instance the way in which the mid IR peak depends on temperature.
Alternatively the sound contribution to the density-density response can been
measured directly by EELS mitranoAnomalousDensityFluctuations2018 ;
husainCrossoverChargeFluctuations2019 , with the caveat that sound is promoted
to a plasmon in the presence of dynamical electromagnetism.This may be hard,
because the plasmon is damped stronger in strange metallic states than
ordinary Fermi liquids Gran:2017jht ; Mauri:2018pzq ; Romero-Bermudez:2018etn
; Andrade:2019bky . The results are at this moment inconclusive, and need to
still be found consistent with the AC optical conductivity.

Perhaps the most delicate issue relates to the connection with microscopic
chaos. The connection with Planckian dissipation requires a saturation of the
Maldacena-Shenker-Stanford bound on the Lyapunov exponent of the OTOC
$\lambda\leq 2\pi T$. It appears that a necessary condition for this to happen
is in the form of dense many body entanglement. One may argue that this is the
secret of the experimental strange metals: these are born from strongly
interacting fermion systems at a finite density and it may well be that the
concomitant sign problem enforces dense entanglement in the non-Fermi-liquids
zaanenLecturesQuantumSupreme2021a . But this may not be a sufficient
condition. The chaos bound is known to saturate in matrix large $N$ systems at
strong coupling with a holographic dual as well as the disorder averaged SYK
models. These systems are characterized by dense matrix interactions.

However, the Hubbard models that are the community standard as microscopic
point of departure for the cuprate electrons are characterized by local
interactions and the associated Hamiltonians correspond with rather sparse
matrices. As with regard to the transport properties, the present benchmark is
in the form of finite temperature quantum Monte Carlo computations for the
resistivity huang2019strange . The temperatures that can be reached are still
quite high ($\simeq 1000$ K) but arguably approaching the linear resistivity
regime. However, the outcomes are quite different from what we find.

The Hubbard models are of course in their own way toy models, capturing the
largest scales in the problem but ignoring a lot of other physics. Could it be
that long range interactions arising e.g. from Coulomb interactions and/or
phonon mediated interactions are crucial to support the rapid scrambling near
the Lyapunov bound given their non-local nature Xu:2022vko ? Could there be a
direct relation to SYK physics associated with the observation of spin glass
physics Tsuji:2018mth ; Christos:2022lma  , with the obvious difficulty that
this has only been observed in the spin striped 214 system?

At the least, holography inspires to ask quite unusual questions to
experiment: it suggests a physics that is tantalizingly different from the
usual Fermi-liquid quasiparticle physics. Eventually, it should be possible by
targeted experimentation to reach a verdict. This is not easy: the cuprates
have been subjected to unprecedented experimental scrutiny over the last 35
years but on basis of the available information it is still impossible to
decide the issue.

An example of this law-of-Murphy that insightful results may be the hardest to
obtain experimentally is the thermal transport. So much is clear that the
thermal conductivity $\kappa$ of the GR metal acquires a universal temperature
dependence that is up to very high lattice potentials independent of the
potential strength. Numerically we observe that $\kappa\sim T^{2}$. But this
in gross contrast with the thermal conductivity in a Fermi liquid, where
$D_{T}\sim\tau_{c}$ where $\tau_{c}\sim 1/T^{2}$ is the quasiparticle
collision time such that $\kappa\sim 1/T$ abrikosovTheoryFermiLiquid1959 .
There is a large difference of the order $T^{3}$ in the temperature dependence
of the thermal conductivity between the holographic metal and a Fermi liquid!

This should be the smoking gun but why can this not be used? The reason is
that at the high temperatures where the strange metal is realized ($>100$ K)
the thermal transport is rather completely dominated by the phonons. The
phonon heat conduction short circuits the heat transport and it is virtually
impossible to extract the electronic contributions. The same problem is there
for a measurement a charge transport without heat $\sigma_{Q=0}$. Aside from
the experimental hurdle of zeroing out heat transport cleanly, the definition
of $\sigma_{Q=0}=\sigma-T\alpha^{2}/\bar{\kappa}$ implicitly refers to the
electronic component of the heat transport only.

Finally, there is one thermo-electrical transport coefficient that is readily
available experimentally: the Seebeck coefficient enumerating the thermopower.
This is given by $s=\alpha/\sigma$. According to Fig. 1, $\alpha\sim T^{0}$
and $\sigma\sim 1/T$, and we predict $s\sim T$: although for different reason
this is the same temperature dependence generic for a Fermi-liquid (the Mott
formula), this is indeed the scaling that has been observed in cuprate strange
metals , e.g. collignonThermopowerPhaseDiagram2021 ;
gourgoutSeebeckCoefficientCuprate2022 .

###### Acknowledgements.

We thank R. Davison, B. Goutéraux, S. Grozdanov, S. Hartnoll and A. Krikun for
discussions. This research was supported in part by the Dutch Research Council
(NWO) project 680-91-116 (Planckian Dissipation and Quantum Thermalisation:
From Black Hole Answers to Strange Metal Questions.), the FOM/NWO program 167
(Strange Metals), and by the Dutch Research Council/Ministry of Education.
K.G. acknowledges funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No
101024967. The numerical computations were carried out on the Dutch national
Cartesius and Snellius national supercomputing facilities with the support of
the SURF Cooperative as well as on the ALICE-cluster of Leiden University. We
are grateful for their help.

##  Appendix A AdS RN and GR black holes

We will be interested in perturbations of both Reissner-Nordström and Gubser-
Rocha black holes.

###  A.1 Reissner-Nordström

The RN black holes start from the Einstein-Maxwell action

$S=\int\text{d}^{4}x\,\sqrt{-g}\left[\dfrac{L^{2}}{2\kappa^{2}}\left(R-2\Lambda\right)-\dfrac{L^{2}}{4e^{2}}F_{\mu\nu}F^{\mu\nu}\right]~{},$
(38)

with $2\kappa^{2}=e^{2}=L^{2}=1$ and $\Lambda=-3$. The equations of motion are

$\displaystyle R_{\mu\nu}-\Lambda g_{\mu\nu}$
$\displaystyle=\dfrac{1}{2}\left[F_{\mu\rho}F_{\nu}^{\rho}-\dfrac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\right]~{},$
(39) $\displaystyle\nabla_{\mu}F^{\mu\nu}$ $\displaystyle=0~{}.$

These equations admit an electrically charged black hole solution, the AdS-
Reissner-Nordström (RN) solution in asymptotically AdS4 space-time, for which
the metric and gauge field are given by131313Sometimes, it is more convenient
to make a change of variable $z\rightarrow 1-(1-r)^{2}$
donosMinimallyPackedPhases2016 .

$\displaystyle\text{d}s^{2}$
$\displaystyle=g_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}=\dfrac{1}{z^{2}}\left[-f(z)\text{d}t^{2}+\dfrac{\text{d}z^{2}}{f(z)}+\text{d}x^{2}+\text{d}y^{2}\right]~{},$
(40) $\displaystyle A$ $\displaystyle=A_{t}(z)\text{d}t~{},$

where

$f(z)=\left(1-z\right)\left(1+z+z^{2}-\frac{\mu^{2}z^{3}}{4}\right),\quad
A_{t}(z)=\mu(1-z).$ (41)

The radial coordinate $z$ can be scaled such that the horizon is located at
$z_{h}=1$ and the boundary of the space-time is at $z=0$. The temperature of
the black hole can be computed by considering the surface gravity of the
horizon, and is given by

$T_{\mathrm{RN}}=\left|\frac{f^{\prime}(z_{h})}{4\pi}\right|=\frac{12-\mu^{2}}{16\pi}$
(42)

###  A.2 Einstein-Maxwell-Dilaton

For the dilatonic black holes, we depart from the Einstein-Maxwell-Dilaton
actiongubserPeculiarPropertiesCharged2010 ;
lingHolographicLatticeEinsteinMaxwellDilaton2013

$S=\frac{1}{2\kappa^{2}}\int\text{d}^{4}x\sqrt{-g}\left[R-\frac{Z(\phi)}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}\left(\partial_{\mu}\phi\right)^{2}+V(\phi)\right].$
(43)

The potentials $Z,V$ are given by

$Z(\phi)=\exp(\phi/\sqrt{3}),\qquad
V(\phi)=\frac{6}{L^{2}}\cosh(\phi/\sqrt{3}).$ (44)

The cosmological constant is given by $\Lambda=-V(0)/2=-3$. Setting
$2\kappa^{2}=L^{2}=1$, the equations of motion for this system are given by

$\displaystyle R_{\mu\nu}$
$\displaystyle=\dfrac{Z(\phi)}{2}\left[F_{\mu}{}^{\rho}F_{\nu\rho}-\dfrac{1}{4}g_{\mu\nu}F^{2}\right]+\dfrac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi+\dfrac{1}{2}g_{\mu\nu}V(\phi)~{},$
(45) $\displaystyle\nabla_{\mu}\left[Z(\phi)F^{\mu\nu}\right]$
$\displaystyle=0~{},$ $\displaystyle\Box\phi$
$\displaystyle=V^{\prime}(\phi)+\dfrac{Z^{\prime}(\phi)}{4}F^{2}~{},$

where we used that on-shell $R=-2V(\phi)+\dfrac{1}{2}(\partial\phi)^{2}$. This
setup also has an analytic solution which is given by a metric, gauge field
and non-trivial scalar $\phi$ in the form of

$\displaystyle ds^{2}$
$\displaystyle=g_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}=\frac{1}{z^{2}}\left[-h(z)\text{d}t^{2}+\frac{1}{h(z)}\text{d}z^{2}+g(z)(\text{d}x^{2}+\text{d}y^{2})\right]$
(46) $\displaystyle A$
$\displaystyle=\frac{\sqrt{3Qz_{h}(1+Qz_{h})}}{z_{h}}\frac{(1-z/z_{h})}{1+Qz}\text{d}t$
$\displaystyle\phi$ $\displaystyle=\frac{\sqrt{3}}{2}\log\left(1+Qz\right)$

where

$\displaystyle h(z)$
$\displaystyle=\frac{(1-z/z_{h})}{g(z)}\left[1+(1+3Qz_{h})\frac{z}{z_{h}}+\left(1+3Qz_{h}(1+Qz_{h})\right)\left(\frac{z}{z_{h}}\right)^{2}\right],$
(47) $\displaystyle g(z)$ $\displaystyle=(1+Qz)^{3/2}.$

The parameter $Q$ encodes the charge of the black hole. The chemical potential
is given by $\mu=\sqrt{3Qz_{h}(1+Qz_{h})}/z_{h}$. The near-horizon form of the
potentials in equation (43) corresponds to a scaling behavior of
$z,-\theta\rightarrow\infty$, also identified by $\gamma,-\delta=1/\sqrt{3}$
in charmousisEffectiveHolographicTheories2010 . The temperature here is given
by

$T_{\mathrm{GR}}=\left|\frac{h^{\prime}(1)}{4\pi}\right|=\frac{3\sqrt{1+Qz_{h}}}{4\pi
z_{h}}.$ (48)

###  A.3 Lattice Backgrounds

The translational symmetry of the black hole backgrounds is broken by applying
a periodic ionic lattice in the boundary field theory through the modulation
of the chemical potential horowitzOpticalConductivityHolographic2012 . In the
gravitational theory, this corresponds to imposing a boundary condition on the
gauge field

$A_{t}(x,y,z=0)=\bar{\mu}\left(1+A_{x}\cos(G_{x}x)+A_{y}\cos(G_{y}y)\right).$
(49)

$A_{x,y}$ parameterize the strength of the lattice, while $G_{x,y}$ are the
reciprocal lattice dimensions, respectively. Our computational domain in
$(x,y)$ is chosen to always contain a whole number of lattice periods, i.e.
$x\sim x+2\pi n_{x}/G_{x}$, $y\sim y+2\pi n_{y}/G_{y}$ where
$n_{x},n_{y}\in\mathbb{Z}$. Throughout this work, we take $G_{x}=G_{y}\equiv
G$ and $A_{x}=A_{y}\equiv A_{0}/2$ in a 2D lattice and $A_{x}\equiv
A_{0},A_{y}=0$ for a 1D lattice.

This breaking of translational symmetry influences the solutions dramatically.
The additional curvature generated by the periodic lattice means that in
principle all the off-diagonal components of the metric as well as all
components of the gauge field will become non-trivial.

For RN, the ansatz for the fields is adapted from reference
donosMinimallyPackedPhases2016 :

$\displaystyle ds^{2}$
$\displaystyle=\frac{1}{z^{2}}\left(-Q_{tt}f(z)\eta_{t}^{2}+Q_{xx}\eta_{x}^{2}+Q_{yy}\eta_{y}^{2}+\frac{Q_{zz}}{f(z)}\eta_{z}^{2}\right),$
(50) $\displaystyle\eta_{t}$ $\displaystyle=\text{d}t,$
$\displaystyle\eta_{x}$
$\displaystyle=\text{d}x+Q_{xy}\text{d}y+Q_{xz}\text{d}z,$
$\displaystyle\eta_{y}$ $\displaystyle=\text{d}y+Q_{yz}\text{d}z,$
$\displaystyle\eta_{z}$ $\displaystyle=\text{d}z,$ $\displaystyle A$
$\displaystyle=\mu(1-z)A_{t}\text{d}t$

Our EMD ansatz looks similar and is given by

$\displaystyle ds^{2}$
$\displaystyle=\frac{1}{z^{2}}\left(-Q_{tt}h(z)\eta_{t}^{2}+g(z)\left(Q_{xx}\eta_{x}^{2}+Q_{yy}\eta_{y}^{2}\right)+\frac{Q_{zz}}{h(z)}\eta_{z}^{2}\right),$
(51) $\displaystyle\eta_{t}$ $\displaystyle=\text{d}t,$
$\displaystyle\eta_{x}$
$\displaystyle=\text{d}x+Q_{xy}\text{d}y+Q_{xz}\text{d}z,$
$\displaystyle\eta_{y}$ $\displaystyle=\text{d}y+Q_{yz}\text{d}z,$
$\displaystyle\eta_{z}$ $\displaystyle=\text{d}z,$ $\displaystyle A$
$\displaystyle=\frac{\mu(1-z)}{1+Qz}A_{t}\text{d}t,\qquad\phi=\frac{3}{2}\log\left(1+\varphi
Qz\right).$

For both types of solutions, we are interested in stationary solutions, and
therefore all functions $F=\left\\{Q_{ij},A_{i},\varphi\right\\}$ are
functions of $(x,y,z)$, each periodic in $(x,y)$ with a periodicity of
$L_{x,y}=2\pi n_{x,y}/G_{x,y}$. The equations of motion in equation (39) and
(45) form very complicated systems of non-linear partial differential
equations in three dimensions, which in general cannot be solved analytically.
For numerical convenience, the DeTurck trick and another gauge fixing term for
the gauge field can be used to turn this set of equations into an elliptic
boundary value problem headrickNewApproachStatic2010 ;
adamNumericalApproachFinding2012 ; donosMinimallyPackedPhases2016 . The UV
boundary conditions on the radial coordinate come from imposing an
asymptotically AdS solution while imposing that the dilaton be a marginal
operator with no source, as was highlighted in
chagnetProperQuantizationGubserRocha2022 . The horizon boundary conditions
arise from requiring regularity at the horizon, which means that in a series
expansion in powers of $(z-1)$ we can relate each $\partial_{z}F_{i}$ to all
functions and their tangential derivatives at the horizon,141414If the change
of coordinates in footnote 13 is used, this simplifies to
$\partial_{r}F_{i}=0\,\forall\,i$, as only even powers of $r$ will appear in
the near-horizon expansion. This comes at the cost of accuracy near the
horizon. together with the condition that
$Q_{tt}\bigr{|}_{z=1}=Q_{zz}\bigr{|}_{z=1}$.151515This ensures a constant
temperature across the (corrugated) horizon.

The boundary value problems are solved using a self-developed software package
in C, using the PETSc library balayEfficientManagementParallelism1997 ;
balayPETScTAOUsers2021 . A Newton line-search algorithm employing second- and
third order finite difference schemes on rectangular grids is used to find
solution to the non-linear problem. The computational grids are either
uniformly spaced or have the radial coordinate run over the Chebyshev-Lobatto
nodes for increased accuracy near the boundaries of the problem. Typical grid
sizes for the simulations run between $N_{x}\times N_{y}\times N_{z}=40\times
40\times 60$ to $80\times 80\times 120$. For convergence checks, the vanishing
of the norm of the DeTurck vector provides a good measure
adamNumericalApproachFinding2012 . Due to the large number of degrees of
freedom involved ($O(10^{7})$ for the largest lattices) most of the numerical
work was done using the ALICE cluster at Leiden University and the Dutch
national Cartesius and Snellius supercomputers with the support of SURF
Cooperative. The code is publicly available floris_balm_2022_7284816 ;
floris_balm_2022_7285050 .

###  A.4 DC Conductivity

The DC conductivity is computed by solving a Stokes flow problem on the black
hole horizon donosThermoelectricDCConductivities2014 ;
banksThermoelectricDCConductivities2015 ; donosDCConductivityMagnetised2016 ;
donosHolographicDCConductivity2017 . Using a set of time-independent
perturbations, one can show that the bulk linear response problem of computing
(thermo)electric DC conductivities can be reduced to a linearized version of
the Navier-Stokes equations for an auxiliary fluid that lives on a static
black hole horizon background. The equations take a similar form for both EMD
and RN black holes, and can be written as donosDCConductivityMagnetised2016

$\displaystyle\eta^{(0)}\bigl{(}-2\nabla^{j}\nabla_{(i}v_{j)}+$ $\displaystyle
3v^{j}\nabla_{j}\phi^{(0)}\nabla_{i}{\phi}^{(0)}\bigr{)}-d\chi_{ij}^{(0)}Q^{j}-F^{(0)}_{ij}J^{j}=$
(52)
$\displaystyle\rho^{(0)}\left(E_{i}+\nabla_{j}w\right)+Ts^{(0)}\left(\zeta_{i}-\nabla_{i}\frac{p}{4\pi
T}\right)$ $\displaystyle\partial_{i}Q^{i}=0,$
$\displaystyle\qquad\qquad\partial_{i}J^{i}=0.$

The superscript $(0)$ indicates that these are background quantities evaluated
at the horizon. These are the values we extract from the numerical solutions
to the background lattices described above.161616For the RN black holes, one
should take $\phi=0,Z(\phi)=1,V(\phi)=6$. The Stokes equations (52) is then a
set of four equations for the four unknown functions $v_{x},v_{y},w,p$. The
currents $Q,J$ and transport coefficients
$\rho^{(0)},\eta^{(0)},s^{(0)},\chi^{(0)}$ can be written in terms of these
four functions, the background horizon quantities and the induced metric on
the horizon $h^{(0)}_{ij}$donosDCConductivityMagnetised2016 .
$E_{i},\zeta_{i}$ source the electric field and thermal gradient, and are
taken to be constant over the unit cell. The thermo-electric DC conductivities
are then extracted by evaluating

$\begin{pmatrix}\vec{J}\\\ \vec{Q}\end{pmatrix}=\begin{pmatrix}\sigma&\alpha
T\\\ \bar{\alpha}T&\bar{\kappa}T\end{pmatrix}\begin{pmatrix}\vec{E}\\\
\vec{\zeta}\end{pmatrix},$ (53)

where $\vec{J},\vec{Q}$ are here the spatial averages of the solutions when
evaluating equation (52). These averaged values do not renormalize when lifted
to the boundary to be interpreted in the field theory. As a result, the
thermo-electric conductivity matrix defined in equation (53) is that of the
field theory.

In previous works, e.g. donosHolographicDCConductivity2017 , these equations
have been used to study simpler systems that do not fully break spatial
translational symmetry or break it in a homogeneous way. That simplification
allows for a largely analytic treatment of these equations. The systems we are
interested in do not permit us such luxuries and therefore we have to solve
this coupled linear PDE in two periodic dimensions numerically. For this, we
developed a package in Python which can solve coupled (non-)linear partial
differential equations for backgrounds as well as perturbations. This package
is rather flexible, in that it can make use of both (pseudo)spectral and
arbitrary-order finite difference methods to solve the equations. This package
will be made available publicly at a later date.

##  Appendix B Semi-local criticality and an induced IR length scale

Semi-local quantum liquids can be defined by a “self-energy” that is either a
power-law in frequency $\Sigma\sim\omega^{2\nu(k)}$ or exponential
$\Sigma\sim\exp(-\frac{k^{z/(z-1)}}{\omega^{z-1}})$ with $z$ the dynamical
critical exponent. In the limit $z\rightarrow\infty$ the latter reduces to the
former faulknerHolographicNonFermiLiquid2011 . Both $\omega$ and $k$ are
dimensionless frequencies and momenta in units of the chemical potential
$\mu$. As emphasized in iqbalSemilocalQuantumLiquids2012  the spatial
structure of such semi-local quantum liquids is that the spread of local
perturbations decays very rapidly and is bounded by an emergent length scale
$\xi\propto\mu^{-1}$.

Though the emergence of this semi-local physics is poorly understood from a
conventional point of view, its emergence bound is surprisingly clear from a
dual holographic perspective. It is a direct consequence of the existence of a
maximal distance, $x_{\rm max}\sim 1/\xi_{\mu}$ that two light-rays emitted
from near the AdS black hole horizon can spread
iqbalSemilocalQuantumLiquids2012 . It implies that a local perturbation in the
IR can only originate from/influence a finite spatial region (in the UV
variables).

This supplementary section shows how this maximal distance arises. A light-ray
parametrized by $X^{\mu}(\tau)$ follows a null geodesic, i.e.

$g_{\mu\nu}\dot{X}^{\mu}\dot{X}^{\nu}=0\,.$ (54)

Consider a generic $z=\infty$ metric

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{r}{R}\right)^{-\frac{2\eta}{d}}\left[-\mu^{2}R^{2}r^{2}dt^{2}+\frac{R_{2}^{2}}{r^{2}}dr^{2}+\mu^{2}R^{2}dx_{i}dx^{i}\right]$
(55)

The parameter $\eta=-\frac{\theta}{z}$ is the remnant of the hyperscaling
violation exponent $\theta$ in the limit $z\rightarrow\infty$ and
$R_{2}=R/\sqrt{6}$ is the emergent AdS2 radius. For a geodesic emanating from
the horizon we can use the radial $r$ coordinate as the affine parameter
$\tau$, and the physical gauge $X^{r}=r$. Then solving the geodesic equation
$D_{\tau}\dot{X}^{\mu}=0$ subject to the null length constraint, one finds for
the motion in the transverse directions

$\displaystyle X_{i,\pm}(r)$
$\displaystyle=x^{(0)}_{i}\pm\frac{v_{i}R_{2}}{\mu
Rv}\tan^{-1}\left(\frac{rv}{\sqrt{v_{t}^{2}-r^{2}v^{2}}}\right)\,.$ (56)

with $v^{2}=\sum_{i}v_{i}^{2}$. Two light-rays starting from the same point
$x_{i}^{(0)}$ one pointing to the left and one to the right therefore arrive
at the boundary ($r=\infty$) a distance $2\frac{v_{i}}{v}\frac{R_{2}}{\mu
R}\frac{1}{\tan(i)}$ apart.

We are now interested in the intersection of two lightcones $x_{i,+}$ and
$x_{i,-}$, which can be found from

$x_{i,+}(r_{0};x^{(0)}=0)=x_{i,-}(r_{0};x^{(0)})\implies
r_{0}=\frac{v_{t}}{v}\sin\left(\frac{\mu
Rvx_{i}^{(0)}}{2v_{i}R_{2}}\right)\,.$ (57)

After combining (56) with (57), we find that the maximal allowed distance is

$x^{\rm(max)}_{1}=\frac{R_{2}}{R\mu}\pi\cos\theta\,,\qquad
x^{\rm(max)}_{2}=\frac{R_{2}}{R\mu}\pi\sin\theta\,.$ (58)

where we have chosen the parametrization for the initial velocity components
along $\left(x_{1},x_{2}\right)$ as $v_{1}=v\,\cos\theta\,,\quad
v_{2}=v\,\sin\theta\,,$ where $\theta\in[0,\pi/2]$ is the initial angle,
measured with respect to the $x_{1}$-axis.

The relative initial distance between the two geodesics $\Delta s$ reads

$\Delta
s=\sqrt{x^{\rm(max)}_{1}{}^{2}+x^{\rm(max)}_{2}{}^{2}}=\frac{R_{2}}{R\mu}\pi\,,$
(59)

which is universal and does not depend on the initial conditions. It coincides
with the result presented in iqbalSemilocalQuantumLiquids2012 .

In figure 16, we plot the causal structure for two light-rays separated by a
certain initial distance. For an initial separation larger than the critical
distance, (58), both light-rays are not causally connected anymore. To
illustrate this, we have chosen as a dialing parameter the external time $t$.
After some computations, we get

$t(x_{i})=-\frac{R_{2}v}{\mu Rv_{t}}\cot\left[\frac{\mu
Rv}{v_{i}R_{2}}\left(x_{i}-x_{i}^{(0)}\right)\right]\,.$ (60)

which is plotted in the second figure in 16. From here, we highlight that any
geodesic that starts at an initial relative distance $\Delta s\leq\pi$, will
be causally connected, whereas if $\Delta s>\pi$, it will be causally
disconnected.

Based on (59) and on the fact that the 2-point correlation function $G\sim
1/\xi_{\rm m}\sim\pi$, we conclude that the maximal correlation distance in
Planckian dissipation is related to the existence of this maximal causality
distance in geodesic.

Figure 16: Left figure: Causal structure for two light-rays separated at a
relative distance $x_{0}=\pi$. The $x$-axis corresponds to the $x_{1}(t)$,
while the $y$-axis to the $r(t)$ coordinate. The external time has been taken
as the dialing parameter, along the $z$-axis in the figure. For the present
purposes, we have considered no motion along the $x_{2}$ direction, and we
have also set $R_{2}=R=\mu=1$. Right figure: Causal structures for three
light-cones as functions of $\left(x_{1}(t),x_{2}(t)\right)$. The $z$-axis
corresponds to time, for which we have set $t=1$ as the time that the
geodesics reach the boundary. Those geodesics that start at any point within
the disk of radius $\pi$ will be causally connected, while disconnected if
otherwise.

##  Appendix C Four pole fitting formula

The full 4-pole fitting formula that can fit the four poles nearest to the
origin in the complex frequency plane is given by the following nine-parameter
formula

$\displaystyle\sigma(\omega)=\sigma_{0}+$ $\displaystyle
Z\frac{\Omega-i\omega}{(\Omega-i\omega)(\Gamma-i\omega)+\omega_{0}^{2}}+$ (61)
$\displaystyle\left(\frac{Z_{s,1}+iZ_{s,2}}{\omega-(\omega_{s,1}+i\omega_{s,2})}+\text{time-
reversed}\right)~{}.$

The weights and positions of the sound poles is constrained by time reversal
symmetry, which dictates that

$\sigma^{*}(-\omega^{*})=\sigma(\omega)~{}.$ (62)

##  Appendix D Memory matrix formalism

The correlation functions of the homogeneous GR and RN fluids are well
described by the standard hydrodynamics of relativistic conformal fluids with
U(1) charge (see kovtunLecturesHydrodynamicFluctuations2012 ). To compute
(29), we simply need the correlator $G_{J^{t}J^{t}}$ which is given by

$G_{J^{t}J^{t}}(\omega,k)=\dfrac{\sigma_{Q}k^{2}}{D_{\rho}k^{2}-i\omega}-\dfrac{k^{2}\omega_{p}^{2}}{\omega^{2}+iD_{\pi}k^{2}\omega-
c_{s}^{2}k^{2}}~{}.$ (63)

This form quite readily shows how this dynamical response has both a
convective part (sound) and a dissipative part. At low frequencies, this
correlator can be expanded as

$G_{J^{t}J^{t}}(\omega,k)=\dfrac{\omega_{p}^{2}}{c_{s}^{2}}+\dfrac{\sigma_{Q}}{D_{\rho}}+i\omega\left[\dfrac{\sigma_{Q}}{D_{\rho}k^{2}}+\omega_{p}^{2}\dfrac{D_{\pi}}{c_{s}^{4}}\right]+\mathcal{O}(\omega^{2})~{}.$
(64)

The leading term is entirely real and will not contribute to the imaginary
part. Therefore, we can eventually obtain (29) as

$\Gamma_{\mathrm{mom.rel.}}=\dfrac{\bar{\mu}^{2}A^{2}}{2(\bar{\epsilon}+\bar{p})D_{\rho}\alpha_{n}}+\dfrac{\bar{\mu}^{2}A^{2}D_{\pi}\bar{n}^{2}}{2c_{s}^{4}(\bar{\epsilon}+\bar{p})^{2}}G^{2}=\Gamma_{d}+\Gamma_{\eta}~{},$
(65)

where we recognize the quantities $\Gamma_{d},\Gamma_{\eta}$ introduced in
(III).

##  Appendix E Scaling of hydrodynamical relaxation rates

Consider an equation of state
$P(T,\mu)/\mu^{3}=a_{0}+a_{1}(T/\mu)^{\hat{\eta}+1}$ where
$\hat{\eta}=(d-\theta)/z$ is the generic effective dimension in the presence
of a dynamical critical exponent $z$ and hyperscaling violation exponent
$\theta$. This equation of state will be a valid approximation for the low-
temperature regime of the holographic Einstein-Maxwell-Dilaton systems, such
as RN and GR. Then, the entropy and charge density one obtains from this
pressure are $s/\mu^{2}=(\hat{\eta}+1)a_{1}(T/\mu)^{\hat{\eta}}$ and
$n/\mu^{2}=3a_{0}-(\hat{\eta}+2)a_{1}(T/\mu)^{\hat{\eta}+1}$. We will now look
at the momentum relaxation rate (29) for a relativistic charged fluid such
that the viscosity saturates the minimal viscosity bound $\eta=s/(4\pi)$ and
we will take the EMD T-scaling
$\sigma_{Q}=\hat{\sigma}_{Q}(T/\mu)^{\hat{\eta}+2}$
davisonIncoherentTransportClean2015 . From the integrated first law
$\epsilon+P=sT+\mu n$, we find $\epsilon=2P$ for this choice such that we
still have a conformal system and therefore $\zeta=0$.

The relaxation rate has two contributions, one $G$-dependent and one
$G$-independent, reminiscent of our result (III), which we will by analogy
name $\Gamma_{\eta}$ and $\Gamma_{d}$. In the general non-conformal case we
have now introduced, these therefore take the form

$\displaystyle\Gamma_{\eta}/\mu$ $\displaystyle=A^{2}(G/\mu)^{2}\dfrac{\pi
a_{1}(\hat{\eta}+1)}{6a_{0}}\left(\dfrac{T}{\mu}\right)^{\hat{\eta}}\dfrac{\left(1-\frac{a_{1}(\hat{\eta}-2)}{3a_{0}}(T/\mu)^{\hat{\eta}+1}\right)^{2}}{\left(1+\frac{a_{1}}{a_{0}}(T/\mu)^{\hat{\eta}+1}\right)^{3}}~{},$
(66) $\displaystyle\Gamma_{d}/\mu$
$\displaystyle=A^{2}\dfrac{a_{1}^{2}(\hat{\eta}+1)^{2}}{6a_{0}\hat{\sigma}_{Q}}\left(\dfrac{T}{\mu}\right)^{\hat{\eta}}\dfrac{\left(\hat{\eta}+\frac{a_{1}(\hat{\eta}-2)}{3a_{0}}(T/\mu)^{\hat{\eta}+1}\right)^{2}}{\left(1+\frac{a_{1}}{a_{0}}(T/\mu)^{\hat{\eta}+1}\right)^{3}}~{}.$

The leading order of $\Gamma_{\eta}$ can therefore be obtained as

$\Gamma_{\eta}/\mu\sim A^{2}(G/\mu)^{2}\dfrac{\pi
a_{1}(\hat{\eta}+1)}{6a_{0}}\left(\dfrac{T}{\mu}\right)^{\hat{\eta}}\sim
A^{2}(G/\mu)^{2}\dfrac{\pi}{2}\dfrac{s}{n}~{}.$ (67)

This shear drag contribution is therefore entirely determined by the entropy
at low temperature. The other contribution, $\Gamma_{d}$, is slightly less
straightforward. When $\hat{\eta}>0$, a similar behavior arises

$\Gamma_{d}/\mu\sim
A^{2}\dfrac{a_{1}^{2}(\hat{\eta}+1)^{2}\hat{\eta}^{2}}{6a_{0}\hat{\sigma}_{Q}}\left(\dfrac{T}{\mu}\right)^{\hat{\eta}}\sim\dfrac{A^{2}}{2n/\mu^{2}}\left(T\dfrac{\partial
s}{\partial T}\right)^{2}\sigma_{Q}^{-1}\left(\dfrac{T}{\mu}\right)^{2}~{}.$
(68)

Therefore in this general case, which encompasses the GR case $\hat{\eta}=1$,
$\Gamma_{d}$ and $\Gamma_{\eta}$ have the same temperature dependence although
$\Gamma_{d}$ is more sensible to the susceptibilities like the specific heat
$T\frac{\partial s}{\partial T}$ and the hydrodynamic transport coefficient
$\sigma_{Q}$. A counterexample of this general rule however arises when
$\hat{\eta}=0$, as it is for the RN black hole for instance, where the leading
order of $\Gamma_{d}$ vanishes and instead one must expand to second order to
have

$\Gamma_{d}/\mu\overset{\hat{\eta}=0}{\sim}A^{2}\dfrac{a_{1}^{4}}{3a_{0}^{3}\hat{\sigma_{Q}}}\left(\dfrac{T}{\mu}\right)^{2}~{}.$
(69)

Finally, we can explain how this $(T/\mu)^{2}$ factor in (68) arises naturally
from the $\alpha_{n}$ factor introduced in (9). To do so, consider the
quantity
$D_{\rho}\Gamma_{d}=A^{2}\frac{\bar{\mu}^{2}}{2(\bar{\epsilon}+\bar{p})\alpha_{n}}$.
We will relax here our assumptions about the equation of state and only assume
some Sommerfeld entropy $s=\gamma(\bar{\mu})\bar{T}$ and $n=n_{0}(\bar{\mu})$
at low temperature. The scaling of $D_{\rho}\Gamma_{d}$ is therefore entirely
determined by that of $\alpha_{n}^{-1}$ which can be determined using

$\displaystyle\left(\dfrac{\partial\bar{n}}{\partial\bar{T}}\right)_{\bar{\mu}}$
$\displaystyle=\left(\dfrac{\partial\bar{s}}{\partial\bar{\mu}}\right)_{\bar{T}}\sim\gamma^{\prime}(\bar{\mu})\bar{T}~{},$
(70)
$\displaystyle\left(\dfrac{\partial\bar{n}}{\partial\bar{\mu}}\right)_{\bar{T}}$
$\displaystyle\sim n_{0}^{\prime}(\bar{\mu})~{},$
$\displaystyle\left(\dfrac{\partial\bar{\epsilon}}{\partial\bar{T}}\right)_{\bar{\mu}}$
$\displaystyle=\bar{T}\left(\dfrac{\partial\bar{s}}{\partial\bar{T}}\right)_{\bar{\mu}}+\bar{\mu}\left(\dfrac{\partial\bar{n}}{\partial\bar{T}}\right)_{\bar{\mu}}\sim\left(\gamma(\bar{\mu})+\bar{\mu}\gamma^{\prime}(\bar{\mu})\right)\bar{T}~{},$
$\displaystyle\left(\dfrac{\partial\bar{\epsilon}}{\partial\bar{\mu}}\right)_{\bar{T}}$
$\displaystyle=\bar{T}\left(\dfrac{\partial\bar{s}}{\partial\bar{\mu}}\right)_{\bar{T}}+\bar{\mu}\left(\dfrac{\partial\bar{n}}{\partial\bar{\mu}}\right)_{\bar{T}}\sim\bar{\mu}n_{0}^{\prime}(\bar{\mu})+\gamma^{\prime}(\bar{\mu})\bar{T}^{2}~{}.$

Then, we can plug these relations into Eqs. (9) and obtain

$\alpha_{n}\sim\dfrac{\bar{T}^{2}\left(\gamma(\bar{\mu})+2\bar{\mu}\gamma^{\prime}(\bar{\mu})\right)+\bar{\mu}n_{0}^{\prime}(\bar{\mu})}{\bar{T}\left[n_{0}^{\prime}(\bar{\mu})\left(\gamma(\bar{\mu})+\bar{\mu}\gamma^{\prime}(\bar{\mu})\right)\bar{T}-\gamma^{\prime}(\bar{\mu})\bar{T}\left(\bar{\mu}n_{0}^{\prime}(\bar{\mu})+\gamma^{\prime}(\bar{\mu})\bar{T}^{2}\right)\right]}\sim\dfrac{\bar{\mu}}{\bar{T}^{2}\gamma(\bar{\mu})}~{}.$
(71)

Therefore, given Sommerfeld entropy, we naturally get that
$D_{\rho}\Gamma_{d}\sim T^{2}$. Provided then that $D_{\rho}\sim T$, which is
the case for the GR holographic metal, you recover the scaling $\Gamma_{d}\sim
T$.

##  Appendix F Lorentz oscillator decoupling

Consider a system of modes coupled to one another in the following way

$\displaystyle\partial_{t}J_{1}+\Gamma_{1}J_{1}+\gamma_{1}J_{2}$
$\displaystyle=E_{1}~{},$ (72)
$\displaystyle\partial_{t}J_{2}+\Gamma_{2}J_{2}-\gamma_{2}J_{1}$
$\displaystyle=E_{2}~{},$

where $\Gamma_{1,2}$ are relaxation rates for the currents $J_{1,2}$,
$E_{1,2}$ are explicit sourcing and $\gamma_{1,2}$ couple the two modes to one
another. In matrix notation $\partial_{t}J_{a}+M_{ab}J_{b}=E_{a}$, this leads
to the following evolution matrix

$M_{\mathrm{LO}}=\begin{pmatrix}\Gamma_{1}&\gamma_{1}\\\
-\gamma_{2}&\Gamma_{2}\\\ \end{pmatrix}~{}.$ (73)

We can then solve this dynamic system and obtain, in frequency space,

$\displaystyle J_{1}(\omega)$
$\displaystyle=\dfrac{\left(\Gamma_{2}-i\omega\right)E_{1}}{(\Gamma_{1}-i\omega)(\Gamma_{2}-i\omega)+\gamma_{1}\gamma_{2}}~{},$
(74) $\displaystyle J_{2}(\omega)$
$\displaystyle=\dfrac{\gamma_{2}E_{1}}{(\Gamma_{1}-i\omega)(\Gamma_{2}-i\omega)+\gamma_{1}\gamma_{2}}~{},$

where we have set $E_{2}=0$ as we are only interested in externally sourcing
one of the currents. Critically, we will be interested in a total current $J$
which overlaps with both $J_{1}$ and $J_{2}$ through

$J=\sigma_{0}E_{1}+aJ_{1}+bJ_{2}$ (75)

where $\sigma_{0}$ is some explicit contribution by the external sourcing.
Therefore, the conductivity associated to this current is

$\sigma=J/E_{1}=\sigma_{0}+\dfrac{a\left(\Gamma_{2}-i\omega\right)+b\gamma_{2}}{(\Gamma_{1}-i\omega)(\Gamma_{2}-i\omega)+\gamma_{1}\gamma_{2}}~{}.$
(76)

This form is very reminiscent of (18) with

$\begin{gathered}Z_{\mathrm{eff}}=a~{},\quad\Omega=\Gamma_{2}+\frac{b}{a}\gamma_{2}\\\
\Gamma=\Gamma_{1}-\frac{b}{a}\gamma_{2}~{},\quad\omega_{0}^{2}=\gamma_{1}\gamma_{2}-\frac{b}{a}\gamma_{2}\left[\Gamma_{1}-\Gamma_{2}-\frac{b}{a}\gamma_{2}\right]~{}.\end{gathered}$
(77)

Let us now compare to the matrix $M$ (11) describing the dynamical
hydrodynamic system in the small lattice expansion. From this system of $4$
coupled fields, it is possible to decouple two by taking the large speed of
sound limit $c_{s}\to\infty$171717To take this limit carefully, one needs to
rescale the momentum modes $\delta\epsilon^{(C)},\delta\pi_{x}^{(C)}$ by a
factor of $c_{s}^{2}$ beforehand. which formally just encodes the assumption
that the sound poles live far from the two poles close to origin. While this
is a relatively simple limit to illustrate the qualitative behavior of the
isolated two pole sector, we must emphasize that this limit will not reproduce
quantitatively the mapping (19) exactly, and that is because there are higher
order effects of the coupling to the sound sector which should be more
carefully disentangled. It will be however a helpful illustration of the
dynamics of the low frequency sector. The two currents remaining $J_{1,2}$ are
then the momentum current density $\delta\pi_{x}^{(0)}$ and the parity-odd
charge density $\delta n^{(S)}$.

The decoupled system then takes the form

$M=\begin{pmatrix}0&AG\bar{\mu}/2\\\
-\frac{AG\bar{\mu}}{(\bar{\epsilon}+\bar{p})\alpha_{n}}&D_{\rho}G^{2}\end{pmatrix}=\begin{pmatrix}0&AG\bar{\mu}/2\\\
-\dfrac{2D_{\rho}G\Gamma_{d}}{A\bar{\mu}}&D_{\rho}G^{2}\end{pmatrix}~{},$ (78)

while the total current of interest is
$J=\sigma_{Q}E_{x}+\omega_{p}^{2}\delta\pi_{x}^{(0)}-\frac{\bar{\mu}A}{2}\omega_{p}^{2}D_{\rho}G\delta
n^{(S)}$. Thus, we deduce from this that $Z_{\mathrm{eff}}=\omega_{p}^{2}$
while the effective momentum relaxation rates and effective couplings are

$\displaystyle\Omega$
$\displaystyle=D_{\rho}G^{2}\left[1-D_{\rho}\Gamma_{d}\right]~{},$ (79)
$\displaystyle\Gamma$ $\displaystyle=(D_{\rho}G)^{2}\Gamma_{d}~{},$
$\displaystyle\omega_{0}^{2}$
$\displaystyle=D_{\rho}G^{2}\Gamma_{d}\left[1-(D_{\rho}G)^{2}+D_{\rho}^{3}G^{2}\Gamma_{d}\right]~{}.$

As expected, there is a discrepancy between Eqs. (19) and Eqs. (79) which just
highlights that the limit $c_{s}\to\infty$ should be refined. However, this
correctly predicts the leading order in $A$ of every coefficient and gives a
very close, qualitative estimate of the corrections at the next order.

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