased, the information velocity increases as well, at all
values of $f$. For small $J_{xox}\lesssim 0.2$, however, this effect is
suppressed or even reversed, as $v_{I}$ is either unchanged or even slightly
decreased for small $f$. This behavior is far more pronounced in the case of
$J_{zoz}$, which shows a clear nonmonotonicity in the behavior of $v_{I}$ as a
function of $J_{zoz}$, at all values of $f$. This behavior is displayed more
clearly in figure 4, and we will discuss it further below.

Figure 4: Left: Information velocity as a function of nonlocal coupling
($J_{xox}$ or $J_{zoz}$) at fixed $f=0$. The dashed lines in this panel are an
independent computation of $v_{E}$ values in the presence of the same nonlocal
couplings. Right: Information velocity as a function of nonlocal coupling
($J_{xox}$ or $J_{zoz}$) at fixed $f=1$. The dashed lines in this panel are an
independent computation of $v_{B}$ values in the presence of the same nonlocal
couplings.

As a confirmation that the information velocity $v_{I}(f)$ interpolates
between $v_{E}$ at $f=0$ and $v_{B}$ at $f=1$, we have computed both of these
velocities independently for the same set of system parameters, using basic
methods. Results are shown as the dashed lines in figure 4. To compute
$v_{E}$, we tracked the rate of growth entanglement entropy on half of the
system, evolving the state $\psi_{L}^{f=0}$ until saturation (see equation
(3)) and performing a linear fit on the early linear growth portion of this
curve. The $v_{B}$ computations consisted of taking the trace of the squared
commutator of local Heisenberg operators over a grid of space and time
displacements, and finding the rate of growth of the region over which the
result is $\mathcal{O}(1)$. In the case of the $v_{E}$ computation, the values
obtained are particular to the state (3) with $f=0$, though they should not
substantially differ from those obtained by averaging over random unentangled
states. On the other hand the $v_{B}$ computation is a state-independent
estimation of the butterfly velocity at infinite temperature, as is
appropriate for comparisons with states of energy $\expectationvalue{H}=0$.
The approximate match of these curves with the behavior of $v_{I}$, shown in
figure 4, is good support for the analysis of Couch:2019zni  and a good
overall consistency check. It is also a confirmation of the non-monotonic
behavior seen in the $v_{I}$ curves, which we now discuss.

The nonmonotonicity evident in figure 4 is somewhat counter-intuitive, as it
indicates that in some cases the inclusion of nonlocal terms in the
Hamiltonian can suppress the rate of information propagation. Such behavior is
not observed in any of the holographic systems investigated in section 4. For
the case of $J_{xox}$ couplings, the suppressive effect is almost small enough
to be dismissed as an artifact of finite size effects or procedural
uncertainties, with $v_{I}$ never falling below about 6 percent less than its
value at $J_{xox}=0$ (though even “flatness” of the $v_{I}(J_{xox})$ curve for
small $J_{xox}$ would be noteworthy). The case of $J_{zoz}$ shows a much
larger suppressive effect, peaking around $J_{zoz}\sim 0.4$, where $v_{I}$ is
nearly 35 percent lower than its value with no nonlocal coupling. We focus on
this effect in the left panel of figure 5, plotting the ratio of the minimum
$v_{I}$ (with respect to $J_{\text{nonlocal}}$) and $v_{I}$ when
$J_{\text{nonlocal}}=0$. The effect is most stark at $f=0$.

In the right hand side of figure 5, we have plotted the ratio $v_{B}/v_{E}$,
as inferred from $v_{I}(f=0)/v_{I}(f=1)$, at various values of nonlocal
coupling. The precise form of the curves emerging from this data are surely
dependent on our specific choices of Hamiltonian and couplings, but we include
this plot for comparison with holographic results (see figures 8 and 11, left
panels), where the analogous ratios are monotonically increasing and
relatively featureless.

Figure 5: Left: Ratio of the minimum information velocity to the information
velocity at zero nonlocal coupling ($J_{xox}=J_{zoz}=0$), denoted $v_{I}^{0}$,
as a function of entanglement fraction $f$. Right: Ratio of the butterfly
velocity to the entanglement velocity as a function of the nonlocal coupling
($J_{xox}$ or $J_{zoz}$).

###  3.2 More nonlocality: fixed width scans

We now move beyond the case of next-nearest neighbor interactions, and
consider adding a higher degree of nonlocality. We fix all
$J_{xox},J_{xoox},...,J_{xo...ox}$ couplings, up to a fixed “width”, to the
same $J_{\text{nonlocal}}$ value and turn them on simultaneously. Results for
$v_{I}$ as a function of $J_{\text{nonlocal}}$ are displayed in 6. In this
case, the affect of additional nonlocal terms is simply to raise the
$v_{I}(J_{\text{nonlocal}})$ curves, moreso with the inclusion of each
additional “width” of nonlocal terms. This also has the affect of washing out
the nonmonotonic behavior observed in section 3.1 and approaching behavior
more similar to the holographic cases we consider in section 4. It is
interesting that the growth of $v_{I}$ begins to level off near
$J_{\text{nonlocal}}=1$. In principle these coupling parameters could be made
arbitrarily large, but we have chosen to limit $J_{\text{nonlocal}}\leq 1$ in
order to leave a comfortable margin before the Hamiltonian level-spacing
departs from the Wigner-Dyson distribution expected of chaotic systems.

Figure 6: Left: Information velocity as a function of nonlocal $\sigma_{x}$
couplings, at $f=0$. Nonlocal terms are turned on up to fixed “width”. Right:
Information velocity as a function of nonlocal $\sigma_{z}$ couplings, at
$f=0$. Nonlocal terms are turned on up to fixed “width”.

###  3.3 Membrane tension function and nonlocality

We now approximate the membrane tension function for the systems of section
3.1 with variable next-nearest neighbor couplings, using the method outlined
in section 2.3.1 with $L=13$. Results are displayed in figure 7. Finite size
affects are substantial, so these curves can only be considered schematic
representations of the effect of the nonlocal coupling on ${\cal E}(v)$ (i.e.
they do not give precise values of $v_{E}={\cal E}(0)$ or $v_{B}={\cal
E}(v_{B})$). They are obtained by computing ${\cal E}_{\text{eff}}(t,v)$ over
a series of $t$ values up to the $v_{B}$ crossing time for each Hamiltonian
and fitting the boundary of the resulting data set with an even polynomial of
degree 12. The central portion (toward $v=0$) of the curves, which in an
infinite system would converge through late time values of ${\cal
E}_{\text{eff}}(t,v)$, tend to be overestimated. This effect is larger for
higher (true) values of $v_{E},v_{B}$, because finite size effects prohibit a
good estimate from ${\cal E}_{\text{eff}}(t,v)$ at much lower values of $t$.
All these caveats aside, in qualitative terms we find good agreement with the
results of the previous section: the effect of adding $J_{xox}$ is initially
fairly flat, but past $J_{xox}\sim 0.25$ dramatically raises ${\cal E}(v)$
upward at all $v$ (entailing an increase of $v_{B}$, $v_{E}$, and $v_{I}$).
Increasing $J_{zoz}$ from zero initially has a small suppressive effect, but
by $J_{zoz}\sim 0.5$ the curve raises to match original values ($J_{zoz}=0$
values) and continues to increase uniformly.

${\cal E}$(v)${\cal E}$(v)

Figure 7: Numerical estimation of the membrane tension function ${\cal E}(v)$
for various the spin chain with a single nonlocal coupling turned on. Left:
The overall affect of increasing the $J_{xox}$ coupling is to raise the
membrane tension function. Right: Increasing the $J_{zoz}$ coupling initially
has a small suppressive effect on the membrane tension function, but beyond
$J_{zoz}\simeq 0.5$ it surpasses the $J_{zoz}=0$ values.

##  4 Information propagation in nonlocal holographic systems

Inspired by the lattice results of the previous section, we now turn our
attention to large-$N$ gauge theories with holographic duals. We will consider
two prototypical theories with different types of nonlocality: non-commutative
SYM (NCSYM) theory and dipole-deformed SYM (DDSYM) theory. The former is an
example of a nonlocal theory with UV/IR mixing. A simple model that realizes
non-commutativity consists of a set of oppositely charged particles (dipoles)
moving in a strong magnetic field Bigatti:1999iz . The UV/IR mixing then
implies that the transverse size of the dipoles grows with their longitudinal
momentum, which may in turn posit obstructions for renormalizability
Minwalla:1999px ; VanRaamsdonk:2000rr . Even though this theory is ultimately
finite, noncommutativity is arguably not the simplest way to introduce
nonlocality. We thus consider a second theory, inspired by Bergman:2000cw ,
which introduces a set of fundamental dipoles of _constant_ length $L$, hence,
resembling more closely the lattice systems studied in section 3. As a result,
one obtains a nonlocal theory without the issue of UV/IR mixing.

In the following, we will review the basic features of the holographic duals
of NCSYM and DDSYM, realized as top-down constructions in string theory. We
will also comment on their finite temperature black brane solutions, which are
needed to tackle the questions pertaining to thermalization and information
spreading.999To our understanding, this is the first time that a black brane
solution is derived for the gravity dual of DDSYM theory. Finally, we will use
the tools developed in 2.3.2 to write down membrane tension functions for both
systems and analyze the results.

###  4.1 Theories of interest

####  4.1.1 Gravity dual of NCSYM

The gravity dual of maximally supersymmetric non-commutative SYM (NCSYM) is
given by type IIB supergravity with non-zero NS-NS $B$-field. Here and below
we assume that the non-commutative parameter is non-vanishing only in the
$(x^{2},x^{3})$-plane, i.e., $[x^{2},x^{3}]\sim i\theta$. This amounts to
replace all multiplication in the Lagrangian of the $\mathcal{N}=4$ SYM theory
with a noncommutative star product

$(f\star
g)(x_{2},x_{3})=e^{\frac{i}{2}\theta\left(\partial_{\xi_{2}}\partial_{\zeta_{3}}-\partial_{\xi_{3}}\partial_{\zeta_{2}}\right)}f(x_{2}+\xi_{2},x_{3}+\xi_{3})g(x_{2}+\zeta_{2},x_{3}+\zeta_{3})|_{\xi_{2}=\xi_{3}=\zeta_{2}=\zeta_{3}=0}\,.$
(51)

The gravity background dual to this theory, in the string frame is
Hashimoto:1999ut ; Maldacena:1999mh :

$\displaystyle ds_{\textrm{\tiny st}}^{2}$ $\displaystyle=$ $\displaystyle
R^{2}\left[r^{2}\left(-f(r)dt^{2}+dx_{1}^{2}+h(r)\left(dx_{2}^{2}+dx_{3}^{2}\right)\right)+\frac{dr^{2}}{r^{2}f(r)}+d\tilde{\Omega}_{5}^{2}\right],$
$\displaystyle e^{2\Phi}$ $\displaystyle=$ $\displaystyle{\hat{g}}^{2}h(r)\,,$
$\displaystyle B_{23}$ $\displaystyle=$ $\displaystyle R^{2}a^{2}r^{4}h(r)\,,$
(52) $\displaystyle C_{01}$ $\displaystyle=$
$\displaystyle\frac{R^{2}a^{2}}{\hat{g}}r^{4}\,,$ $\displaystyle F_{0123r}$
$\displaystyle=$ $\displaystyle\frac{4R^{4}}{{\hat{g}}}r^{3}h(r)\,,$

with

$f(r)=1-\left(\frac{r_{h}}{r}\right)^{4}\,,\qquad r_{h}=\pi T\,,$ (53)

and

$h(r)=\frac{1}{1+a^{4}r^{4}}\,,\qquad a=\lambda^{1/4}\sqrt{\theta}\,.$ (54)

In the above $d\tilde{\Omega}_{5}^{2}$ denotes the metric on a unit $S^{5}$,
$R^{4}=4\pi\hat{g}N\alpha^{\prime 2}$, $\hat{g}$ denotes the string coupling
and $\alpha^{\prime}$ is the string tension, which is related to the ’t Hooft
coupling through the standard relation $\sqrt{\lambda}=R^{2}/\alpha^{\prime}$.

To use the results for the membrane theory we first express the metric in
Einstein frame,101010The metrics in the Einstein and string frames are related
via the standard relation: $ds_{\textrm{\tiny
E}}^{2}=e^{-\Phi/2}ds_{\textrm{\tiny st}}^{2}$.

$ds_{\textrm{\tiny
E}}^{2}=\frac{R^{2}}{{\hat{g}}^{1/2}h(r)^{1/4}}\left[r^{2}\left(-f(r)dt^{2}+dx_{1}^{2}+h(r)\left(dx_{2}^{2}+dx_{3}^{2}\right)\right)+\frac{dr^{2}}{r^{2}f(r)}+d\tilde{\Omega}_{5}^{2}\right]\,,$
(55)

and go to a radial coordinate $z=1/r$, so that:

$ds_{\textrm{\tiny
E}}^{2}=\frac{R^{2}}{{\hat{g}}^{1/2}h(z)^{1/4}z^{2}}\left[-f(z)dt^{2}+dx_{1}^{2}+h(z)\left(dx_{2}^{2}+dx_{3}^{2}\right)+\frac{dz^{2}}{f(z)}+z^{2}d\tilde{\Omega}_{5}^{2}\right]\,,$
(56)

where

$f(z)=1-\left(\frac{z}{z_{h}}\right)^{4}\,,\qquad z_{h}=\frac{1}{\pi T}\,,$
(57)

and

$h(z)=\frac{1}{1+\left(\frac{a}{z}\right)^{4}}\,,\qquad
a=\lambda^{1/4}\sqrt{\theta}\,.$ (58)

We further define the infalling time coordinate $v$ such that

$dt=dv+\frac{dz}{f(z)}\,,$ (59)

hence the metric becomes:

$ds_{\textrm{\tiny
E}}^{2}=\frac{R^{2}}{{\hat{g}}^{1/2}h(z)^{1/4}z^{2}}\left[-f(z)dv^{2}-2dvdz+dx_{1}^{2}+h(z)\left(dx_{2}^{2}+dx_{3}^{2}\right)+z^{2}d\tilde{\Omega}_{5}^{2}\right]\,.$
(60)

This metric is of the form (29), with the following identifications:

$g_{vv}=\frac{R^{2}f(z)}{{\hat{g}}^{1/2}h(z)^{1/4}z^{2}}\,,\quad
g_{vz}=g_{11}=\frac{R^{2}}{{\hat{g}}^{1/2}h(z)^{1/4}z^{2}}\,,\quad
g_{22}=g_{33}=\frac{R^{2}h(z)^{3/4}}{{\hat{g}}^{1/2}z^{2}}\,.$ (61)

Furthermore, the metric of the compact space is

$d\Omega_{5}^{2}=\frac{R^{2}}{{\hat{g}}^{1/2}h(z)^{1/4}}d\tilde{\Omega}_{5}^{2}\,,$
(62)

so we obtain that

$V_{\Omega}(z)=\frac{R^{5}}{{\hat{g}}^{5/4}h(z)^{5/8}}\,,\qquad\text{Vol}(\tilde{\Omega}_{5})=\pi^{3}\,.$
(63)

With the above identifications, we are now ready to apply our general results
of section 2.3.2 specializing to this system.

####  4.1.2 Gravity dual of DDSYM

The gravity dual of the dipole deformed maximally supersymmetric SYM (DDSYM)
is given by type IIB supergravity with non-zero NS-NS $B$-field and a deformed
compact space. Here and below we assume that the dipoles have fixed length $L$
and are all oriented along the $x^{3}$-direction. This amounts to replace all
multiplication in the Lagrangian of the $\mathcal{N}=4$ SYM theory with the
star product

$(f\star
g)(x_{3})=f\left(x_{3}-\frac{L}{2}\right)g\left(x_{3}+\frac{L}{2}\right)\,.$
(64)

The gravity background dual to this theory, in the string frame is
Bergman:2001rw :111111The metric presented in Bergman:2001rw  is the zero
temperature version of (4.1.2), where $f(r)=1$. To our understanding, this is
the first time that a black brane solution was derived for the gravity dual of
the dipole theory.

$\displaystyle ds_{\textrm{\tiny st}}^{2}$ $\displaystyle=$ $\displaystyle
R^{2}\left[r^{2}\left(-f(r)dt^{2}+dx_{1}^{2}+dx_{2}^{2}+h(r)dx_{3}^{2}\right)+\frac{dr^{2}}{r^{2}f(r)}+d\mathrm{\bar{\Omega}}_{5}^{2}\right],$
$\displaystyle e^{2\Phi}$ $\displaystyle=$ $\displaystyle{\hat{g}}^{2}h(r)\,,$
$\displaystyle B_{3\psi}$ $\displaystyle=$ $\displaystyle
R^{2}b\,r^{2}h(r)\,,$ (65) $\displaystyle F_{0123r}$ $\displaystyle=$
$\displaystyle 4R^{4}r^{3}\,,$

with

$f(r)=1-\left(\frac{r_{h}}{r}\right)^{4}\,,\qquad r_{h}=\pi T\,,$ (66)

and

$h(r)=\frac{1}{1+b^{2}r^{2}}\,,\qquad b=\frac{\lambda^{1/2}L}{2\pi}\,.$ (67)

In the above $d\bar{\Omega}_{5}^{2}$ denotes the metric on a _deformed_ unit
$S^{5}$. This compact space has the structure of an $S^{1}$ (Hopf) fibration
over a base $\mathbb{C}\mathbb{P}^{2}$. The global angular 1-form of the Hopf
fibration is denoted as $\psi$. This fiber acquires an $r$-dependent radius
$h(r)^{1/2}$. The volume of the $\mathbb{C}\mathbb{P}^{2}$ is constant and
given by $\pi^{2}/2$, so the total volume of the compact manifold is
$\pi^{3}h(r)^{1/2}$.

To use the results for the membrane theory we first express the metric in the
Einstein frame

$ds_{\textrm{\tiny
E}}^{2}=\frac{R^{2}}{{\hat{g}}^{1/2}h(r)^{1/4}}\left[r^{2}\left(-f(r)dt^{2}+dx_{1}^{2}+dx_{2}^{2}+h(r)dx_{3}^{2}\right)+\frac{dr^{2}}{r^{2}f(r)}+d\mathrm{\bar{\Omega}}_{5}^{2}\right]\,,$
(68)

and go to a radial coordinate $z=1/r$:

$ds_{\textrm{\tiny
E}}^{2}=\frac{R^{2}}{{\hat{g}}^{1/2}h(z)^{1/4}z^{2}}\left[-f(z)dt^{2}+dx_{1}^{2}+dx_{2}^{2}+h(z)dx_{3}^{2}+\frac{dz^{2}}{f(z)}+z^{2}d\bar{\Omega}_{5}^{2}\right]\,,$
(69)

where

$f(z)=1-\left(\frac{z}{z_{h}}\right)^{4}\,,\qquad z_{h}=\frac{1}{\pi T}\,,$
(70)

and

$h(z)=\frac{1}{1+\left(\frac{b}{z}\right)^{2}}\,,\qquad
b=\frac{\lambda^{1/2}L}{2\pi}\,.$ (71)

We further define the infalling time coordinate $v$ such that

$dt=dv+\frac{dz}{f(z)}\,,$ (72)

hence the metric becomes:

$ds_{\textrm{\tiny
E}}^{2}=\frac{R^{2}}{{\hat{g}}^{1/2}h(z)^{1/4}z^{2}}\left[-f(z)dv^{2}-2dvdz+dx_{1}^{2}+dx_{2}^{2}+h(z)dx_{3}^{2}+z^{2}d\bar{\Omega}_{5}^{2}\right]\,.$
(73)

This metric is of the form (29), with the following identifications:

$g_{vv}=\frac{R^{2}f(z)}{{\hat{g}}^{1/2}h(z)^{1/4}z^{2}}\,,\quad
g_{vz}=g_{11}=g_{22}=\frac{R^{2}}{{\hat{g}}^{1/2}h(z)^{1/4}z^{2}}\,,\quad
g_{33}=\frac{R^{2}h(z)^{3/4}}{{\hat{g}}^{1/2}z^{2}}\,.$ (74)

Furthermore, the metric of the compact space is

$d\Omega_{5}^{2}=\frac{R^{2}}{{\hat{g}}^{1/2}h(z)^{1/4}}d\bar{\Omega}_{5}^{2}\,.$
(75)

From the discussion below (67) we know that
$\text{Vol}(\bar{\Omega}_{5})=\pi^{3}h(z)^{1/2}$. Extracting all $z$-dependent
terms into $V_{\Omega}(z)$, we obtain that

$V_{\Omega}(z)=\frac{R^{5}}{{\hat{g}}^{5/4}h(z)^{1/8}}\,,\qquad\text{Vol}(\tilde{\Omega}_{5})=\pi^{3}\,.$
(76)

Again, with these identifications, we can now apply our general results of
section 2.3.2 specializing to this system.

###  4.2 Membrane theory and results for information spreading

####  4.2.1 Membrane theory for NCYSM

We can analyze two cases, depending on the orientation of the strip. In both
cases we find:

$P(\zeta)=\frac{R^{16}}{{\hat{g}}^{4}\zeta^{6}}\,,$ (77)

however, the membrane tensions differ.

  * •

Commutative strip ($\xi=\xi_{1}$ and
$\vec{\xi}_{\perp}=\\{\xi_{2},\xi_{3}\\}$)

First, note that (36) can be written as:

$v^{2}=c_{\text{c}}(\zeta)\,,\qquad
c_{\text{c}}(\zeta)=f(\zeta)-\frac{1}{6}\zeta
f^{\prime}(\zeta)=1-\frac{\zeta^{4}}{3\zeta_{h}^{4}}\,,$ (78)

and this equation can be inverted analytically to give:

$\zeta_{\text{c}}=c_{\text{c}}^{-1}(v^{2})=\zeta_{h}\left[3(1-v^{2})\right]^{1/4}\,.$
(79)

We have added a subindices “c” to indicate that we are working out the
commutative strip. Finally, the membrane tension is

${\cal
E}_{\text{c}}\left(v\right)=\zeta_{h}^{3}\sqrt{-\frac{f^{\prime}(\zeta)}{6\zeta^{5}}}\Bigg{|}_{\zeta_{\text{c}}}=\sqrt{\frac{2}{3}}\frac{\zeta_{h}}{\zeta}\Bigg{|}_{\zeta_{\text{c}}}=\frac{\sqrt{2}}{3^{3/4}(1-v^{2})^{1/4}}\,.$
(80)

This is exactly the same result for a strip in Schwarzschild-AdS5. It
satisfies all the desired properties, including:

${\cal E}_{\text{c}}(0)=v_{E}\,,\qquad{\cal
E}_{\text{c}}^{\prime}(0)=0\,,\qquad{\cal
E}_{\text{c}}(v_{B})=v_{B}\,,\qquad{\cal E}_{\text{c}}^{\prime}(v_{B})=1\,,$
(81)

with

$v_{E}^{\text{(c)}}=\frac{\sqrt{2}}{3^{3/4}}\,,\qquad
v_{B}^{\text{(c)}}=\sqrt{\frac{2}{3}}\,.$ (82)

  * •

Non-commutative strip ($\xi=\xi_{2}$ and
$\vec{\xi}_{\perp}=\\{\xi_{1},\xi_{3}\\}$)

In this case (36) can be written as:

$v^{2}=c_{\text{nc}}(\zeta)\,,\qquad
c_{\text{nc}}(\zeta)=\frac{f(\zeta)}{h(\zeta)}-\frac{\zeta
f^{\prime}(\zeta)}{6h(\zeta)}+\frac{\zeta
f(\zeta)h^{\prime}(\zeta)}{6h(\zeta)^{2}}=1-\frac{\zeta^{4}}{3\zeta_{h}^{4}}+\frac{5a^{4}}{3\zeta^{4}}-\frac{a^{4}}{\zeta_{h}^{4}}\,,$
(83)

and can also be inverted analytically to yield:121212There are 8 roots (4
imaginary and 4 real), but only one coincides with the commutative case in the
limit $a\to 0$.

$\zeta_{\text{nc}}=c_{\text{nc}}^{-1}(v^{2})=\frac{\zeta_{h}}{2^{1/4}}\left[3\left(1-v^{2}-\frac{a^{4}}{\zeta_{h}^{4}}\right)+\sqrt{9\left(1-v^{2}-\frac{a^{4}}{\zeta_{h}^{4}}\right)^{2}+\frac{20a^{4}}{\zeta_{h}^{4}}}\right]^{1/4}\,.$
(84)

We have added a subindices “nc” to indicate that we are working out the non-
commutative strip. Finally, the membrane tension is

$\displaystyle{\cal E}_{\text{nc}}\left(v\right)$ $\displaystyle=$
$\displaystyle\zeta_{h}^{3}\sqrt{-\frac{f^{\prime}(\zeta)h(\zeta)-f(\zeta)h^{\prime}(\zeta)}{6\zeta^{5}h(\zeta)^{2}}}\Bigg{|}_{\zeta_{\text{nc}}}=\sqrt{\frac{2}{3}\left(1+\frac{a^{4}\zeta_{h}^{4}}{\zeta^{8}}\right)}\frac{\zeta_{h}}{\zeta}\Bigg{|}_{\zeta_{\text{nc}}}$
(85) $\displaystyle=$
$\displaystyle\frac{2^{3/4}\\!\\!\left[\left(3\left(1-v^{2}-\frac{a^{4}}{\zeta_{h}^{4}}\right)+\sqrt{9\left(1-v^{2}-\frac{a^{4}}{\zeta_{h}^{4}}\right)^{2}+\frac{20a^{4}}{\zeta_{h}^{4}}}\right)^{2}+\frac{4a^{4}}{\zeta_{h}^{4}}\right]^{1/2}}{\sqrt{3}\left[3\left(1-v^{2}-\frac{a^{4}}{\zeta_{h}^{4}}\right)+\sqrt{9\left(1-v^{2}-\frac{a^{4}}{\zeta_{h}^{4}}\right)^{2}+\frac{20a^{4}}{\zeta_{h}^{4}}}\right]^{5/4}}\,.$

We can check that in the $a\to 0$ limit we recover (80). It also satisfies all
the desired properties:

${\cal E}_{\text{nc}}(0)=v_{E}\,,\qquad{\cal
E}_{\text{nc}}^{\prime}(0)=0\,,\qquad{\cal
E}_{\text{nc}}(v_{B})=v_{B}\,,\qquad{\cal E}_{\text{nc}}^{\prime}(v_{B})=1\,,$
(86)

now with

$v_{E}^{\text{(nc)}}=\frac{2^{3/4}\\!\\!\left[\left(3\left(1-\frac{a^{4}}{\zeta_{h}^{4}}\right)+\sqrt{9\left(1-\frac{a^{4}}{\zeta_{h}^{4}}\right)^{2}+\frac{20a^{4}}{\zeta_{h}^{4}}}\right)^{2}+\frac{4a^{4}}{\zeta_{h}^{4}}\right]^{1/2}}{\sqrt{3}\left[3\left(1-\frac{a^{4}}{\zeta_{h}^{4}}\right)+\sqrt{9\left(1-\frac{a^{4}}{\zeta_{h}^{4}}\right)^{2}+\frac{20a^{4}}{\zeta_{h}^{4}}}\right]^{5/4}}\,,\qquad
v_{B}^{\text{(nc)}}=\sqrt{\frac{2}{3}\left(1+\frac{a^{4}}{\zeta_{h}^{4}}\right)}\,.$
(87)

Note that in all the above formulas, the non-commutative parameter $\theta$
only appears through the dimensionless combination:

$\vartheta\equiv\frac{a^{4}}{\zeta_{h}^{4}}=\lambda\pi^{4}T^{4}\theta^{2}\,.$
(88)

We can verify that $v_{B}^{\text{(nc)}}>v_{E}^{\text{(nc)}}$ for all
$\vartheta$ Fischler:2018kwt . Also, both $v_{E}^{\text{(nc)}}\to\infty$ and
$v_{B}^{\text{(nc)}}\to\infty$ as $\vartheta\to\infty$, but this is the
expected behavior for an “infinitely” nonlocal theory. Even so, they diverge
with the same power of $\vartheta$ ($v_{B}^{\text{(nc)}}\propto\vartheta^{2}$,
$v_{E}^{\text{(nc)}}\propto\vartheta^{2}$) so their ratio remains finite.
Expanding this ratio in the limits of strong and weak non-commutativity, we
obtain:

$\frac{v_{B}^{\text{(nc)}}}{v_{E}^{\text{(nc)}}}=\begin{cases}\displaystyle
3^{1/4}\left[1+\frac{\vartheta}{3}+\mathcal{O}(\vartheta^{2})\right]&\displaystyle\qquad(\vartheta\ll
1)\,,\\\\[12.91663pt]
\displaystyle\left(\frac{5}{3}\right)^{5/4}\left[1-\frac{1}{3\vartheta}+\mathcal{O}\left(\frac{1}{\vartheta^{2}}\right)\right]&\displaystyle\qquad(\vartheta\gg
1)\,.\end{cases}$ (89)

These results are illustrated in figure 8. From the plots we can confirm the
monotonicity of $v_{B}^{\text{(nc)}}/v_{E}^{\text{(nc)}}$ with the strength of
the non-commutative parameter. More specifically, the ratio interpolates
between $v_{B}^{\text{(nc)}}/v_{E}^{\text{(nc)}}\to 3^{1/4}\approx 1.316$ (for
$\vartheta\to 0$) to
$v_{B}^{\text{(nc)}}/v_{E}^{\text{(nc)}}\to(5/3)^{5/4}\approx 1.894$ (for
$\vartheta\to\infty$).

$v_{B}^{\text{(nc)}}/v_{E}^{\text{(nc)}}$$\mathcal{E}_{\text{nc}}(v)$$\vartheta$$v$

Figure 8: Left: ratio between the butterfly velocity and the entanglement
velocity as a function of the dimensionless combination
$\vartheta=\lambda\pi^{4}T^{4}\theta^{2}$. We also show the asymptotic value
$v_{B}/v_{E}\to(5/3)^{5/4}\approx 1.894$, depicted in orange. Right: Membrane
tensions for $\vartheta\in\\{0,1/2,1,2,5,10\\}$, from bottom to top,
respectively. In each case, we indicate the point at which the curve touches
the straight line $\mathcal{E}(v_{B})=v_{B}$.

#### Information velocity

Finally, we can obtain the information velocity $v_{I}$ as a function of the
entanglement fraction $f$ for . In order to do so, invert equation (47) (to
obtain $v_{I}(f)$) and then plug this into (48). Unfortunately, it is not
possible to invert (47) analytically, even for the commutative strip. So we
proceed numerically. In figure 9 we show our findings for this quantity. In
general we obtain that the $v_{I}$ increases monotonically both with $f$ and
with $\vartheta$, as opposed to the 1-D spin chain systems we studied
previously. We believe that, in fact, large-$N$ is responsible for washing out
the initial suppression of $v_{I}$ with respect to the nonlocal scale.

$v_{I}^{\text{(c)}}$$v_{I}^{\text{(nc)}}$$f$$f$

Figure 9: Left: Information velocity as a function of the entanglement
fraction $f$ for the commutative strip. We also show the limiting values,
$v_{I}^{\text{(c)}}(f=0)=v_{E}^{\text{(c)}}$ and
$v_{I}^{\text{(c)}}(f=1)=v_{B}^{\text{(c)}}$, depicted in red and orange,
respectively. Right: Information velocity for the non-commutative strip as a
function of $f$ for various values of $\vartheta\in\\{0,1/2,1,2,5,10\\}$, from
bottom to top, respectively.

####  4.2.2 Membrane theory for DDSYM

We can analyze two cases, depending on the orientation of the strip. In both
cases we find:

$P(\zeta)=\frac{R^{16}}{{\hat{g}}^{4}\zeta^{6}}\,,$ (90)

however, the membrane tensions differ.

  * •

Standard strip ($\xi=\xi_{1}$ and $\vec{\xi}_{\perp}=\\{\xi_{2},\xi_{3}\\}$)

This is completely analogous to the commutative strip. Here we repeat the
analysis for completeness. First, note that (36) can be written as:

$v^{2}=c_{\text{s}}(\zeta)\,,\qquad
c_{\text{s}}(\zeta)=f(\zeta)-\frac{1}{6}\zeta
f^{\prime}(\zeta)=1-\frac{\zeta^{4}}{3\zeta_{h}^{4}}\,,$ (91)

and this equation can be inverted analytically to give:

$\zeta_{\text{s}}=c_{\text{s}}^{-1}(v^{2})=\zeta_{h}\left[3(1-v^{2})\right]^{1/4}\,.$
(92)

We have added a subindices “s” to indicate that we are working out the
standard strip. Finally, the membrane tension is

${\cal
E}_{\text{s}}\left(v\right)=\zeta_{h}^{3}\sqrt{-\frac{f^{\prime}(\zeta)}{6\zeta^{5}}}\Bigg{|}_{\zeta_{\text{s}}}=\sqrt{\frac{2}{3}}\frac{\zeta_{h}}{\zeta}\Bigg{|}_{\zeta_{\text{s}}}=\frac{\sqrt{2}}{3^{3/4}(1-v^{2})^{1/4}}\,.$
(93)

This is also the exact same result for a strip in Schwarzschild-AdS5. It
satisfies all the desired properties, including:

${\cal E}_{\text{s}}(0)=v_{E}\,,\qquad{\cal
E}_{\text{s}}^{\prime}(0)=0\,,\qquad{\cal
E}_{\text{s}}(v_{B})=v_{B}\,,\qquad{\cal E}_{\text{s}}^{\prime}(v_{B})=1\,,$
(94)

with

$v_{E}^{\text{(s)}}=\frac{\sqrt{2}}{3^{3/4}}\,,\qquad
v_{B}^{\text{(s)}}=\sqrt{\frac{2}{3}}\,.$ (95)

  * •

Dipole oriented strip ($\xi=\xi_{3}$ and
$\vec{\xi}_{\perp}=\\{\xi_{1},\xi_{2}\\}$)

In this case (36) can be written as:

$v^{2}=c_{\text{d}}(\zeta)\,,\qquad
c_{\text{d}}(\zeta)=\frac{f(\zeta)}{h(\zeta)}-\frac{\zeta
f^{\prime}(\zeta)}{6h(\zeta)}+\frac{\zeta
f(\zeta)h^{\prime}(\zeta)}{6h(\zeta)^{2}}=1-\frac{\zeta^{4}}{3\zeta_{h}^{4}}+\frac{4b^{2}}{3\zeta^{2}}-\frac{2b^{2}z^{2}}{3\zeta_{h}^{4}}\,.$
(96)

This expression can also be inverted analytically.131313There are 6 roots (4
complex and 2 real), but only one coincides with the standard case in the
limit $b\to 0$. However, the final expression is lengthy and not very
illuminating so we will not transcribe it here. To have a flavor of the
effects due to nonlocality, we expand the resulting functions in two limits.
Defining a dimensionless parameter,

$\delta\equiv\frac{b^{2}}{\zeta_{h}^{2}}=\frac{\lambda L^{2}T^{2}}{4}\,,$ (97)

we find that

$\zeta_{\text{d}}=c_{\text{d}}^{-1}(v^{2})=\begin{cases}\displaystyle\zeta_{h}\left[3(1-v^{2})\right]^{1/4}\left[1-\frac{(1-3v^{2})\delta}{6\sqrt{3}(1-v^{2})^{3/2}}+\mathcal{O}\left(\delta^{2}\right)\right]&\displaystyle\qquad(\delta\ll
1)\,,\\\\[12.91663pt]
\displaystyle\zeta_{h}\,2^{1/4}\left[1+\frac{(1-3v^{2})}{8\sqrt{2}\delta}+\mathcal{O}\left(\frac{1}{\delta^{2}}\right)\right]&\displaystyle\qquad(\delta\gg
1)\,.\end{cases}$ (98)

We have added a subindices ‘d” to indicate that we are working out the dipole
oriented strip. In this case, the membrane tension is

$\displaystyle{\cal E}_{\text{d}}\left(v\right)$ $\displaystyle=$
$\displaystyle\zeta_{h}^{3}\sqrt{-\frac{f^{\prime}(\zeta)h(\zeta)-f(\zeta)h^{\prime}(\zeta)}{6\zeta^{5}h(\zeta)}}\Bigg{|}_{\zeta_{\text{d}}}=\sqrt{\frac{2}{3}\left[1+\frac{\zeta_{h}^{2}(\zeta^{4}+\zeta_{h}^{4})\delta}{2\zeta^{6}}\right]}\frac{\zeta_{h}}{\zeta}\Bigg{|}_{\zeta_{\text{d}}}\,,$
(99) $\displaystyle=$
$\displaystyle\begin{cases}\displaystyle\frac{\sqrt{2}}{3^{3/4}(1-v^{2})^{1/4}}\left[1+\frac{(2-3v^{2})\delta}{4\sqrt{3}(1-v^{2})^{3/2}}+\mathcal{O}(\delta^{2})\right]&\displaystyle\qquad(\delta\ll
1)\,,\\\\[12.91663pt]
\displaystyle\frac{\sqrt{\delta}}{2}\left[1+\frac{(1+v^{2})}{\sqrt{2}\delta}+\mathcal{O}\left(\frac{1}{\delta^{2}}\right)\right]&\displaystyle\qquad(\delta\gg
1)\,.\end{cases}$

We can check that in the $\delta\to 0$ limit we recover (93). At finite
$\delta$ we can also verify (numerically, or analytically in some limits) that
is satisfies all the desired properties:

${\cal E}_{\text{nc}}(0)=v_{E}\,,\qquad{\cal
E}_{\text{nc}}^{\prime}(0)=0\,,\qquad{\cal
E}_{\text{nc}}(v_{B})=v_{B}\,,\qquad{\cal E}_{\text{nc}}^{\prime}(v_{B})=1\,,$
(100)

now with

$v_{E}^{\text{(d)}}$$v_{B}^{\text{(d)}}$$\delta$$\delta$

Figure 10: Left: Entanglement velocity and Right: Butterfly velocity as a
function of the dimensionless combination $\delta=\lambda L^{2}T^{2}/4$. In
both plots we have shown in orange the standard values
$v_{E}=\sqrt{2}/3^{3/4}\approx 0.620$ and $v_{B}=\sqrt{2/3}\approx 0.816$
obtained in the limit $\delta\to 0$, and the expansions for $\delta\ll 1$ and
$\delta\gg 1$, depicted in red. We notice that both velocities become
superluminal at around $\delta\sim\mathcal{O}(1)$.

$v_{E}^{\text{(d)}}=\begin{cases}\displaystyle\frac{\sqrt{2}}{3^{3/4}}\left[1+\frac{\delta}{2\sqrt{3}}+\mathcal{O}(\delta^{2})\right]&\displaystyle\qquad(\delta\ll
1)\,,\\\\[12.91663pt]
\displaystyle\frac{\sqrt{\delta}}{2}\left[1+\frac{1}{\sqrt{2}\delta}+\mathcal{O}\left(\frac{1}{\delta^{2}}\right)\right]&\displaystyle\qquad(\delta\gg
1)\,,\end{cases}$ (101)

and

$v_{B}^{\text{(d)}}=\begin{cases}\displaystyle\sqrt{\frac{2}{3}}\left[1+\frac{\delta}{2}+\mathcal{O}(\delta^{2})\right]&\displaystyle\qquad(\delta\ll
1)\,,\\\\[12.91663pt]
\displaystyle\sqrt{\frac{2\delta}{3}}\left[1+\frac{1}{2\delta}+\mathcal{O}\left(\frac{1}{\delta^{2}}\right)\right]&\displaystyle\qquad(\delta\gg
1)\,.\end{cases}$ (102)

In fact, we can obtain a very compact expression for $v_{B}$ by noticing that
it can alternatively be obtained from (27) and (96):

$v_{B}=c(\zeta_{h})^{1/2}=\sqrt{\frac{2}{3}(1+\delta)}\,.$ (103)

Unfortunately, if we try to use (28) to recover $v_{E}$ we run into similar
problems as before, since the expression for the Hartman-Maldacena
$\zeta_{\text{HM}}$ turns out to have a very complicated dependence with
respect to $\delta$. We will therefore refrain from writing down a closed
expression for $v_{E}$. Figure 10 illustrate our main findings. In general, we
notice that both (101) and (102) provide very good fits in their regime of
applicability. We also observe that both velocities undergo a transition from
subluminal to superluminal in the intermediate regime, where
$\delta\sim\mathcal{O}(1)$ and ultimately diverge in the limit of very strong
nonlocality $\delta\to\infty$. This is in qualitative agreement with the
results we obtained for the NCSYM theory. Finally, we can also verify that
$v_{B}^{\text{(d)}}>v_{E}^{\text{(d)}}$ for all $\delta$. Since both
quantities diverge with the same power of $\delta$
($v_{B}^{\text{(d)}}\propto\delta^{1/2}$,
$v_{E}^{\text{(d)}}\propto\delta^{1/2}$) their ratio remains finite at large
$\delta$. Expanding this ratio in the two limits we obtain:

$\frac{v_{B}^{\text{(d)}}}{v_{E}^{\text{(d)}}}=\begin{cases}\displaystyle
3^{1/4}\left[1+\frac{(3-\sqrt{3})\delta}{6}+\mathcal{O}(\delta^{2})\right]&\displaystyle\qquad(\delta\ll
1)\,,\\\\[12.91663pt] \displaystyle
2\sqrt{\frac{2}{3}}\left[1-\frac{(\sqrt{2}-1)}{2\delta}+\mathcal{O}\left(\frac{1}{\delta^{2}}\right)\right]&\displaystyle\qquad(\delta\gg
1)\,.\end{cases}$ (104)

These results are illustrated in figure 11. From the plots we can confirm the
monotonicity of $v_{B}^{\text{(d)}}/v_{E}^{\text{(d)}}$ with the strength of
nonlocality. More specifically, the ratio interpolates between
$v_{B}^{\text{(d)}}/v_{E}^{\text{(d)}}\to 3^{1/4}\approx 1.316$ (for
$\delta\to 0$) to $v_{B}^{\text{(d)}}/v_{E}^{\text{(d)}}\to 2\sqrt{2/3}\approx
1.633$ (for $\delta\to\infty$). We see here that the nonlocal effects in the
NC case are a bit more severe (in that case
$v_{B}^{\text{(nc)}}/v_{E}^{\text{(nc)}}\to(5/3)^{5/4}\approx 1.894$ as
$\vartheta\to\infty$), which presumably can be traced back to the fact that in
that case the nonlocality affects two of the spatial directions, while for the
dipole theory only one of the directions.

$v_{B}^{\text{(d)}}/v_{E}^{\text{(d)}}$$\mathcal{E}_{\text{d}}(v)$$\delta$$v$

Figure 11: Left: ratio between the butterfly velocity and the entanglement
velocity as a function of the dimensionless combination $\delta=\lambda
L^{2}T^{2}/4$. We also show the asymptotic value $v_{B}/v_{E}\to
2\sqrt{2/3}\approx 1.633$, depicted in orange, and the expansions for
$\delta\ll 1$ and $\delta\gg 1$, depicted in red. Right: Membrane tensions for
$\delta\in\\{0,1/2,1,2,5,10\\}$, from bottom to top, respectively. In each
case, we indicate the point at which the curve touches the straight line
$\mathcal{E}(v_{B})=v_{B}$.

#### Information velocity

Finally, we can obtain the information velocity $v_{I}$ as a function of the
entanglement fraction $f$ for the DDSYM theory. In order to do so, invert
equation (47) (to obtain $v(f)$) and then plug this into (48). Again, we
proceed numerically here because it is not possible to invert (47) even for
the standard strip. In figure 12 we show our findings for this quantity. We
obtain that the $v_{I}$ increases monotonically both with $f$ and with
$\delta$, which was also found for the NCSYM system. This suggests that in
large-$N$ systems the interference effects between the local and nonlocal
couplings are effectively washed out.

$v_{I}^{\text{(s)}}$$v_{I}^{\text{(d)}}$$f$$f$

Figure 12: Left: Information velocity as a function of the entanglement
fraction $f$ for the standard strip. This reproduces the result for the
commutative strip in figure 9. We also show the limiting values,
$v_{I}^{\text{(s)}}(f=0)=v_{E}^{\text{(s)}}$ and
$v_{I}^{\text{(s)}}(f=1)=v_{B}^{\text{(s)}}$, depicted in red and orange,
respectively. Right: Information velocity for the dipole oriented strip as a
function of $f$ for various values of $\delta\in\\{0,1/2,1,2,5,10\\}$, from
bottom to top, respectively. This quantity behaves qualitatively the same as
for the non-commutative strip, provided we identify the values of
$\delta\leftrightarrow\vartheta$. However, there are very minor numerical
differences for $f<1$ which make the curves for the dipole case slightly more
concave. For $f=1$ they are exactly the same, because the functional
dependence of $v_{B}$ with respect to $\delta$ or $\vartheta$ is the same for
both theories.

##  5 Conclusions and future work

In this paper we have considered the possibility of speeding up the
information transfer in a variety of systems by turning on mild nonlocalities.
Our studies include 1-dimensional spin chains as well as strongly-coupled,
large-$N$ theories with classical holographic duals. The nonlocal interactions
that we consider only act below certain length scale and hence, and can be
integrated out by a coarse groaning procedure. Thus, in the large-distance,
late-time regime we do recover an effective notion of locality. Nevertheless,
we find that these nonlocal interactions induce an clear imprint in this
regime, which in most cases translates to an enhancement in the rates of
information transfer. Known systems that mimic this type of interactions
include certain materials with high polarizability, which in the presence of a
strong electromagnetic field undergo an ordered phase with their molecular
dipoles aligned. We thus believe that our results could have practical
applications in the areas of quantum computation and quantum communications.

The first part of the paper focused on the study of 1-dimensional spin chain
systems with the addition of next-nearest neighbor and next-next-nearest
neighbor interactions. We have computed the information velocity for states of
uniform entanglement fraction, which confers not only the rate at which
information propagates in these systems, but also the entanglement velocity
$v_{E}$ and the butterfly velocity $v_{B}$, characterizing the rates of
entanglement generation and operator growth, respectively. Our results provide
further support for the theory of information spreading of Couch:2019zni  and
extend it to systems with mild nonlocalities. We find that the addition of
these extra couplings have a nontrivial effect on the information velocity,
either suppressive (for very weak nonlocality), or enhancing $v_{I}$, though
the generic behavior seems to be to raise $v_{I}$ (and $v_{E}$, $v_{B}$). The
second part of the paper focused on holographic systems dual to strongly-
coupled nonlocal gauge theories. In this case, we found that the suppressive
behavior is completely washed out, in agreement with the earlier results in
Fischler:2018kwt . This behavior can possibly be explained as an effect of the
large-$N$ limit. Indeed, preliminary results suggest that increasing the size
of the local Hilbert space in the spin chain systems (e.g. considering qudits
instead of qubits) seems to suppress the interference effects workinprogress .

Some interesting directions for future work include:

  1. 1.

Investigate the origin and nature of the interference effect found for some
instances of small nonlocal couplings (see section 3.1) and understand how to
maximize the information velocity $v_{I}$ from first principles.

  2. 2.

Explore the precise form of the information cone wavefront in the systems
studied in this paper, along the lines of the $v_{B}$ wavefront analysis of
Xu:2018dfp .

  3. 3.

Test the effects of other forms of nonlocality chosen to match experimentally
realizable setups, and explore more in detail possible practical applications.

  4. 4.

Investigate the effect of increasing the size of the local Hilbert space (e.g.
qudits instead of qubits) to see what aspects of the phenomenology are brought
closer to the large-$N$ limit represented in holographic systems
workinprogress .

  5. 5.

Study the universality of our holographic results (i.e. the suppression of the
interference effects) in other nonlocal models. As an example, one may
consider the dual to the near horizon limit of a stack of NS5-branes
Aharony:1998ub , also known as “little string theory.”

We hope to report back on some of these points in the near future.

## Acknowledgements

The authors would like to thank Josiah Couch and Phuc Nguyen for early
collaboration on related topics. We thank Mark Mezei for useful notes on
employing the membrane method to compute $v_{I}$ in holography, and Shenglong
Xu for helpful comments regarding spin chain computations. We also acknowledge
the Texas Advanced Computing Center (TACC) at The University of Texas at
Austin for providing HPC resources that have contributed to the research
results reported within this paper, TACC. The work of SE, WF, TG and SR is
supported by the National Science Foundation under Grant Number PHY-1914679.
JFP is supported by the Simons Foundation through _It from Qubit: Simons
Collaboration on Quantum Fields, Gravity, and Information_.

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