aainstitutetext: Institute of Theoretical Physics, School of Physics, Dalian
University of Technology, Dalian 116024, China.bbinstitutetext: Wilczek
Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China.ccinstitutetext: Shanghai Research Center
for Quantum Sciences, Shanghai 201315.

# On the universality of AdS2 diffusion bounds and the breakdown of linearized
hydrodynamics

Ning Wu b,c Matteo Baggioli a Wei-Jia Li<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The chase of universal bounds on diffusivities in strongly coupled systems and
holographic models has a long track record. The identification of a universal
velocity scale, independent of the presence of well-defined quasiparticle
excitations, is one of the major challenges of this program. A recent
analysis, valid for emergent IR fixed points exhibiting local quantum
criticality, and dual to IR AdS2 geometries, suggests to identify such a
velocity using the time and length scales at which hydrodynamics breaks down –
the equilibration velocity. The latter relates to the radius of convergence of
the hydrodynamic expansion and it is extracted from a collision between a
hydrodynamic diffusive mode and a non-hydrodynamic mode associated to the IR
AdS2 region. In this short note, we confirm this picture for holographic
systems displaying the spontaneous breaking of translational invariance.
Moreover, we find that, at zero temperature, the lower bound set by quantum
chaos and the upper one defined by causality and hydrodynamics exactly
coincide, determining uniquely the diffusion constant. Finally, we comment on
the meaning and universality of this newly proposed prescription.

##  1 Introduction

> The more specific we are, the more universal something can become.
>
> Jaqueline Woodson

The search for universal features in the transport properties of many-body
quantum systems, strongly coupled materials and holographic models has a long
history. In this context, universality is intended as insensitivity to the
specific microscopic details of the system and it therefore resonates nicely
with the concept of hydrodynamics landau2013fluid . Hydrodynamics is an
effective description controlling the long time and large scales dynamics,
where all the short-lived operators carrying the microscopic information get
washed out. In this sense, the universal properties remaining are carried by
the long-lived quantities and they are consequently related to the so-called
hydrodynamic modes and the corresponding conservation equations. Importantly,
using this broad terminology, hydrodynamics can be applied to any physical
systems and is not restricted to the description of fluids PhysRevA.6.2401 .

In this ballpark, a milestone result has been the identification of a
universal lower bound on the ratio of shear viscosity $\eta$ to entropy
density $s$, which supposedly holds for any system in nature. The resulting
inequality

$\frac{\eta}{s}\,\geq\,\frac{\hbar}{4\,\pi\,k_{B}}$ (1)

takes the name of Kovtun-Son-Starinets (KSS) bound Policastro:2001yc  and it
has been derived using a dual gravitational description in terms of the black
hole horizon dynamics and the gravitons absorption rate therein Gubser:1997yh
; Klebanov:1997kc . Notice how this bound immediately connects with
hydrodynamics since the $\eta/s$ ratio coincides exactly with the transverse
momentum dimensionless diffusion constant in a neutral relativistic fluid
Kovtun:2012rj . Indeed, for neutral relativistic systems, the KSS bound can be
re-written as:

$D_{\text{shear}}\,\geq\,\frac{c^{2}}{4\,\pi}\,\tau_{pl}$ (2)

where $D_{\text{shear}}$ is the diffusion constant of the shear mode, $c$ the
restored speed of light and $\tau_{pl}\equiv\hbar/k_{B}T$ the so-called
Planckian time Zaanen2004 ; 10.21468/SciPostPhys.6.5.061 . This last quantity
plays an important role and it has been involved in several discussions and
experimental observations about universality and transport Bruin804 ;
Behnia_2019 ; PhysRevX.5.041025 ; PhysRevLett.123.066601 ; Policastro:2001yc ;
Maldacena:2015waa ; Mousatov2020 ; PhysRevLett.124.076801 ;
PhysRevLett.120.125901 ; Zhang19869 ; Lucas:2018wsc ; Hartman:2017hhp

Despite the great success of the KSS bound even when confronted with realistic
experimental data Schafer:2009dj ; Cremonini:2011iq ; Luzum:2008cw ;
Nagle:2011uz ; Shen:2011eg , it soon became clear that the inequality in
Eq.(1) could be violated by breaking explicitly and/or spontaneously spacetime
symmetries, such as translations and rotations111Curiously, this is slightly
imprecise for the case of rotations. Indeed, contrary to the explicit breaking
scenario, the spontaneous breaking of rotations does not imply a violation of
the KSS bound ERDMENGER2011301 . The references reporting on these violations
are indeed several Alberte:2016xja ; Hartnoll:2016tri ; Burikham:2016roo ;
Rebhan:2011vd ; Ge:2018lzo ; Gochan:2018eez ; Figueroa:2020tya . From the
physical point of view, these cases were accompanied by the observation that
the ratio $\eta/s$ plays a very special role only in relativistic neutral
fluids, while it is not connected with any specific transport properties
otherwise. A clear example is that of a non-relativistic system in which the
KSS bound can be violated just by increasing the number of different species
PhysRevLett.99.021602 . In view of these facts, the universal character of the
KSS bound has been recently discredited Baggioli:2020ljz .

In a parallel line of investigation Trachenkoeaba3747 ; Baggioli:2020lcf ;
Trachenko:2020jgr , the superior (in the sense of more general and universal)
role of the diffusion constants (compared to the $\eta/s$ ratio for example)
has been outlined in the context of realistic liquids and simple bounds on
momentum and energy diffusion have been derived in terms of few fundamental
physical constants.

Inspired by the equivalent formulation of the KSS bound in terms of the shear
diffusion constant expressed in Eq.(2), a more general universal bound was
later proposed in Hartnoll:2014lpa . The idea is that any diffusive process in
nature is bounded from below by a certain combination of two unknown velocity
and time scales as:

$D\,\geq\,v^{2}_{?}\,\tau_{?}\,.$ (3)

This last expression recovers immediately the KSS bound by setting
$D=D_{\text{shear}}$, $v_{?}=c$ and $\tau_{?}=\tau_{pl}$.
The inequality in Eq.(3) applies to physical diffusion constants, it is very
general and it can be consistently defined for any system possessing a
diffusive hydrodynamic process which corresponds to the time evolution of a
certain conserved quantity. Nevertheless, it appears quite void and not
practical unless one specifies in detail which are the scales appearing in the
r.h.s. of Eq.(3). Additionally, one would call such an expression universal
only if the same velocity and time scales bounded all the diffusive processes
in the system. On the contrary, a statement like Eq.(3) would become quite
poor if, for any diffusion constant $D_{i}$, different scales in the r.h.s.
had to be used. Finally, following this logic, one would expect the scales in
the r.h.s. of Eq.(3) to be infrared (IR) quantities, independent of the
ultraviolet (UV) microscopic physics, and therefore universal.

A first, and partially successful, attempt to make the bound in Eq.(3) more
concrete has originated from the idea of identifying the scales in the r.h.s.
using physical observables from quantum chaos. In particular, Refs.
Blake:2016wvh ; Blake:2016sud  have proposed to identify:

$v_{?}\,=\,v_{B}\,,\qquad\tau_{?}\,=\,\tau_{L},$ (4)

where $v_{B}$ is the butterfly velocity and $\tau_{L}$ the Liapunov time. Both
these quantities can be directly extracted using the out-of-time-order
correlator (OTOC) 1969JETP…28.1200L . In summary, the final proposal coming
from Blake:2016wvh ; Blake:2016sud  was that any diffusion constant $D_{i}$
has to be bounded from below as follows:

$D_{i}\,\geq\,\\#_{i}\,v_{B}^{2}\,\tau_{L},$ (5)

where $\\#_{i}$ is an $\mathcal{O}(1)$ number which depends on the specific
diffusive process as well as IR fixed point considered.
Despite the considerable success of this proposal Davison:2018ofp ; Gu:2017njx
; Ling:2017jik ; Gu:2017ohj ; Blake:2016jnn ; Blake:2017qgd ; Wu:2017mdl ;
Li:2019bgc ; Ge:2017fix ; Li:2017nxh ; Ahn:2017kvc ; Baggioli:2017ojd ;
Kim:2017dgz ; Aleiner:2016eni ; Patel:2017vfp ; Patel:2016wdy ; Bohrdt:2016vhv
; 2017arXiv170507895W ; Kim:2017dgz ; Ahn:2017kvc ; Chen:2020bvf ;
Lucas:2018wsc , it was soon realized that in the case of charge diffusion
$D_{i}\equiv D_{Q}=\sigma/\chi_{\rho\rho}$ (with $\sigma$ the electric
conductivity and $\chi_{\rho\rho}$ the charge susceptibility) the bound in
Eq.(5) could be violated Lucas1608 ; Baggioli:2016pia . One more time, this is
not surprising from a physical point of view. In fact, the quantum chaos data
($v_{B}$ and $\lambda_{L}$) are extracted holographically from the
gravitational sector of fluctuations and in general are totally agnostic about
the charge sector to which the charge diffusion constant attains. When the
diffusive process considered is that of energy, $D_{i}\equiv
D_{\epsilon}=\kappa/c_{v}$ (with $\kappa$ the thermal conductivity and $c_{v}$
the specific heat), the bound in Eq.(5) is much more robust and hard to break.
Nevertheless, there are at least two known cases Lucas:2016yfl ;
wu2021classical  where this happens.

Figure 1: Given an arbitrary diffusive process and its related diffusion
constant $D$, does $D$ obey any universal lower/upper bounds and in terms of
which physical quantities? This is the question we address in this note.

Taking a similar perspective, one could ask whether the diffusion constants
are also bounded from above or they can grow indefinitely (see cartoon in
Fig.1). It turns out that causality, and in particular the requirement of
avoiding superluminal propagation, imposes a strong upper bound on diffusion
which takes the form Hartman:2017hhp :

$D_{i}\,\leq\,v^{2}_{\text{lightcone}}\,\tau_{eq},$ (6)

where $v_{\text{lightcone}}$ is the velocity, setting the causal lightcone in
the theory, and $\tau_{eq}$ the equilibration time at which the system
thermalizes, and after which hydrodynamics starts to apply. The equilibration
time can be universally defined using the imaginary part of the first damped,
and therefore non-hydrodynamic, mode as:

$\tau^{-1}_{eq}\,\equiv\,\omega_{eq}=|\mathrm{Im}\,\omega_{1}|,$ (7)

where $\omega_{1}$ is the frequency of the lowest of those modes.

Contrary to the equilibration time, the definition of the lightcone velocity
related to the causal structure is far from trivial in systems which do not
enjoy relativistic invariance and/or systems with emergent IR lightcone
structures. In relativistic systems, the lightcone velocity is obviously set
by the speed of light $c$. One simple example is given by Israel-Stewart
relativistic hydrodynamics ISRAEL1979341 . There, the lightcone speed is
immediately identified with the speed of light $c$ and the equilibration time
with the IR relaxation time $\tau_{\pi}$ which is pheomenologically introduced
in the framework. Indeed, within the Israel-Stewart formalism, the absence of
superluminality can be re-written exactly as an upper bound on the shear
diffusion constant $D_{\text{shear}}<c^{2}\tau_{\pi}$ Hartman:2017hhp ;
Baggioli:2020ljz .

A first check of the upper bound in Eq.(6) was performed in
Ref.Baggioli:2020ljz  using different velocity scales. A more formal
derivation, based on technical mathematical properties of the hydrodynamic
perturbative expansion, was discussed in Grozdanov:2020koi . Interestingly, if
one considers the momentum diffusion constant, instead of the $\eta/s$ ratio,
all the known violations related to the breaking of spacetime symmetries
disappear Baggioli:2020ljz .

Given the important role of hydrodynamics, recently, Ref.Arean:2020eus
proposed to connect the lightcone velocity with the equilibration velocity. In
particular, Ref.Arean:2020eus  proposed a new bound which takes the following
form:

$D_{i}\,\leq\,v^{2}_{i,eq}\,\tau_{eq}\,.$ (8)

Here, $v_{i,eq}$ is the equilibration velocity for the $i^{th}$ diffusive
mode, defined as:

$v_{eq}\,\equiv\,\frac{\omega_{eq}}{k_{eq}}\,.$ (9)

where, to avoid clutter, the $i$ index has been dropped.
This idea uses the recent definition of the radius of convergence of
hydrodynamics presented in PhysRevLett.122.251601 ; Grozdanov2019 ;
Withers:2018srf  and discussed further in Abbasi:2020ykq ; Jansen:2020hfd ;
Baggioli:2020loj . In particular, the pair $(\omega_{eq},k_{eq})$ corresponds
to the location of the first (closest to the origin) critical point of the
hydrodynamic perturbative series. This point coincides with the collision (in
general in the complex plane) between a first (in this case diffusive)
hydrodynamic mode and a nearby non-hydrodynamic mode (or a tower of them), and
it determines the radius of convergence of the hydrodynamic series. More
precisely, we utilize the following definitions:

$k_{eq}\,\equiv\,|k^{*}|\,,\qquad\omega_{eq}\,\equiv\,|\omega^{*}|\,,$ (10)

where $(\omega^{*},k^{*})$ is the first critical point – the position of the
collision for complex frequency and momentum, $k^{*},\omega^{*}\in\mathbb{C}$.
In simple words, such a point determines the scale at which considering only
conserved quantities is not enough anymore and the hydrodynamics description
must be improved.

Few comments are in order. (I) The definition of the equilibration velocity is
specific to the diffusive process considered. In this sense, it is quite a
stretch to consider the bound in Eq.(8) as universal. Notice for example the
crucial difference with the butterfly velocity proposal in Eq.(5), in which
the velocity scale on the r.h.s. is the same for all the diffusion constants
considered. (II) It is not clear how the bound in Eq.(8) connects with that in
Eq.(6). In particular, Eq.(8) does not follow from the requirement of
causality and furthermore $v_{eq}$ does not define nor the lightcone velocity
nor the causal structure of any propagating process. (III) The equal sign in
Eq.(8) follows trivially from assuming that the diffusive dispersion relation:

$\omega\,=\,-i\,D\,k^{2}$ (11)

is valid until the critical point $(\omega^{*},k^{*})$.
In particular simple algebra gives

$\displaystyle\omega_{eq}\,=\,\,D\,k_{eq}^{2}\,\rightarrow\,1\,=\,\,D\,\frac{\omega_{eq}}{v_{eq}^{2}}\,\rightarrow\,D=v_{eq}^{2}\tau_{eq}\,.$
(12)

That said, the observation of Arean:2020eus  is interesting and it boils down
to understand the following questions:

  * •

Are there situations where the corrections to the hydrodynamic dispersion
relation can be neglected until the critical point determining the radius of
convergence of linearized hydrodynamics?

  * •

Which conditions ensure the existence of such a scenarios and what is their
meaning?

In this note, we consider the proposal of Ref.Arean:2020eus  in homogeneous
holographic models with long-range order, i.e. with spontaneously broken
translational invariance. These systems display an $\text{AdS}_{2}$ IR near-
horizon geometry and a peculiar new diffusive mode labelled crystal diffusion
Donos:2019txg ; Baggioli:2020nay ; Baggioli:2020haa . In these holographic
models, the lower bound Eq.(5) for crystal diffusion has already been verified
in Baggioli:2020ljz . Our task now is to determine whether the inequivalence
in Eq.(8) applies also to the same mode and which are the scales involved.
Moreover, we analyze the connections and interplay between the lower bound on
diffusion dictated by quantum chaos and this new upper bound determined by the
breakdown of the hydrodynamic perturbative expansion. Finally, we provide some
comments and thoughts for the future.

##  2 The holographic model

Figure 2: The longitudinal spectrum of quasinormal modes for $N=3$ and
$m/T=500$. For simplicity, the attenuation constant of the longitudinal sound
modes has been manually removed. The gray region emphasizes the location of
the collision between the crystal diffusion mode and the first non-
hydrodynamic $\text{AdS}_{2}$ mode, from which the parameters
$\omega_{eq},k_{eq}$ are extracted as shown by the dashed lines. The tower of
non-hydrodynamic modes follow the expected trend $\omega_{n}=2\pi
T(n+\Delta)$, where in this case $\Delta=2$. Similar pictures can be obtained
for different values $N$ and of $m/T$.

We consider a large class of holographic axion models Baggioli:2021xuv
introduced and discussed in Baggioli:2014roa ; Alberte:2015isw ;
Baggioli:2016rdj ; Baggioli:2019rrs  and defined as follows:

$S\,=\,\int d^{4}x\sqrt{-g}\left[\frac{R}{2}+3-\,m^{2}\,V(X)\right]\,,$ (13)

where
$X\equiv\frac{1}{2}\,g^{\mu\nu}\,\partial_{\mu}\phi^{I}\partial_{\nu}\phi^{I}$
and we have set the AdS radius to be unit. We choose an isotropic profile for
the axion fields given by

$\phi^{I}\,=\,x^{I}\,,$ (14)

which represents a trivial solution of the equations of motion because of the
global shift symmetry $\phi^{I}\rightarrow\phi^{I}+b^{I}$ of the action (13).
The background geometry in Eddington-Filkenstein coordinates is written as:

$ds^{2}=\frac{1}{u^{2}}\left[-f(u)\,dt^{2}-2\,dt\,du+dx^{2}+dy^{2}\right]\,,$
(15)

where $u\in[0,u_{h}]$ is the radial holographic direction going from the
boundary $u=0$ to the horizon, $f(u_{h})=0$. Finally, we have:

$f(u)=u^{3}\int_{u}^{u_{h}}dv\;\left[\frac{3}{v^{4}}-\frac{m^{2}\,V(v^{2})}{v^{4}}\right]\,,$
(16)

and consequently the temperature $T$ is defined as

$T=-\frac{f^{\prime}(u_{h})}{4\pi}=\frac{6-2m^{2}\,V\left(u_{h}^{2}\right)}{8\pi\,u_{h}}\,,$
(17)

while the entropy density is given by $s=2\pi/u_{h}^{2}$.

In the rest of the note, we will focus on the monomial form:

$V(X)\,=\,X^{N}\,.$ (18)

We refer to the previous literature Alberte:2017cch ; Alberte:2017oqx ;
Baggioli:2018vfc ; Andrade:2019zey ; Ammon:2019wci ; Ammon:2019apj ;
Baggioli:2019abx ; Ammon:2020xyv  for more details concerning these models and
their properties.
In these holographic models, the UV expansion of the $\phi^{I}$ bulk fields
reads:

$\phi^{I}(t,x,u)\,=\,\phi_{(0)}^{I}(t,x)\,+\,\phi_{(1)}^{I}(t,x)\,u^{5-2N}\,+\,\dots$
(19)

Assuming standard quantization, one could verifiy that for $N<5/2$ the
background solution $\phi^{I}=x^{I}$ plays the role of an external source,
while for $N>5/2$ it describes a finite expectation value for the operators
$\mathcal{O}^{I}$, dual to the axion fields. Following this argument, the
breaking of translations is explicit for $N<5/2$ and spontaneous for $N>5/2$.
In this note, we will only consider the case $N>5/2$.

##  3 Results

The longitudinal spectrum of systems with spontaneously broken translations
features a peculiar mode with diffusive dispersion relation which is usually
labelled “crystal diffusion”. Using the correct hydrodynamic description
Armas:2019sbe , the diffusion constant of such a mode reads:

$D\,=\,\xi\,\frac{\left(B+G-\mathcal{P}\right)\,\chi_{\pi\pi}}{s^{\prime}\,T^{2}\,v_{L}^{2}}\,.$
(20)

The various parameters appearing in the equation above are: the Goldstones
diffusion constant $\xi$, the bulk modulus $B$, the shear modulus $G$, the
momentum susceptibility $\chi_{\pi\pi}$, the temperature derivative of the
entropy $s^{\prime}$ and finally the so called crystal pressure $\mathcal{P}$.
The hydrodynamic formula Eq.(20) has been successfully matched to the
holographic results Ammon:2020xyv ; Baggioli:2019abx  after some initial, and
then resolved, tension Ammon:2019apj .

The dispersion relation of the crystal diffusion mode is shown in Fig.2 for
very low values of temperature. The characteristic quadratic scaling and the
corresponding slope are consistent with the previous theoretical computations.
In Fig.2, the lowest non-hydrodynamic modes $\omega=-i\,\omega_{n}$ are shown
as well. As explained in the previous literature Faulkner:2009wj ;
Edalati:2010hk  and discussed in Arean:2020eus , these modes have a quite flat
dispersion relation at low temperatures and they appear equally separated. In
particular, this tower of non-hydrodynamic modes belongs to the IR
$\text{AdS}_{2}$ spectrum and it is given by:

$\omega_{n}\,=\,2\,\pi\,T\,\left(\Delta_{0}\,+\,n\right)\qquad\text{for}\qquad
T\rightarrow 0\,,$ (21)

where $n$ is the index labelling the modes and the constant $\Delta_{0}$
corresponds to the conformal dimension of the lowest operator in the
$\text{AdS}_{2}$ IR fixed point evaluated at zero momentum,
$\Delta_{0}=\Delta(k=0)$Arean:2020eus .
Given the presence of these modes, we can immediately identify the value of
the lowest ($n=0$) frequency in Eq.(21) with the equilibration timescale:

$\omega_{eq}=\tau^{-1}_{eq}=\omega_{0}=2\pi\,\Delta_{0}\,T\,.$ (22)

We plot the value of the normalized equilibration frequency in function of
$m/T$ at low temperatures in Fig.3. The curves approach nicely a common
asymptotic $T=0$ value which is given by $\Delta_{0}=2$. Importantly, this
value is completely independent of the choice of the potential and in
particular the value of the power $N$ 222We are grateful to Hyun-Sik Jeong,
Keun-Young Kim and Ya-Wen Sun for pointing out a mistake in a previous version
of our manuscript which was due to numerical inaccuracies. For an analytic
derivation, and extension, of our results we refer to their forthcoming work..

Figure 3: The equilibration frequency $\omega_{eq}$ in function of the
dimensionless temperature for various $N=3,4,5$. All the curves approach
asymptotically a constant value given by $\Delta_{0}=2$.

Let us now move to discuss the interplay between the diffusive hydrodynamic
mode and the first non-hydro mode. We have zoomed in the area in which the
crystal diffusion mode and the first non-hydrodynamic mode approach each
other, which is indicated with a gray shaded region in Fig.2. The results are
more clearly shown in Fig.4. From there, it is evident that the two modes
display an avoided crossing dynamics as noticed already in Arean:2020eus . In
particular, we numerically observe (see Arean:2020eus  for an analytic proof)
that the avoided crossing mechanism becomes more and more evident by
increasing the temperature. This is consistent with the observation that the
collision between the diffusive mode and the first non-hydrodynamic mode
happens for complex momentum, but the imaginary part of the critical point
tends to zero with the temperature $T$. Therefore, at low temperature, we can
approximate the equilibration scale with the real value of the critical
momentum:

$k_{eq}\,\equiv\,|k^{*}|\,\approx\mathrm{Re}\,k^{*}$ (23)

which, together with the critical frequency $\omega^{*}$, is easily readable
from the dispersion relation of the modes in the longitudinal spectrum, as
shown in Fig.2.

Figure 4: Top Left:  The ”almost” collision between the diffusive mode and the
first non-hydrodynamic mode. Top Right: A zoom of the avoided crossing
dynamics for $N=3$ and $m/T=900$. The gap between the two modes becomes
smaller and smaller going to lower temperature. Bottom panels:  The same
figures at larger temperature, $m/T=300$. Here, the avoided crossing is more
pronounced and the separation between the two curves is larger. This implies
that the imaginary part of the critical complex momentum is larger.

In summary, we can extract both parameters, $\omega_{eq},k_{eq}$, simply by
looking at the dispersion relations of the two lowest modes as shown in
Figures 2 and 4. At this point, we can also straightforwardly define the
equilibration velocity as:

$v_{eq}\,\equiv\,\frac{\omega_{eq}}{k_{eq}}\,.$ (24)

We show the behaviour of the non-normalized equilibration velocity in function
of temperature in the left panel of Fig.5. Interestingly, we observed a very
clear $T^{1/2}$ scaling close to zero temperature. Given that $\omega_{eq}\sim
T$ at low temperature (Eq.(21)), we can derive that in such a regime the
radius of convergence of hydrodynamics goes as:

$k_{eq}\,\sim\,T^{1/2}\,.$ (25)

This is consistent with the idea that the convergence properties of
hydrodynamics become worse and worse upon lowering the temperature
Baggioli:2020loj . In other words, hydrodynamics breaks down at larger
distances going towards zero temperature. Eq.(25) is also consistent with
previous results for charged holographic backgrounds and realistic liquids
Abbasi:2020ykq ; Jansen:2020hfd ; Baggioli:2020loj .

Figure 5: Left: The behaviour of the equilibration velocity in function of the
dimensionless temperature $m/T$ for various values of $N=3,4,5$. The dashed
line shows the $T^{1/2}$ scaling. Right:  The comparison with the butterfly
velocity. At low temperatures, the ratio of the two velocities approach a
constant value, $\sqrt{\Delta_{0}}=\sqrt{2}$.

At this stage, we want to compare the behaviour of the equilibration velocity
with that of the butterfly velocity, which has a fundamental role in the
diffusion bounds discussed in the introduction. The butterfly velocity can be
obtained from horizon data and in our background is given by:

$v_{B}^{2}\,=\,\frac{\pi\,T}{u_{h}}\,.$ (26)

In the limit of $m/T\rightarrow\infty$, the radius of the horizon goes to a
constant and therefore we obtain the expected scaling $v_{B}\sim T^{1/2}$.
Interestingly, the ratio between the two velocity scales approaches a
constant, given by $\sqrt{\Delta_{0}}$, in the low temperature limit. We
notice that for canonically normalized operators with unitary conformal
dimension in the $AdS_{2}$ IR fixed point, the two velocities would exactly
coincide at low temperature. More in general, we find that:

$v_{eq}\,>\,v_{B}\,.$ (27)

If one considers the butterfly velocity as an emergent lightcone speed of some
(non-relativistic) quantum chaotic system, the relation just obtained looks
quite dangerous since it would imply a superluminal propagation with speed
$v_{eq}$ outside of the causal lightcone. Nevertheless, the equilibration
velocity $v_{eq}$ does not correspond to any propagating modes. In other
words, in these cases, there is absolutely no excitation propagating at such
speed. In any case, this relation looks certainly interesting and it deserves
further understanding. It also implies that the thermalization speed, defined
as in Eq.(24), is faster than the speed of information scrambling. It would be
important to understand how universal this hierarchy is and which are the
physical consequences.

Another indication that the equilibration speed $v_{eq}$ cannot play the role
of the lightcone velocity in the bound of Hartman:2017hhp  is given by the
fact that in this model the longitudinal speed of sound is much larger than
the equilibration velocity. In this sense, if one had to choose a lightcone
velocity, the sound speed would be the most appropriate.

After having identified all the scales entering in the bound of Eq.(8), we can
finally test its validity for the crystal diffusion mode. In Fig.6, we plot
the dimensionless ratio $D/v_{eq}^{2}\tau_{eq}$ in function of the
dimensionless inverse temperature $m/T$ at small temperatures and for various
potentials, $N=3,4,5$. In all the cases, we notice that this ratio approaches
unity at $T\rightarrow 0$ confirming the validity of the relationship:

$D\,=\,v_{eq}^{2}\tau_{eq}\qquad\text{for}\qquad T\rightarrow 0\,.$ (28)

Moreover, we obtain that in general

$D\,\leq\,v_{eq}^{2}\tau_{eq}$ (29)

and that therefore the bound Eq.(8) proposed in Arean:2020eus  indeed holds.
This constitutes an explicit confirmation that even for the crystal diffusion
mode, the diffusion constant is bounded from above by the equilibration scales
$v_{eq}$ and $\tau_{eq}$.

Figure 6: The dimensionless ratio $D/(v_{eq}^{2}\tau_{eq})$ in function of the
inverse dimensionless temperature for various powers $N=3,4,5$. All the curves
approach the unit value at low temperature. Moreover, all curves are
consistent with the inequality $D\leq v_{eq}^{2}\tau_{eq}$.

##  4 Conclusions

In summary, in this short note, we have confirmed the validity of the
diffusion bound defined in terms of the hydrodynamics breakdown data proposed
in Ref.Arean:2020eus  for the crystal diffusion mode present in all
holographic systems with spontaneously broken translations. Importantly, in
our case, the collision determining the equilibration scales happens for real
values of the momentum $k$ and therefore it is easily extracted from the
dispersion relation of the lowest quasinormal modes.

Moreover, we do find that, for arbitrary values of the temperature, the
diffusion constant is confined in an range determined by:

$v_{B}^{2}\,\tau_{L}\,\leq\,D\,\leq\,v_{eq}^{2}\,\tau_{eq},$ (30)

which is shown in Fig.7. Importantly, approaching the zero temperature limit
this allowed region shrinks and at exactly zero temperature the two limits
collapse on each other. This means that the universal bounds determine
uniquely the value of the diffusion constant at zero temperature:

$\left(v_{B}^{2}\,\tau_{L}\right)|_{T=0}\,=\,D|_{T=0}\,=\,\left(v_{eq}^{2}\,\tau_{eq}\right)|_{T=0}\,.$
(31)

This is a very interesting and new outcome whose universal character must be
investigated further.

##  5 Additional comments

We conclude with several comments and ideas for the future.

Figure 7: A cartoon of the allowed region for the diffusion constant $D$. The
upper edge comes from the scales related to the breakdown of linearized
hydrodynamics. While, the lower one is dictated by the data of quantum chaos
Baggioli:2020ljz . Both curves merge in the limit of $T\rightarrow 0$.

  * •

One interesting point is the connection between the diffusivity bound $D\leq
v_{eq}^{2}\tau_{eq}$ and the higher order corrections to the diffusive
dispersion relation. As we have shown in the main text, if the diffusive
behaviour $\omega=-iDk^{2}$ persists until the collision point, the equality
trivially holds. What happens if that is not the case? In general, the
dispersion relation is given by a perturbative expansion in momentum of the
type:

$\omega\,=\,-i\left(Dk^{2}+a_{1}\,k^{4}\,+\,a_{2}\,k^{6}+\dots\,\right),$ (32)

where for simplicity we have ignored any possible real part and considered a
purely imaginary mode. Let us consider that, upon reaching the critical point
$(\omega^{*},k^{*})$, the first higher order correction cannot be neglected.
Then, we have:

$\omega_{eq}\,=\,Dk_{eq}^{2}+a_{1}\,k_{eq}^{4}$ (33)

which, after some manipulations, gives:

$D\,=\,v_{eq}^{2}\tau_{eq}\,-\,\frac{a_{1}}{v_{eq}^{2}\,\tau_{eq}}\,.$ (34)

The extension to higher order terms is trivial and for simplicity not shown
here. By looking at Eq.(34), one can infer that the validity of the upper
diffusion bound depends crucially on the signs of the higher order
corrections. In particular, the higher order terms in the diffusive dispersion
relations introduce corrections of order
$\mathcal{O}(1/v_{eq}^{2n}\tau_{eq}^{n})$, controlled by the various new
coefficients $a_{n}$. Is there any physical requirement (e.g. causality) that
fixes the sign of these coefficients and therefore the validity of the upper
bound? That seems indeed the case. The univalence property of the
hydrodynamics expansion put stringent bounds on all the higher order
coefficients Grozdanov:2020koi . For example, it constraints the first of
them, labelled $a_{1}$ above, to be positive. In other words, it is very
tempting to claim that the validity of the upper bound proposed in
Arean:2020eus  can be formally derived using mathematical properties of the
hydrodynamic series Grozdanov:2020koi . A simple scenario where this mechanism
appears is the telegrapher equation BAGGIOLI2020 :

$\omega^{2}+i\omega/\tau\,=\,v^{2}\,k^{2}\,.$ (35)

In this case, the purely diffusive dispersion law gets corrected before the
poles collision as:

$\omega\,=\,-\,i\,D\,k^{2}\,-\,i\,v^{4}\,\tau^{3}\,k^{4}\,+\,\dots$ (36)

and the corresponding higher order coefficient reads $a_{1}=v^{4}\,\tau^{3}$.
Stability, and more precisely the requirement of having $\tau>0$ (a relaxation
process and not an “exploding” one), implies that $a_{1}>0$ and therefore that
the diffusion constant is bounded from above, as discussed in the previous
paragraph.

  * •

It would be interesting to understand better the zero temperature relation in
Eq.(31). To the best of our knowledge, the collapse of the two bounds at zero
temperature has not been observed nor discussed before. How universal is this
feature? What can we learn from it? Is it possible to maintain a finite range
of allowed values at zero temperature or the two bounds always collapse?

  * •

In all this discussion, the value of the constant $\Delta_{0}$ plays a
fundamental role. In particular, the concrete value $\sqrt{\Delta_{0}}$
controls the ratio between the equilibration velocity and the butterfly
velocity at low temperature. It would be interesting to understand if any
physical requirement (e.g. unitarity and the corresponding bounds on the
conformal dimensions) constraints $\Delta_{0}>1$ and if not what is the
meaning of this role-reversal phenomenon.

  * •

A priori, it is not clear what is the relation between the equilibration
velocity $v_{eq}$ and the causal structure of the system. In particular, such
a velocity in general does not correspond to any propagating mode.
Interestingly, taking the telegrapher equation (35), one can derive that the
equilibration velocity $v_{eq}$ coincides exactly with the sound speed of the
emergent propagating mode at large momentum. In this simplified scenario, it
does corresponding to a propagating mode at short distance.

  * •

In the main text, we have derived that in the low temperature limit,
$\tau_{eq}^{-1}=2\pi\,\Delta_{0}\,T$. This relaxation time has the same
temperature dependence of the Planckian time and the Liapunov time but with a
different numerical prefactor. In particular we have:

$(\tau_{pl},\tau_{L},\tau_{eq})\,T\,=\,\left(1,\frac{1}{2\pi},\frac{1}{2\pi\,\Delta_{0}}\right)\,.$
(37)

Moreover, the hierarchy of these timescales depends crucially on the value of
$\Delta_{0}$. Also, the situation might be substantially different away from
maximal chaos Choi:2020tdj  where the Maldacena bound Maldacena:2015waa  is
not saturated and $\tau_{L}>\frac{1}{2\pi\,T}$. Which is the order of these
timescales and how can it be changed?

  * •

In Arean:2020eus , the equilibration velocity has been defined using the
critical point which determines the breakdown of the hydrodynamics expansion
and in particular of the diffusive dispersion relation. In principle, there
are other points in the complex plane which assume a particular role – the
pole-skipping points Grozdanov:2017ajz ; Blake:2018leo ; Blake:2017ris ;
Ahn:2020baf . Given a certain Green function, those are the points at which
the zeros and the poles cross, rendering the Green function indeterminate.
Following Arean:2020eus , one could define a pole-skipping velocity:

$v^{\mathcal{O}}_{skip}\,\equiv\,\frac{|\omega_{skip}|}{|k_{skip}|},$ (38)

where $(\omega_{skip},k_{skip})$ indicates the location of the first pole
skipping point in the complex plane and the index $\mathcal{O}$ the operator
whose Green function is considered. Notice that if one considers the energy-
energy correlator, one finds $v^{\epsilon}_{skip}=v_{B}$ and
$\omega_{skip}=\tau_{L}^{-1}$ Grozdanov:2017ajz . In this sense, for the
energy correlator, we have the following identification:

$\frac{v^{2}_{skip}}{\omega_{skip}}\,=\,v_{B}^{2}\,\tau_{L}$ (39)

and therefore one can immediately write down a lower bound of the type:

$D_{\epsilon}\,\geq\,\frac{v^{2}_{skip}}{\omega_{skip}}\,.$ (40)

The question whether a more general bound:

$D_{\mathcal{O}}\,\geq\,\frac{v^{2}_{\mathcal{O},skip}}{\omega_{\mathcal{O},skip}}\,$
(41)

exists for an arbitrary operator $\mathcal{O}$ is valuable and it can be
easily investigated with the existing techniques. Preliminary indications
private  seem to suggest our hypothesis.

  * •

As already mentioned, the speed of longitudinal sound $v_{L}$ in this model is
much larger than the equilibration speed $v_{eq}$. This implies that (I) the
equilibration velocity cannot be taken as the one determining the causal
lightcone and (II) that the bound $D\leq v_{eq}^{2}\tau_{eq}$ is more
stringent than the one coming from causality as in Hartman:2017hhp .

  * •

Finally, it would be interesting to consider IR fixed point with dangerously
irrelevant deformations Davison:2018ofp . There, the equilibration time is
expected to be parametrically longer, $\tau_{eq}\gg T^{-1}$, and the full
picture could change substantially. Hyperscaling-Lifshitz IR geometries are
also a straightforward generalization of this program.

The emerging global picture suggests intriguing and possibly fundamental
connections between transport, quantum chaos, hydrodynamics and pole skipping
which are left to be revealed.

We plan to come back to some of these questions in the near future.

### Acknowledgments

We thank Sebastian Grieninger for providing the numerical codes used in
previous works and for useful comments. We thank Saso Grozdanov, Keun-Young
Kim, Yongjun Ahn and Hyun-Sik Jeong for reading a preliminary version of the
manuscript and providing useful comments and suggestions. We thank Keun-Young
Kim, Yongjun Ahn, Hyun-Sik Jeong and Ya-Wen Sun for sharing with us
unpublished results and for correcting a mistake contained in the previous
version of this manuscript. N.W. and W.J.L. are supported by NSFC No.11905024
and No.DUT19LK20. M.B. acknowledges the support of the Shanghai Municipal
Science and Technology Major Project (Grant No.2019SHZDZX01).

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##  Appendix A Equations for the perturbations

We align the momentum $k$ along the $y$ direction. The perturbations in the
longitudinal sector are given by

$\\{h_{x,\,{s}}=1/2\,(h_{xx}+h_{yy}),\,h_{x,\,{a}}=1/2\,(h_{xx}-h_{yy}),\,\delta\phi_{y},\,h_{tt},\,h_{ty}\\}\,,$
(42)

We use a radial gauge. The final set of equations for the perturbations reads

$\displaystyle
uf^{\prime}\,\delta\phi_{y}^{\prime}\,\dot{V}\,+2\,u^{2}\,f\,\delta\phi_{y}^{\prime}\,\ddot{V}+u\,f\,\delta\phi_{y}^{\prime\prime}\,\dot{V}-2\,f\,\delta\phi_{y}^{\prime}\,\dot{V}-k^{2}\,u\,\delta\phi_{y}\,\dot{V}-k^{2}\,u^{3}\,\delta\phi_{y}\,\ddot{V}$
$\displaystyle+i\,k\,u\,h_{x,\,{a}}\,\dot{V}-i\,k\,u^{3}\,h_{x,\,{s}}\,\ddot{V}+2\,i\,u^{2}\,\omega\,\delta\phi_{y}\,\ddot{V}+u\,h_{ty}^{\prime}\,\dot{V}+2\,i\,u\,\omega\,\delta\phi_{y}^{\prime}\,\dot{V}$
$\displaystyle-2\,i\,\omega\,\delta\phi_{y}\dot{V}-2\,h_{ty}\left(\dot{V}-u^{2}\,\ddot{V}\right)=0$
(43) $\displaystyle
u(f\left(uf^{\prime}\,h_{x,\,{s}}^{\prime}-2\,f\,h_{x,\,{s}}^{\prime}-2\,i\,k\,m^{2}\,u^{3}\,\delta\phi_{y}\,\ddot{V}-u\,h_{tt}^{\prime\prime}+4\,h_{tt}^{\prime}\right)$
$\displaystyle+k\,h_{ty}\left(i\,u\,f^{\prime}-2\,i\,f+2\,u\,\omega\right)+h_{x,\,{s}}\left(2\,m^{2}\,u^{3}\,f\,\ddot{V}+\omega\,\left(i\,u\,f^{\prime}-2\,i\,f+2\,u\,\omega\right)\right))$
$\displaystyle+h_{tt}\left(u\left(-uf^{\prime\prime}+4\,f^{\prime}+2\,m^{2}\,u\,\dot{V}\right)-12\,f+k^{2}\,u^{2}-2\,m^{2}\,V-2\,i\,u\,\omega+6\right)=0$
(44) $\displaystyle
2h_{ty}\left(u\left(f^{\prime}+m^{2}\,u\,\dot{V}\right)-3\,f-m^{2}\,V+3\right)$
$\displaystyle-u\left(u\,f\,h_{ty}^{\prime\prime}-2\,f\,h_{ty}^{\prime}+i\,k\,u\,h_{tt}^{\prime}+k\,u\,\omega\,h_{x,\,{s}}+k\,u\,\omega\,h_{x,\,{a}}-2\,i\,m^{2}\,u\,\omega\,\delta\phi_{y}\,\dot{V}+i\,u\,\omega\,h_{ty}^{\prime}\right)$
$\displaystyle+2\,i\,k\,u\,h_{tt}=0$ (45) $\displaystyle
h_{x,\,{s}}\left(2\,u\left(f^{\prime}+m^{2}\,u\,\dot{V}\right)-6\,f+k^{2}\,u^{2}-2\,m^{2}\,V+4\,i\,u\,\omega+6\right)-u^{2}\,f^{\prime}\,h_{x,\,{s}}^{\prime}$
$\displaystyle-u^{2}\,f^{\prime}\,h_{x,\,{a}}^{\prime}+2\,u\,h_{x,\,{a}}\,f^{\prime}-u^{2}\,f\,h_{x,\,{s}}^{\prime\prime}-u^{2}\,f\,h_{x,\,{a}}^{\prime\prime}+4\,u\,f\,h_{x,\,{s}}^{\prime}+2\,u\,f\,h_{x,\,{a}}^{\prime}-6\,f\,h_{x,\,{a}}$
$\displaystyle+k^{2}\,u^{2}\,h_{x,\,{a}}+2\,i\,k\,u\,h_{ty}+2\,m^{2}\,u^{2}\,h_{x,\,{a}}\,\dot{V}-2\,m^{2}\,h_{x,\,{a}}\,V-2\,i\,u^{2}\,\omega\,h_{x,\,{s}}^{\prime}$
$\displaystyle-2\,i\,u^{2}\,\omega\,h_{x,\,{a}}^{\prime}-2\,u\,h_{tt}^{\prime}+6h_{tt}(u)+2\,i\,u\,\omega\,h_{x,\,{a}}+6h_{x,\,{a}}=0$
(46) $\displaystyle
h_{x,\,{s}}\left(2\,u\left(f^{\prime}+m^{2}\,u\,\dot{V}\right)-6\,f+k^{2}\,u^{2}-2\,m^{2}\,V+4\,i\,u\,\omega+6\right)-u^{2}\,f^{\prime}\,h_{x,\,{s}}^{\prime}+u^{2}\,f^{\prime}\,h_{x,\,{a}}^{\prime}$
$\displaystyle-2\,u\,h_{x,\,{a}}\,f^{\prime}-u^{2}\,f\,h_{x,\,{s}}^{\prime\prime}+u^{2}\,f\,h_{x,\,{a}}^{\prime\prime}+4\,u\,f\,h_{x,\,{s}}^{\prime}-2\,u\,fh_{x,\,{a}}^{\prime}+6\,f\,h_{x,\,{a}}+k^{2}\,u^{2}\,h_{x,\,{a}}$
$\displaystyle-4\,i\,k\,m^{2}\,u^{2}\,\delta\phi_{y}\,\dot{V}-2\,i\,k\,u^{2}\,h_{ty}^{\prime}+6\,i\,k\,u\,h_{ty}-2\,m^{2}\,u^{2}\,h_{x,\,{a}}\,\dot{V}+2\,m^{2}\,h_{x,\,{a}}\,V$
$\displaystyle-2\,i\,u^{2}\,\omega\,h_{x,\,{s}}^{\prime}+2\,i\,u^{2}\,\omega\,h_{x,\,{a}}^{\prime}-2\,u\,h_{tt}^{\prime}+6\,h_{tt}-2\,i\,u\,\omega\,h_{x,\,{a}}-6\,h_{x,\,{a}}=0$
(47)
$\displaystyle-6\,h_{tt}+u\,(u\,f^{\prime}\,h_{x,\,{s}}^{\prime}-2\,f\,h_{x,\,{s}}^{\prime}-2\,i\,k\,m^{2}\,u^{3}\,\delta\phi_{y}\,\ddot{V}+i\,k\,u\,h_{ty}^{\prime}-2\,i\,k\,h_{ty}$
$\displaystyle+2\,m^{2}\,u^{3}\,h_{x,\,{s}}\,\ddot{V}-u\,h_{tt}^{\prime\prime}+4\,h_{tt}^{\prime}+2\,i\,u\,\omega\,h_{x,\,{s}}^{\prime}-2\,i\,\omega\,h_{x,\,{s}})=0$
(48) $\displaystyle
k\,u\left(h_{x,\,{s}}^{\prime}+h_{x,\,{a}}^{\prime}\right)-i\,u\,\left(2\,m^{2}\,\delta\phi_{y}^{\prime}\,\dot{V}+h_{ty}^{\prime\prime}\right)+2\,i\,h_{ty}^{\prime}=0$
(49) $\displaystyle h_{x,\,{s}}^{\prime\prime}=0,$ (50)

where the following notations $\dot{V}\equiv dV(X)/dX,\,\ddot{V}\equiv
d^{2}V(X)/dX^{2}$ are used.
The quasinormal modes are obtained using pseudo-spectral methods. For more
details about the numerical procedure see Grieninger:2020wsb .