††institutetext: Instituto de Física, Universidade Federal do Rio de Janeiro,
RJ 21941-972 – Brazil

# Bosonic and Fermionic Holographic Fluctuation and Dissipation at finite
temperature and density

Nathan G. Caldeira  Carlos A. D. Zarro  Henrique Boschi-Filho
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we investigate some general aspects of fluctuation and
dissipation in the holographic scenario at zero and finite density. We model
this situation with a probe string in a diagonal metric representing a black
brane. The string stretches from the black brane to a probe brane thus
simulating a stochastic driven particle. In this scenario, we compute the
admittance, the diffusion coefficient, the correlation functions and the
regularized mean square displacement, for bosons and fermions, all from the
metric components. We check these calculations with the fluctuation-
dissipation theorem. Further, we show that at finite temperature and density,
the mean square displacement in the limit of short times reproduces the usual
quadratic (ballistic) behavior, for bosons and fermions. For large times, we
find ultraslow diffusive processes in various cases, except for bosons at zero
chemical potential. We apply this general analysis in two different models,
for bosons and fermions, in many interesting particular cases.

###### Keywords:

Fluctuation and dissipation, Holography, AdS/CMT, Finite temperature and
density.

##  1 Introduction

The study of the dynamics of finite temperature and/or finite chemical
potential systems occupies an important part in physical sciences. Of
particular interest is how such systems respond to an external force and its
diffusive process. Huge results in this direction can be made studying the
linear regime of those processes and the related quantities obtained from this
analysis. Thus it is interesting to explore methods to get relevant quantities
in this context.

As a tool for this study we can use the framework of AdS/CFT correspondence in
its broader form, applying its dictionary, for example, to analyze and
describe aspects of this kind of phenomenon.

The study of linear response in the context of Brownian motion using
holographic models started in deBoer:2008gu  where it was proposed a set up
based on an stretched string that goes from the black hole horizon to a probe
brane near the space-time boundary. The presence of a horizon associated with
a black hole allows one to calculate the Hawking temperature and to obtain
some quantities as functions of temperature. One then interpret the string
endpoint as a probe particle in a thermal bath. From the excitation of the
string at the boundary one reads the motion of the particle and finds the
diffusion coefficient from the admittance. They also calculate the mean square
displacement, for short and long times reproducing the ballistic and diffusive
regimes, respectively.

This set up was used to study quantum critical points with Lifshitz symmetry
in Tong:2012nf  and with hyperscaling-Lifshitz symmetry in Edalati:2012tc . In
Giataganas:2018ekx  a description in terms of a metric written as monomials of
the holographic coordinate was presented and applied to many different cases
encompassing the previous results deBoer:2008gu ; Tong:2012nf ; Edalati:2012tc
and discussing further cases. In particular, in Ref. Edalati:2012tc  a
chemical potential at zero temperature was introduced in the case of the
extremal black branes. This was also discussed in Banerjee:2015vmo  where a
very low temperature compared to the chemical potential was introduced.

In Ref. Caldeira:2020sot  an exponential factor was used to deform the AdS-
Schwarzschild metric. This deformation was inspired by the soft-wall model
where a quadratic exponential dilaton is included in the action Karch:2006pv .
This exponential factor was introduced in the metric in Ref. Andreev:2006ct
to guarantee confinement in a quark-antiquark potential. This metric was used
in Ref. Caldeira:2020sot  to obtain the admittance, the diffusion coefficient,
compute the regularized mean square displacement and verify the fluctuation-
dissipation theorem for this set up. In Ref. Caldeira:2020rir , this problem
was reanalyzed including a backreaction from the exponential factor altering
the horizon function, generalizing the results of Ref. Caldeira:2020sot . In
Ref. Caldeira:2021izy , this set up was further extended to include a non-zero
chemical potential allowing the discussion of bosonic and fermionic mean
square displacements and also to verify the fluctuation-dissipation theorem in
both (bosonic and fermionic) cases. In particular, the long time regime for
both cases present a Sinai-like diffusion logarithm behavior sinai-like .

In this work we extend some finite temperature results deBoer:2008gu ;
Tong:2012nf ; Edalati:2012tc ; Giataganas:2018ekx  regarding the linear
response and diffusion in the literature to a general diagonal background
metric. We also include a non-zero chemical potential in this general metric
allowing us to distinguish between the bosonic and fermionic cases. Using the
corresponding statistical distributions we calculate and present the values of
the mean square displacement for bosonic and fermionic cases. After obtaining
closed expressions, we get the limits of short and long times for zero and
non-zero chemical potentials for bosons and fermions.

As an application for these general results we apply our findings on two
holographic models of interest. First, we consider the hyperscaling-Lifshitz
model at finite temperature and zero chemical potential. This background was
studied as a holographic dual for a series of condensed matter systems and it
was used in the holographic study of Fermi surfaces Huijse:2011ef . Second, we
discuss a charged dilatonic AdS black hole presented in Ref. Gubser:2009qt ,
as another application of our general results. This is a top-down model with
finite chemical potential which describes dual Fermi liquids with massless
charged fermionic modes. In this model, the entropy is a linear function of
the temperature, for low $T$.

This work is organized as follows. In Section 2, we start setting the class of
metrics explored here and calculating the Hawking temperature. From the Nambu-
Goto action we find the general equation of motion in the linear regime. Then
we solve it using a patching method. In Section 3, we find a general
expression for the linear response function and for the diffusion coefficient
in terms of the general metric elements. In Section 4, we calculate
correlation functions in the bosonic and fermionic cases. In Section 5, we
check our results for the admittance and correlation functions with the
fluctuation-dissipation theorem for bosons and fermions. In section 6, our
goal is to calculate the regularized mean square displacement
$s^{2}_{\text{reg}}(t)$ for multiple scenarios. We begin establishing the
boundary conditions for the solutions and from them we get a general integral
representation for $s^{2}_{\text{reg}}(t)$. After that, we specialize to four
different cases: bosonic and fermionic at zero and non-zero chemical
potentials. For each of these cases we further obtain the regimes of short and
large times. At the end of this section we present a summary of this results
and compare them. Going ahead, in section 7, we apply the previous results to
a hyperscaling violation $d+2$ spacetime metric at finite temperature and zero
chemical potential presenting some interesting cases for particular values of
the parameters of the model. In section 8, we look at a top-down finite
chemical potential model in $2+1$ dimensions introduced in Gubser:2009qt . We
use it as an illustration for our general finite chemical results in the
bosonic and fermionic cases presenting interesting properties. Finally in the
last Section 9, we show our conclusions making general comments about the
results. Some technical calculations are presented in two appendices.

##  2 Nambu-Goto action and equations of motion

In this section we study the dynamics of the probe string, finding its
equations of motion and their solutions. The set up we are going to use
deBoer:2008gu  considers a probe string in bulk with a black hole. One
endpoint of the string interacts with the horizon (IR) and the other to a
brane near the boundary (UV). The endpoint near the UV behaves as a particle
simulating a stochastic motion. This set up was used to describe Lifshitz as
well as hyperscaling violation Tong:2012nf ; Edalati:2012tc ;
Giataganas:2018ekx .

We start with a general diagonal metric

$ds^{2}=-g_{tt}dt^{2}+g_{rr}dr^{2}+\sum_{i=1}^{d}g_{ii}dx_{i}^{2},$ (1)

where we defined

$g_{tt}\equiv a(r)f(r)\,;\qquad g_{rr}\equiv\frac{b(r)}{f(r)}\,;\qquad
g_{ii}\equiv c(r)\,;\quad(i=1,\cdots,d).$ (2)

Note that $d$ is the number of spatial dimensions and $r$ is the holographic
coordinate. The horizon function $f(r)$ of the black hole is assumed to have a
simple zero at $r=r_{h}$ and goes to 1 for $r\to\infty$. In the particular
case of asymptotically AdS spaces, $a(r)$, $c(r)$ and $1/b(r)$ go to $r^{2}$
as $r\to\infty$, where the boundary is located.

The corresponding Hawking temperature is given by

$T=\frac{1}{4\pi}\sqrt{\frac{a(r_{h})}{b(r_{h})}[f^{\prime}(r_{h})]^{2}}=\frac{|f^{\prime}(r_{h})|}{4\pi}\sqrt{\frac{a(r_{h})}{b(r_{h})}}$
(3)

In order to describe the motion of the probe string we use the Nambu-Goto
action:

$S_{NG}=-\frac{1}{2\pi\alpha^{\prime}}\int d\tau d\sigma\sqrt{-\gamma}\,,$ (4)

where $\alpha^{\prime}$ is the string tension, $\gamma={\rm
det}(\gamma_{\alpha\beta})$ and
$\gamma_{\alpha\beta}=g_{mn}\partial_{\alpha}X^{m}\partial_{\beta}X^{n}$ is
the induced metric on the worldsheet with $m,n=0,1,\cdots,d+1$.

We choose the usual static gauge, where $t=\tau$, $r=\sigma$ and
$X^{m}=X^{m}(\tau,\sigma)$. By using the general metric Eq.(1) and expanding
the Nambu-Goto action keeping up to quadratic terms $\dot{X}^{2}$, $X^{\prime
2}$, one gets:

$S_{NG}\approx-\frac{1}{4\pi\alpha^{\prime}}\int d\tau
d\sigma\left[\;\frac{\dot{X}^{2}\,g_{ii}(r)\sqrt{g_{tt}(r)g_{{rr}}(r)}}{g_{tt}(r)}-\frac{X^{\prime
2}\,g_{ii}(r)\sqrt{g_{tt}(r)g_{{rr}}(r)}}{g_{{rr}}(r)}\;\right]\,,$ (5)

where $\dot{X}=\partial_{\tau=t}X$ and $X^{\prime}=\partial_{\sigma=r}X$. From
this action one obtains the EoM and define $g_{xx}=g_{ii}$ (without sum over
$i$ from now on) for any $i$,

$\displaystyle\frac{\partial}{\partial
r}\left(\frac{X^{\prime}(t,r)\left(g_{xx}(r)\sqrt{g_{tt}(r)g_{{rr}}(r)}\right)}{g_{{rr}}(r)}\right)+\frac{\ddot{X}(t,r)\left(g_{xx}(r)\sqrt{g_{{tt}}(r)g_{{rr}}(r)}\right)}{g_{{tt}}(r)}=0\,.\qquad$
(6)

Then, by using the decomposition $X(t,r)=e^{i\omega t}h_{\omega}(r)$, the EoM
reads:

$\displaystyle\frac{\partial}{\partial
r}\left(h_{\omega}^{\prime}(r)\frac{g_{xx}(r)\sqrt{g_{{tt}}(r)}}{\sqrt{g_{{rr}(r)}}}\right)+\frac{\omega^{2}g_{xx}(r)\sqrt{g_{rr}(r)}}{\sqrt{g_{tt}(r)}}h_{\omega}(r)=0\,.$
(7)

For a general background metric this equation cannot be solved analytically.
Then, we will apply a standard patching method to obtain approximate
analytical solutions, to be presented in the next section.

###  2.1 General solution in the hydrodynamic limit

In this section we are going to consider approximate solutions for Eq. (7)
considering three different particular cases:

A) Near the horizon (IR);

B) Hydrodynamic limit: $\omega\to 0$;

C) Far from the horizon (UV).

Then, we will match these expressions to get an approximate solution for the
Eq. (7) near UV, but keeping essential data from the IR and the hydrodynamic
limit. This is useful to describe the motion of the string endpoint simulating
the stochastic behavior.

First, we consider the solution for region A:

$h_{A\omega}(r)=\frac{A_{1}}{\sqrt{g_{xx}(r_{h})}}\left[1-i\frac{\omega}{f^{\prime}(r_{h})}\sqrt{\frac{b(r_{h})}{a(r_{h})}}\log\left(\frac{r}{r_{h}}-1\right)\right],$
(8)

where $A_{1}$ is a normalization constant. This solution is obtained in
Appendix A.

Now we solve the equation of motion for the hydrodynamic limit which is a
general solution in $r$ but only up to order $\omega$. In this case one can
neglect the term proportional to $\omega^{2}$ in Eq. (7)

$\frac{d}{dr}\left(\frac{g_{xx}\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\frac{dh_{\omega}}{dr}\right)=0.$
(9)

Solving it we obtain

$\displaystyle h^{B}_{\omega}(r)$ $\displaystyle=$ $\displaystyle
B_{1}(\omega)\int^{r}\frac{\sqrt{g_{rr}}}{g_{xx}\sqrt{g_{tt}}}dr^{\prime}+B_{2}(\omega)$
(10) $\displaystyle=$ $\displaystyle
B_{1}(\omega)\int^{r}\frac{\sqrt{b(r^{\prime})}}{g_{xx}(r^{\prime})f(r^{\prime})\sqrt{a(r^{\prime})}}dr^{\prime}+B_{2}(\omega)\,,$
(11)

where $B_{1}$ and $B_{2}$ are integration constants with respect to $r$ but
are functions of $\omega$. This can be seen as the general solution up to
order $\omega^{2}$ of the equation of motion.

One can approximate the integral above in the IR using the fact that $f(r)$
has a simple zero at the horizon. Then we can write

$\displaystyle h_{\omega\rm(IR)}^{B}$ $\displaystyle\approx$
$\displaystyle\frac{\sqrt{b(r_{h})}B_{1}(\omega)}{g_{xx}(r_{h})\sqrt{a(r_{h})}}\int^{r}\frac{1}{f(r^{\prime})}dr^{\prime}+B_{2}(\omega)$
(12) $\displaystyle=$
$\displaystyle\frac{\sqrt{b(r_{h})}B_{1}(\omega)}{g_{xx}(r_{h})\sqrt{a(r_{h})}f^{\prime}(r_{h})}\log\left(\frac{r}{r_{h}}-1\right)+B_{2}(\omega)\,.$
(13)

In order to obtain the functions $B_{1}$ and $B_{2}$ of $\omega$, we compare
this expression with the solution Eq. (8) for the deep IR region:

$B_{1}(\omega)=-iA_{1}\omega\sqrt{g_{xx}(r_{h})},\quad
B_{2}(\omega)=\frac{A_{1}}{\sqrt{g_{xx}(r_{h})}}\,,$ (14)

where the constant $A_{1}$ can be fixed by a convenient normalization.

For the case far from the horizon (case C: UV region) we use the solution for
the case B (10) considering the approximation $f(r)\approx 1$ so that the
solution becomes

$\displaystyle h_{\omega}^{C}(r)\approx h^{B}_{\omega(\rm UV)}(r)$
$\displaystyle=$ $\displaystyle
B_{1}(\omega)\int^{r}\frac{\sqrt{b(r^{\prime})}}{g_{xx}(r^{\prime})\sqrt{a(r^{\prime})}}dr^{\prime}+B_{2}(\omega)$
(15) $\displaystyle=$
$\displaystyle\frac{A_{1}}{\sqrt{g_{xx}(r_{h})}}\left(1-i\omega
g_{xx}(r_{h})\int^{r}\frac{\sqrt{b(r^{\prime})}}{g_{xx}(r^{\prime})\sqrt{a(r^{\prime})}}dr^{\prime}\right).$
(16)

In the last expression we are interested mainly in the asymptotic behavior of
$h_{\omega}^{C}(r)$ for large $r$.

##  3 Admittance and diffusion coefficient

In this section we proceed to calculate the admittance from which we find the
diffusion coefficient. The admittance is the linear response of the system to
an external force. Thus, first we need to introduce an small external force
acting on the boundary particle. With this purpose, we turn on an
electromagnetic potential $A_{\mu}$ on the UV brane. This will not change the
bulk dynamics but will introduce an external force to the string endpoint. The
linearized Nambu-Goto action, Eq. (5), then becomes (with unit charge)

$\displaystyle S=-\frac{1}{4\pi\alpha^{\prime}}\int
dtdr\left[\;\dot{X}^{2}\frac{g_{xx}\sqrt{g_{rr}}}{\sqrt{g_{tt}}}-X^{\prime
2}\frac{g_{xx}\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\right]+\int
dt\left(A_{t}+\vec{A}\cdot\vec{\dot{x}}\right)\Big{|}_{r=r_{b}}\,.\quad$ (17)

From this expression we get a modified equation of motion in the brane
position given by

$\displaystyle\frac{\sqrt{g_{tt}(r_{b})}}{\sqrt{g_{rr}(r_{b})}}g_{xx}(r_{b})X^{\prime}|_{r=r_{b}}-2\pi\alpha^{\prime}F=0\,,$
(18)

with $F=\partial_{x}A_{t}-\partial_{t}A_{x}$. Since $X^{\prime}=\partial_{r}X$
and $X(t,r)=e^{i\omega t}h^{C}_{\omega}(r)$, we can write from Eq. (15)

$\displaystyle
F(\omega)=\frac{-iA_{1}\omega\sqrt{g_{xx}(r_{h})}}{2\pi\alpha^{\prime}}f(r_{b})\approx\frac{-iA_{1}\omega\sqrt{g_{xx}(r_{h})}}{2\pi\alpha^{\prime}}.$
(19)

Note that for this regime of small frequencies and large string energy (large
$r_{b}$, $f(r_{b})\approx 1$) the force on the particle depends only on the IR
structure of the metric. Then, the admittance can be written as

$\displaystyle\chi(\omega)=\frac{h^{C}_{\omega}(r_{b})}{F(\omega)}=2\pi\alpha^{\prime}\frac{\left(1-i\omega
g_{xx}(r_{h})\int^{r_{b}}\frac{\sqrt{b(r^{\prime})}}{g_{xx}(r^{\prime})\sqrt{a(r^{\prime})}}dr^{\prime}\right)}{-i\omega
g_{xx}(r_{h})}\,.$ (20)

So, its imaginary part is

$\displaystyle\Im{\chi(\omega)}=\frac{2\pi\alpha^{\prime}}{\omega
g_{xx}(r_{h})}\,.$ (21)

The results expressed in Eqs. (20)-(21) are valid for any metric in the form
of Eq. (1) in the regime of small frequencies.

From the response function $\chi(\omega)$, Eq. (20), we can calculate the
diffusion coefficient $D$. It is given by Tong:2012nf  :

$\displaystyle D=\frac{1}{\beta}\lim_{\omega\to
0}(-i\omega\chi(\omega))=\frac{2\pi\alpha^{\prime}}{\beta
g_{xx}(r_{h})}=\frac{2\pi\alpha^{\prime}T}{g_{xx}(r_{h})}$ (22)

We can rewrite this result just in terms of the metric using (3) as

$D=\frac{\alpha^{\prime}|f^{\prime}(r_{h})|}{2g_{xx}(r_{h})}\sqrt{\frac{a(r_{h})}{b(r_{h})}}\,.$
(23)

At this stage we have found the linear response and the diffusion coefficient
as functions of the black hole radius $r_{h}$. Also we have the temperature as
function of the same parameter, so in principle we can invert the equations to
get results as functions of temperature. On the other hand, as temperature
depends of different parts of the metric compared with for example admittance
we can not write at this point a general expression for it as function of
temperature. We will need then to specialize to some particular model to get
such relation.

Before we conclude this section, let us comment that the real part of the
admittance $\chi(\omega)$, Eq. (20), is given by

$\displaystyle\Re\chi(\omega)=2\pi\alpha^{\prime}\int^{r_{b}}\frac{\sqrt{b(r^{\prime})}}{g_{xx}(r^{\prime})\sqrt{a(r^{\prime})}}dr^{\prime}\,.$
(24)

This response goes beyond the hydrodynamical limit Zaanen:2015oix . In
particular, if the metric parameters are related as $a(r)=[b(r)]^{-1}$, as
happens in AdS-like models, one finds

$\displaystyle\Re\chi(\omega)=2\pi\alpha^{\prime}\int^{r_{b}}\frac{1}{g_{xx}(r^{\prime})a(r^{\prime})}dr^{\prime}\,.$
(25)

##  4 Correlation functions

The mean square displacement is a measure of the variance of the random walk
of the particle in the thermal bath from the motion of the probe string. In
the following, we are going to calculate in general grounds this quantity
using the results from previous sections.

First, in order to obtain the mean square displacement one needs to impose
ingoing and outgoing boundary conditions near the horizon

$\displaystyle h^{IR}_{\omega}(r)$ $\displaystyle=$
$\displaystyle\frac{A}{\sqrt{g_{xx}(r_{h})}}\left[e^{i\omega
r_{*}}+Be^{-i\omega r_{*}}\right]$ (26) $\displaystyle=$
$\displaystyle\frac{A}{\sqrt{g_{xx}(r_{h})}}\left[e^{i\frac{\omega}{f^{\prime}(r_{h})}\sqrt{\frac{b(r_{h})}{a(r_{h})}}\log\left(\frac{r}{r_{h}}-1\right)}+Be^{-i\frac{\omega}{f^{\prime}(r_{h})}\sqrt{\frac{b(r_{h})}{a(r_{h})}}\log\left(\frac{r}{r_{h}}-1\right)}\right]\,.$
(27)

In the UV region the general solution considering these modes is

$\displaystyle
h^{UV}_{\omega}(r)=\frac{A}{\sqrt{g_{xx}(r_{h})}}\left[\left(1-i\omega
g_{xx}(r_{h})\int^{r}\frac{\sqrt{b(r^{\prime})}}{g_{xx}(r^{\prime})\sqrt{a(r^{\prime})}}dr^{\prime}\right)\right.$
(28) $\displaystyle\left.+B\left(1+i\omega
g_{xx}(r_{h})\int^{r}\frac{\sqrt{b(r^{\prime})}}{g_{xx}(r^{\prime})\sqrt{a(r^{\prime})}}dr^{\prime}\right)\right]\,.$
(29)

where the coefficients $A$ and $B$ are the same as in the IR region.

Matching the UV solution with the Neumann boundary condition in $r=r_{b}$, one
obtains the $B$ is a pure phase. On the other side imposing Neumann b.c. at
the IR one obtains a discretization of the frequencies as

$\displaystyle\Delta\omega=\frac{\pi
f^{\prime}(r_{h})}{\log\left(\frac{1}{\epsilon}\right)}\sqrt{\frac{a(r_{h})}{b(r_{h})}}=\frac{4\pi^{2}T}{\log\left(\frac{1}{\epsilon}\right)}\,,$
(30)

analogous to the result found in deBoer:2008gu . For details, see the Appendix
B.

###  4.1 Grand canonical ensemble and correlation functions

In the grand canonical ensemble, the density operator is defined as

$\rho_{0}=\frac{e^{-\beta(\sum_{\omega>0}\omega
a_{\omega}^{\dagger}a_{\omega}-\mu
a_{\omega}^{\dagger}a_{\omega}})}{\text{Tr}\left(\,e^{-\beta(H-\mu
N)}\right)}\,,$ (31)

where $a_{\omega}^{\dagger}$ and $a_{\omega}$ are the usual creation and
annihilation operators which satisfy

$\displaystyle\langle
a^{\dagger}_{\omega}a_{\omega}\rangle=\frac{\delta_{\omega\omega^{\prime}}}{e^{\beta\left(\omega-\mu\right)}\pm
1};\qquad\langle
a^{\dagger}_{\omega_{n}}a^{\dagger}_{\omega}\rangle=0;\qquad\langle
a_{\omega}a_{\omega}\rangle=0\,.$ (32)

Note that the plus (minus) sign corresponds to the fermionic (bosonic) case.
With these operators, one can write down the solution for the equation of
motion near the boundary

$\displaystyle
X(t,r)=\sum_{\omega>0}\left(a_{\omega}h_{\omega}^{UV}(r)e^{-i\omega
t}+a_{\omega}^{\dagger}(h_{\omega}^{UV}(r))^{*}e^{i\omega t}\right)\,,$ (33)

where we used explicitly the quantization of the frequencies obtained in the
previous section. Then, the two point function for the string endpoint reads

$\displaystyle\langle x(t)x(0)\rangle$ $\displaystyle\equiv$
$\displaystyle\langle X(t,r_{b})X(0,r_{b})\rangle$ (34) $\displaystyle=$
$\displaystyle\sum_{\omega>0}\sum_{\omega^{\prime}>0}\Big{(}\frac{h^{UV*}_{\omega}(r_{b})h^{UV}_{\omega^{\prime}}(r_{b})e^{i\omega
t}+h^{UV}_{\omega}(r_{b})h^{UV*}_{\omega^{\prime}}(r_{b})e^{-i\omega
t}}{e^{\beta\left(\omega-\mu\right)}\pm 1}$ (36)
$\displaystyle\qquad\qquad+h^{UV}_{\omega}(r_{b})h^{UV*}_{\omega^{\prime}}(r_{b})e^{-i\omega
t}\Big{)}\delta_{\omega\omega^{\prime}}$ $\displaystyle=$
$\displaystyle\sum_{\omega>0}|h^{UV}_{\omega}(r_{b})|^{2}\Big{(}\frac{2\cos(\omega
t)}{e^{\beta\left(\omega-\mu\right)}\pm 1}+e^{-i\omega t}\Big{)}$ (37)

Substituting in the above equation the expression for $h_{\omega}^{UV}(r_{b})$
given by (28) and disregarding terms of order $\omega^{2}$, we obtain

$\displaystyle\langle x(t)x(0)\rangle$ $\displaystyle=$
$\displaystyle\frac{\pi\alpha^{\prime}f^{\prime}(r_{h})}{g_{xx}(r_{h})\log\left(\frac{1}{\epsilon}\right)}\sqrt{\frac{a(r_{h})}{b(r_{h})}}\sum_{\omega>0}\frac{1}{\omega}\left(\frac{2\cos(\omega
t)}{e^{\beta\left(\omega-\mu\right)}\pm 1}+e^{-i\omega t}\right)\,.$ (38)

Considering the approximation

$\displaystyle d\omega\sim\Delta\omega=\frac{\pi
f^{\prime}(r_{h})}{\log\left(\frac{1}{\epsilon}\right)}\sqrt{\frac{a(r_{h})}{b(r_{h})}},$
(39)

one can rewrite

$\displaystyle\langle x(t)x(0)\rangle$ $\displaystyle=$
$\displaystyle\frac{\alpha^{\prime}}{g_{xx}(r_{h})}\int_{0}^{\infty}\frac{d\omega}{\omega}\left(\frac{2\cos(\omega
t)}{e^{\beta\left(\omega-\mu\right)}\pm 1}+e^{-i\omega t}\right)$ (40)
$\displaystyle=$ $\displaystyle\langle x(0)x(t)\rangle^{*}\,.$ (42)

Analogously, one can obtain

$\displaystyle\langle x(t)x(t)\rangle$ $\displaystyle=$ $\displaystyle\langle
X(t,r_{b})X(t,r_{b})\rangle$ (43) $\displaystyle=$
$\displaystyle\sum_{\omega>0}|h^{UV}_{\omega}(r_{b})|^{2}\Big{(}\frac{2}{e^{\beta(\omega-\mu)}\pm
1}+1\Big{)}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{\prime}}{g_{xx}(r_{h})}\int_{0}^{\infty}\frac{d\omega}{\omega}\left(\frac{2}{e^{\beta(\omega-\mu)}\pm
1}+1\right)$ $\displaystyle=$ $\displaystyle\langle x(0)x(0)\rangle\,.$ (45)

##  5 Fluctuation-dissipation theorem

In this Section, we are going to verify the consistency of our results with
the fluctuation-dissipation theorem, using the admittance obtained in Section
3 and the correlation functions in Section 4.

Starting from the correlation functions $\langle x(t)x(0)\rangle$ and $\langle
x(0)x(t)\rangle$, one can define a symmetric Green’s function as:

$G_{\rm Sym}(t)\equiv\frac{1}{2}\left(\langle x(t)x(0)\rangle+\langle
x(0)x(t)\rangle\right)\,.$ (46)

The fluctuation-dissipation theorem in the presence of a chemical potential
for the bosonic Zubarev  and fermionic Markov:2009ue  cases, can be written as

$\displaystyle G_{\rm
Sym}^{B,F}(t)=\mathcal{F}^{-1}\left[\left(1+2n_{B,F}\right)\Im\chi(\omega)\right],$
(47)

where $\mathcal{F}^{-1}[\cdots]$ denotes the inverse Fourier transform,
$n_{B,F}$ are the Bose-Einstein and Fermi-Dirac distributions, and
$\Im\chi(\omega)$ is the imaginary part of the admittance. Note that the
frequencies in this equation are positive physical quantities.

Then, one can rewrite of the r.h.s of equation (47) as Fourier transforms as

$\displaystyle\mathcal{F}^{-1}\left[\left(1+2n_{B,F}\right)\Im\chi(\omega)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega\left(1+\frac{2}{e^{\beta(|\omega|-\mu)}\pm
1}\right)\Im\chi(\omega)e^{i\omega t}$ (48) $\displaystyle=$
$\displaystyle\frac{\alpha^{\prime}}{g_{xx}(r_{h})}\int_{-\infty}^{\infty}\frac{d\omega}{|\omega|}\left(1+\frac{2}{e^{\beta(|\omega|-\mu)}\pm
1}\right)e^{i\omega t}\,,$ (49)

where we used the admittance, Eq. (21), and the plus (minis) sign represents
fermions (bosons).

On the other hand, from the correlation functions, Eqs. (40), the l.h.s. of
Eq. (46) becomes

$\displaystyle G_{\rm Sym}^{B,F}(t)$ $\displaystyle=$
$\displaystyle\frac{\alpha^{\prime}}{g_{xx}(r_{h})}\int_{0}^{\infty}\frac{d\omega}{\omega}\left(\frac{4\cos(\omega
t)}{e^{\beta\left(\omega-\mu\right)}\pm 1}+e^{-i\omega t}+e^{i\omega
t}\right)$ (50) $\displaystyle=$
$\displaystyle\frac{\alpha^{\prime}}{g_{xx}(r_{h})}\int_{0}^{\infty}\frac{d\omega}{|\omega|}\left(\frac{2\left(e^{-i\omega
t}+e^{i\omega t}\right)}{e^{\beta\left(|\omega|-\mu\right)}\pm 1}+e^{-i\omega
t}+e^{i\omega t}\right).$ (51)

Since

$\displaystyle\int_{0}^{\infty}d\omega f(|\omega|)e^{i\omega
t}+\int_{0}^{\infty}d\omega f(|\omega|)e^{-i\omega
t}=\int_{-\infty}^{\infty}d\omega f(|\omega|)e^{i\omega t}\,,$ (52)

the Eq.(51) becomes

$\displaystyle G_{\rm Sym}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{\prime}}{g_{xx}(r_{h})}\int_{-\infty}^{\infty}\frac{d\omega}{|\omega|}\left(\frac{2}{e^{\beta\left(|\omega|-\mu\right)}\pm
1}+1\right)e^{i\omega t}\,,$ (53)

which coincides with Eq. (48). Therefore, it completes the check of our
calculations with the fluctuation-dissipation theorem.

##  6 Mean square displacements

From the results of Section 4, one can calculate the regularized expression
for the mean square displacement as:

$s^{2}_{\rm
reg}(t)=\langle:[x(t)-x(0)]^{2}:\rangle\equiv\langle:[X(t,r_{b})-X(0,r_{b})]^{2}:\rangle\,,$
(54)

where the double dots notation $:[\cdots]:$ means that we are using normal
ordering. Explicitly,

$\displaystyle s^{2}_{\rm reg}(t)$ $\displaystyle=$
$\displaystyle\frac{\alpha^{\prime}}{g_{xx}(r_{h})}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{4(1-\cos(\omega
t))}{e^{\beta(\omega-\mu)}\pm 1}$ (55) $\displaystyle=$
$\displaystyle\frac{8\alpha^{\prime}}{g_{xx}(r_{h})}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{\sin^{2}\left(\frac{\omega
t}{2}\right)}{e^{\beta(\omega-\mu)}\pm 1}.$ (56)

In the following, we are going to calculate the regularized mean square
displacement in various interesting cases.

###  6.1 Zero chemical potential

In this section, we study the particular case of zero chemical potential from
previous results. This is interesting, for instance, in models which describe
the superfluid Bose-Mott insulator transition Zaanen:2015oix  and massless
Dirac fermions in graphene novoselov ; moriconi .

For zero chemical potential, the mean square displacement, Eq. (55), reads

$\displaystyle s^{2}_{\rm reg}(t)$ $\displaystyle=$
$\displaystyle\frac{8\alpha^{\prime}}{g_{xx}(r_{h})}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{\sin^{2}\left(\frac{\omega
t}{2}\right)}{e^{\beta\omega}\pm 1}\,.$ (57)

This expression is valid for the bosonic $(-)$ and fermionic $(+)$ cases. In
the following we discuss separately these two cases.

####  6.1.1 Bosons

To solve the integral (57) for the bosonic case $(-)$, we follow
Caldeira:2020sot  and find

$\displaystyle s^{2}_{\rm reg}(t)$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{\prime}}{g_{xx}(r_{h})}\log\left(\frac{\sinh(\frac{t\pi}{\beta})}{\frac{t\pi}{\beta}}\right)\,.$
(58)

For the short time approximation $t\ll\beta$, we have

$\displaystyle s^{2}_{\rm
reg}(t)\approx\frac{\pi^{2}\alpha^{\prime}}{3g_{xx}(r_{h})}\frac{t^{2}}{\beta^{2}}\,,$
(59)

which is the expected result for the ballistic regime. Considering now the
approximation for large times $t\gg\beta$, one finds

$\displaystyle s^{2}_{\rm
reg}(t)\approx\frac{2\pi\alpha^{\prime}}{g_{xx}(r_{h})}\frac{t}{\beta}\,,$
(60)

which is the standard diffusion result.

####  6.1.2 Fermions

Now, considering the fermionic case $(+)$ corresponding to the integral (57),
we have

$\displaystyle s^{2}_{\rm reg}(t)$ $\displaystyle=$
$\displaystyle\frac{8\alpha^{\prime}}{g_{xx}(r_{h})}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{\sin^{2}\left(\frac{\omega
t}{2}\right)}{e^{\beta\omega}+1}\,.$ (61)

In order to simplify the following analysis, we perform the change of
variables ${\omega t}/{2}\to x$, rewriting the above integral as

$\displaystyle
I=\int_{0}^{\infty}\frac{dx}{x}\frac{\sin^{2}\left(x\right)}{e^{\frac{2\beta}{t}x}+1}\,.$
(62)

In the limit $\beta\gg t$ (short times) we have $e^{\frac{2\beta}{t}}\gg 1$,
and we can write

$\displaystyle
I\approx\int_{0}^{\infty}dx\frac{\sin^{2}(x)e^{-\frac{2\beta}{t}x}}{x}=\frac{1}{4}\log\left(1+\frac{t^{2}}{\beta^{2}}\right)\approx\frac{t^{2}}{4\beta^{2}}\,,$
(63)

so that

$\displaystyle s^{2}_{\rm reg}(t)$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{\prime}}{g_{xx}(r_{h})}\frac{t^{2}}{\beta^{2}},$
(64)

in agreement with the known result for the ballistic regime.

For large times, or $t\gg\beta$, we can make the approximation

$\displaystyle I$ $\displaystyle\approx$
$\displaystyle\int_{0}^{\infty}\frac{dx}{x}\frac{\sin^{2}\left(x\right)}{2\left(1+\frac{\beta}{t}x\right)}.$
(65)

Asymptotically this is

$\displaystyle I\sim\frac{1}{4}\log\left(\frac{t}{\beta}\right)\,,$ (66)

and then for this case, one finds

$\displaystyle s^{2}_{\rm reg}(t)$ $\displaystyle\sim$
$\displaystyle\frac{2\alpha^{\prime}}{g_{xx}(r_{h})}\log\left(\frac{t}{\beta}\right)\,,$
(67)

which is a Sinai-like subdiffusive regime sinai-like .

###  6.2 Finite chemical potential

In the case of a finite chemical potential the mean square displacement is
given by Eq. (55) which we repeat here for convenience

$\displaystyle s^{2}_{\rm reg}(t)$ $\displaystyle=$
$\displaystyle\frac{8\alpha^{\prime}}{g_{xx}(r_{h})}\int_{0}^{\infty}\frac{d\omega}{\omega}\frac{\sin^{2}\left(\frac{\omega
t}{2}\right)}{e^{\beta(\omega-\mu)}\pm 1}\,.$ (68)

In the following, we specialize to the bosonic $(-)$ and fermionic $(+)$
cases.

####  6.2.1 Bosons

In order to calculate the mean square displacement given by the above equation
for the bosonic case, we take $\mu<0$, and consider the series expansion

$\frac{1}{e^{\beta(\omega-\mu)}-1}=\frac{e^{-\beta(\omega-\mu)}}{1-e^{-\beta(\omega-\mu)}}=\sum_{n=0}^{\infty}e^{-\beta(\omega-\mu)(n+1)}\,.$
(69)

Then,

$s^{2}_{\rm
Breg}(t)=\frac{8\alpha^{\prime}}{g_{xx}(r_{h})}\sum_{n=1}^{\infty}\int_{0}^{\infty}\frac{d\omega}{\omega}e^{-\beta(\omega-\mu)n}\sin^{2}(\frac{\omega
t}{2}).$ (70)

Performing the integral, one gets

$s^{2}_{\rm
Breg}(t)=\frac{2\alpha^{\prime}}{g_{xx}(r_{h})}\sum_{n=1}^{\infty}e^{\beta\mu
n}\log\left(1+\frac{t^{2}}{n^{2}\beta^{2}}\right)\,.$ (71)

This expression can be rewritten in a formal way as

$\displaystyle s^{2}_{\rm
Breg}(t)=\frac{2\alpha^{\prime}}{g_{xx}(r_{h})}\left\\{2{\rm
Li_{0}}^{(1,0)}\left(0,e^{\mu}\right)-e^{\mu}\left[{\Phi}^{(0,1,0)}\left(e^{\mu},0,1+i\frac{t}{\beta}\right)\right.\right.$
(72)
$\displaystyle\left.\left.+{\Phi}^{(0,1,0)}\left(e^{\mu},0,1-i\frac{t}{\beta}\right)\right]\right\\}\,.$
(73)

In this equation we used the following notation: Li${}_{n}^{(1,0)}(x,y)$ is
the first derivative of the polylogarithm function of order $n$ with respect
to its first argument $x$; $\Phi^{(0,1,0)}(x,y,z)$ is the first derivative of
the Lerch transcendent function with respect to the second argument $y$.

First we consider $t\ll\beta$, which is the short time approximation, and then
from Eq. (71), we find

$\displaystyle s^{2}_{\rm Breg}(t)$ $\displaystyle\approx$
$\displaystyle\frac{2\alpha^{\prime}}{g_{xx}(r_{h})}\left(\sum_{n=1}^{\infty}\frac{e^{\beta\mu
n}}{n^{2}}\right)\frac{t^{2}}{\beta^{2}}=\frac{2\alpha^{\prime}}{g_{xx}(r_{h})}\text{Li}_{2}\left(e^{\beta\mu}\right)\frac{t^{2}}{\beta^{2}}\,.$
(74)

In the right hand side of this equation we used the polylogarithm function of
order 2, $\text{Li}_{2}\left(e^{\beta\mu}\right)$. This equation gives the
usual ballistic behavior since it goes like $t^{2}$.

On the other side, the late time approximation $t\gg\beta$ for Eq. (71) gives

$\displaystyle s^{2}_{\rm Breg}(t)$ $\displaystyle\approx$
$\displaystyle\frac{4\alpha^{\prime}}{g_{xx}(r_{h})}\frac{1}{e^{-\beta\mu}-1}\log\left(\frac{t}{\beta}\right)\,,$
(75)

where we disregarded a time independent contribution. This equation
corresponds to a subdiffusive regime which is due to the presence of the non-
zero chemical potential in this case. This result is analogous to what was
found in classical physical systems in sinai-like  or in Lorentz invariant
bosonic theories Caldeira:2021izy .

####  6.2.2 Fermions

Now, we consider the regularized mean square displacement for the fermionic
case. From Eq. (55), with $\mu>0$, we have

$s^{2}_{\rm
Freg}(t)=\frac{8\alpha^{\prime}}{g_{xx}(r_{h})}\int_{0}^{\infty}\frac{d\omega}{\omega}\left(\frac{\sin^{2}(\frac{\omega
t}{2})}{e^{\beta(\omega-\mu)}+1}\right)\,.$ (76)

In order to evaluate this integral we resort to the Sommerfeld expansion:

$\displaystyle\int_{0}^{\infty}\frac{d\omega}{\omega}\left(\frac{\sin^{2}(\frac{\omega
t}{2})}{e^{\beta(\omega-\mu)}+1}\right)$ $\displaystyle=$
$\displaystyle\int_{0}^{\mu}\frac{\sin^{2}(\frac{\omega
t}{2})}{\omega}d\omega+\cdots$ (77) $\displaystyle=$
$\displaystyle\frac{1}{2}(-\text{Ci}(t\mu)+\log(t\mu)+\gamma)+\cdots,$ (78)

where $\text{Ci}(z)$ is the Cosine integral function, $\gamma$ is Euler-
Mascheroni constant and we disregarded terms of order ${1}/{\mu^{2}\beta^{2}}$
and higher, since we are considering the low temperature regime,
$\mu\beta=\frac{\mu}{T}\gg 1$.

The mean square displacement, Eq. (76), for small times $\mu t\ll 1$, can be
approximated as

$s^{2}_{\rm
Freg}(t)\approx\frac{\alpha^{\prime}}{g_{xx}(r_{h})}\mu^{2}t^{2}\,.$ (79)

This expression for $s^{2}$ gives the well known ballistic regime $t^{2}$.

On the other hand, taking the large time approximation $\mu t\gg 1$ in Eq.
(76), we obtain

$s^{2}_{\rm
Freg}(t)\approx\frac{4\alpha^{\prime}}{g_{xx}(r_{h})}\log\left(t\mu\right)$
(80)

which corresponds to a subdiffusive behavior $\log t$, as in the bosonic case
discussed above. Note that this behavior also appears in classical set-ups as
in Ref. sinai-like . In Ref. Caldeira:2021izy  we found a similar fermionic
behavior for a Lorentz invariant context.

###  6.3 Summary and discussions on $s_{\rm reg}^{2}$

We can now summarize the results of this section on Table 1. First, one can
note that for all scenarios we get the usual ballistic regime $s^{2}_{\rm
reg}\sim t^{2}$ for short times. This is expected since for this situation the
particle do not had completely felt the characteristics of the environment.
Another feature for all these results is the proportionality to the inverse of
$g_{xx}(r_{h})$, showing the dependence on the IR of the metric in the
direction of motion of the test particle.

For the bosonic case, the short time behavior at zero chemical potential is
obtained directly just taking the $\mu=0$ limit from the finite chemical
potential result. However, at large times, one can not obtain the zero
chemical potential case by taking the above limit. In this case it is
necessary to go back to Eq. (55) and recalculate this quantity. Then turning
on a chemical potential changes the long time behavior of the regularized mean
square displacement for the bosonic scenario.

In doing the present analysis it is necessary to bear in mind that the term
$g_{xx}(r_{h})$ will bring a non-trivial dependence on the temperature and on
the chemical potential. So, in general, the temperature behavior of these
quantities depends heavily on the particular form of the metric. In the
following sections we apply the results obtained here for two set ups as a
illustration for those aspects.

As a general comment, it is interesting to note that the set up discussed
above for bosons and fermions at zero or non-zero chemical potential verifies
the fluctuation-dissipation theorem. This is done relating the imaginary part
of the admittance with the two point correlation functions calculated above,
as discussed in Section 5.

Description |  $s^{2}_{\text{reg}}(t)$ short times |  $s^{2}_{\text{reg}}(t)$ large times
---|---|---
Bosonic $\mu=0$ | $\frac{\pi^{2}\alpha^{\prime}}{3g_{xx}(r_{h})}\frac{t^{2}}{\beta^{2}}$ |  $\frac{2\pi\alpha^{\prime}}{g_{xx}(r_{h})}\frac{t}{\beta}=Dt$
Bosonic $\mu\not=0$ | $\frac{2\alpha^{\prime}}{g_{xx}(r_{h})}\text{Li}_{2}\left(e^{\beta\mu}\right)\frac{t^{2}}{\beta^{2}}$ |  $\frac{4\alpha^{\prime}}{g_{xx}(r_{h})}\frac{1}{e^{-\beta\mu}-1}\log\left(\frac{t}{\beta}\right)=\frac{1}{\pi}\frac{2\beta D}{e^{-\beta\mu}-1}\log\left(\frac{t}{\beta}\right)$
Fermionic $\mu=0$ | $\frac{2\alpha^{\prime}}{g_{xx}(r_{h})}\frac{t^{2}}{\beta^{2}}$ |  $\frac{2\alpha^{\prime}}{g_{xx}(r_{h})}\log\left(\frac{t}{\beta}\right)=\frac{\beta D}{\pi}\log\left(\frac{t}{\beta}\right)$
Fermionic $\mu\not=0$ | $\frac{\alpha^{\prime}}{g_{xx}(r_{h})}\mu^{2}t^{2}$ |  $\frac{4\alpha^{\prime}}{g_{xx}(r_{h})}\log\left(t\mu\right)=\frac{2\beta D}{\pi}\log\left(t\mu\right)$
Table 1: Mean square displacement for zero and non-zero chemical potential for
bosons and fermions.

##  7 Hyperscaling violation at finite temperature

As a first application of the above discussion we consider an Einstein-
Maxwell-Dilaton gravity given by the action (see for instance Zaanen:2015oix )

$\displaystyle{\cal
L}=R-\frac{1}{4}Z_{0}e^{\alpha\phi}F_{\mu\nu}^{2}-\frac{1}{2}(\partial_{\mu}\phi)^{2}-V_{0}e^{\beta\phi}\,.$
(81)

This implies a metric with hyperscaling violation at finite temperature
(without chemical potential for simplicity):111The notation used here differs
from others like Dong:2012se  where the metric reads (with $f(r)=1$):
$ds^{2}=\tilde{r}^{-2({d-\theta})/{d}}\left(-\tilde{r}^{-2(z-1)}dt^{2}+d\vec{x}^{2}+{d\tilde{r}^{2}}\right)$,
such that the holographic coordinates in both models are related by
$\tilde{r}=1/r$. In the particular case where $z=1$ and $\theta/d=1$, this
metric with $\tilde{r}$ is usually called a Poincaré patch.

$\displaystyle
ds^{2}=r^{-\frac{2\theta}{d}}\left(-r^{2z}f(r)dt^{2}+r^{2}d\vec{x}^{2}+\frac{dr^{2}}{f(r)r^{2}}\right),$
(82)

where $z$, $d$ and $\theta$ (which are functions of $\alpha$ and $\beta$) are
the Lifshitz and hyperscaling violation parameters respectively. The horizon
function is given by

$\displaystyle f(r)=1-\left(\frac{r_{h}}{r}\right)^{d+z-\theta}.$ (83)

This model was studied as a holographic gravity dual for some non-relativistic
systems. This was applied in some condensed matter models with a Fermi surface
such as strange metals (see for example Huijse:2011ef ; Zaanen:2015oix ;
Hartnoll:2018xxg ; Gubser:2000mm ; Dong:2012se ; Ogawa:2011bz ; Kachru:2008yh
).

The above horizon function implies that the Hawking temperature reads

$\displaystyle T=\frac{\left|d+z-\theta\right|}{4\pi}\,r_{h}^{z}\;.$ (84)

Using this result and the ones from the previous sections, we can write the
admittance from Eq. (21) as

$\displaystyle\Im{\chi}(\omega)=\frac{2\pi\alpha^{\prime}}{\omega}\left(\frac{4\pi}{\left|d+z-\theta\right|}\right)^{-{2(d-\theta)}/{zd}}T^{-{2(d-\theta)}/{zd}}.$
(85)

In order to have a positive specific heat we need to impose the condition
Dong:2012se

$\displaystyle\frac{d-\theta}{z}\geq 0\,.$ (86)

This condition implies a negative exponent for the temperature dependence of
the imaginary part of the admittance, which is the usual behavior for other
holographic systems such as the ones investigated in Ref. deBoer:2008gu .

Using Eq. (22), the diffusion coefficient for this model is

$\displaystyle
D=2\pi\alpha^{\prime}\left(\frac{4\pi}{\left|d+z-\theta\right|}\right)^{-{2(d-\theta)}/{zd}}T^{(-{2(d-\theta)+zd)}/{zd}}.$
(87)

In the particular case where $d=\theta$ one has the usual Einstein diffusion
proportional to $T$. On the other side, for $d-\theta=zd$ one has the
diffusion coefficient proportional to $T^{-1}$. This behavior was also found
in Ref. deBoer:2008gu  for pure AdS space. These cases are thermodynamically
stable accordingly to the condition given by Eq. (86).

###  7.1 Bosons

The regularized mean square displacement for the bosonic case without chemical
potential ($\mu=0$), Eq. (58), on a Lifshitz and hyperscaling violation metric
reduces to

$\displaystyle s^{2}_{\rm Breg}(t)$ $\displaystyle=$
$\displaystyle{2\alpha^{\prime}}{\left(\frac{4\pi
T}{|d+z-\theta|}\right)^{-{2(d-\theta)}/{zd}}}\log\left(\frac{\sinh({Tt\pi})}{{Tt\pi}}\right)\,.$
(88)

For short times one reobtains the ballistic regime

$\displaystyle s^{2}_{\rm Breg}(t)$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{\prime}}{3}{\left(\frac{4\pi
T}{|d+z-\theta|}\right)^{-{2(d-\theta)}/{zd}}}(Tt\pi)^{2}\,$ (89)

and for long times we find

$\displaystyle s^{2}_{\rm Breg}(t)$ $\displaystyle=$
$\displaystyle{2\alpha^{\prime}}{\left(\frac{4\pi
T}{|d+z-\theta|}\right)^{-\frac{2}{d}\frac{d-\theta}{z}}}\left(Tt\pi\right)$
(90) $\displaystyle=$
$\displaystyle{2\pi\alpha^{\prime}}{\left(\frac{4\pi}{|d+z-\theta|}\right)^{-\frac{2}{d}\frac{d-\theta}{z}}}\left(T\right)^{-\frac{2}{d}\frac{d-\theta}{z}+1}\,t$
(91)

which is the usual diffusive behavior in time.

###  7.2 Fermions

Now, we are going to analyze the fermionic case with zero chemical potential
within a Lifshitz and hypescaling violation metric. So, here we will calculate
the mean square displacement for this system. We get from Eq. (64) that for
short times one has

$\displaystyle s^{2}_{\rm
Freg}(t)\approx\frac{2\alpha^{\prime}}{g_{xx}(r_{h})}\frac{t^{2}}{\beta^{2}}=2\alpha^{\prime}\left(\frac{4\pi
T}{|d+z-\theta|}\right)^{-{2(d-\theta)}/{zd}}\frac{t^{2}}{\beta^{2}}\,,$ (92)

which is the typical ballistic behavior while for long times, in turn, from
Eq. (67), one finds

$\displaystyle s^{2}_{\rm Freg}(t)$ $\displaystyle\sim$
$\displaystyle\frac{2\alpha^{\prime}}{g_{xx}(r_{h})}\log\left(\frac{t}{\beta}\right)=2\alpha^{\prime}\left(\frac{4\pi
T}{|d+z-\theta|}\right)^{-{2(d-\theta)}/{zd}}\log\left(\frac{t}{\beta}\right)\,,$
(93)

which corresponds to a fermionic subdiffusive regime analogous to the
classical (Boltzmann) ones found in Ref. sinai-like .

Now, we are going to discuss two special cases: $\theta=d$ and $\theta=d-1$ in
the next two sections.

###  7.3 The particular case $\theta=d$

In this section, we discuss the Lifshitz and hyperscaling violation at zero
chemical potential ($\mu=0$) presented above for the particular case of
$\theta=d$. In this case, the admittance is independent of the temperature

$\displaystyle\chi(\omega)|_{\theta=d}=\frac{2\pi\alpha^{\prime}}{\omega}\,,$
(94)

and the diffusion coefficient becomes

$\displaystyle D|_{\theta=d}=2\pi\alpha^{\prime}T,$ (95)

which is the usual expected behavior for Brownian motion.

####  7.3.1 Bosons

The regularized mean square displacement for bosons, Eq. (88), in this case is
given by

$\displaystyle s^{2}_{\rm Breg}(t)|_{\theta=d}$ $\displaystyle=$
$\displaystyle{2\alpha^{\prime}}\log\left(\frac{\sinh({Tt\pi})}{{Tt\pi}}\right)\,.$
(96)

Taking this expression in the limit of short times, one finds the usual
ballistic regime

$\displaystyle s^{2}_{\rm Breg}(t)|_{\theta=d}$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{\prime}}{3}(Tt\pi)^{2},$ (97)

while for long times one finds

$\displaystyle s^{2}_{\rm Breg}(t)|_{\theta=d}=2\pi\alpha^{\prime}Tt=Dt\,,$
(98)

which is also the usual Einstein diffusive regime.

####  7.3.2 Fermions

In this case ($\theta=d$) the fermionic mean square displacement from (92) for
small times reads the natural ballistic profile

$\displaystyle s^{2}_{\rm Freg}(t)|_{\theta=d}\approx
2\alpha^{\prime}\frac{t^{2}}{\beta^{2}}\,.$ (99)

From the diffusive regime (93), we get

$\displaystyle s^{2}_{\rm Freg}(t)|_{\theta=d}\approx
2\alpha^{\prime}\log\left(\frac{t}{\beta}\right),$ (100)

which is also a subdiffusive behavior as the ones found in Ref. sinai-like .

It is interesting to note that in this case $\theta=d$ there is no dependence
on the dynamical exponent $z$ in the admittance, although the diffusion
coefficient and the mean square displacement depend on the temperature.

In ref. Dong:2012se  the authors consider $d-\theta$ as an effective spatial
dimension so that one obtains a $0+1$ dimensional system regarding the entropy
of the system. However, even for $\theta=d$ one finds non zero spatial
correlations, as we found above.

###  7.4 The particular case $\theta=d-1$

In the case $\theta=d-1$ without chemical potential, the imaginary part of the
admittance, Eq. (85), gives

$\displaystyle\Im{\chi}(\omega)|_{\theta=d-1}=\frac{2\pi\alpha^{\prime}}{\omega}\left(\frac{4\pi
T}{\left|z+1\right|}\right)^{-{2}/{zd}}$ (101)

By the condition (86) which keeps the specific heat positive one finds that in
this case one has $z>0$, so that the imaginary part of the admittance is
proportional to some inverse power of the temperature. It is also remarkable
that for $z\to\infty$ the imaginary part of the admittance becomes a constant
independent of the temperature.

The diffusion coefficient, Eq. (87), will be now

$\displaystyle
D|_{\theta=d-1}=2\pi\alpha^{\prime}\left(\frac{4\pi}{\left|z+1\right|}\right)^{-{2}/{zd}}T^{1-2/zd}.$
(102)

If $z\to\infty$ this diffusion coefficient becomes proportional to $T$, and if
$zd=2$ it becomes a constant.

####  7.4.1 Bosons

From Eq. (88) one finds for $\theta=d-1$

$\displaystyle s^{2}_{\rm Breg}(t)$ $\displaystyle=$
$\displaystyle{2\alpha^{\prime}}{\left(\frac{4\pi
T}{|z+1|}\right)^{-{2}/{zd}}}\log\left(\frac{\sinh({Tt\pi})}{{Tt\pi}}\right)\,.$
(103)

Then, for short times one has the ballistic regime

$\displaystyle s^{2}_{\rm Breg}(t)|_{\theta=d-1}$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{\prime}}{3}{\left(\frac{4}{|z+1|}\right)^{-{2}/{zd}}}\left(\pi
T\right)^{2-{2}/{zd}}t^{2}\,$ (104)

and for long times the diffusive response

$\displaystyle s^{2}_{\rm Breg}(t)|_{\theta=d-1}$ $\displaystyle=$
$\displaystyle{2\alpha^{\prime}}{\left(\frac{4}{|z+1|}\right)^{-\frac{2}{dz}}}\left(\pi
T\right)^{-\frac{2}{dz}+1}t.$ (105)

Note that for $z>2/d$ the mean square displacement is proportional to
$T^{\alpha}$ with $\alpha>0$, in both regimes. If $z\to\infty$, then the
ballistic regime goes with $T^{2}$ and the diffusion as $T$.

####  7.4.2 Fermions

For fermions in this special case and for short times the mean square
displacement, Eq. (92), becomes

$\displaystyle s^{2}_{\rm Freg}(t)|_{\theta=d-1}\approx
2\alpha^{\prime}\left(\frac{4\pi
T}{|z+1|}\right)^{-{2}/{zd}}\frac{t^{2}}{\beta^{2}}\,.$ (106)

In the diffusive regime (long times) we have, from Eq. (93), that

$\displaystyle s^{2}_{\rm Freg}|_{\theta=d-1}=2\alpha^{\prime}\left(\frac{4\pi
T}{|z+1|}\right)^{-{2}/{zd}}\log\left(\frac{t}{\beta}\right)\,.$ (107)

In Ref. Huijse:2011ef  the case $\theta=d-1$ was discussed and one finds a
compressible state with hidden Fermi surfaces.

##  8 Charged dilatonic AdS black hole

We consider here the Gubser-Rocha model Gubser:2009qt ; Zaanen:2015oix , which
is a top-down construction from eleven to four spacetime dimensions. In this
model there are fermionic modes which indicate the presence of Fermi surfaces.
The lagrangian in four dimensions is given by

$\displaystyle{\cal
L}=R-\frac{1}{4}e^{\phi/\sqrt{3}}F_{\mu\nu}^{2}-\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{6}{L^{2}}\cosh\left(\frac{\phi}{\sqrt{3}}\right)\,,$
(108)

and it is a particular case of an Einstein-Maxwell-dilaton system. The
solution of the corresponding equations of motion are Cvetic:1999xp ;
Gubser:2000mm

$\displaystyle
ds^{2}=\frac{r^{2}}{L^{2}}\left(\frac{Q}{r}+1\right)^{3/2}\left(-f(r)dt^{2}+d\vec{x}^{2}\right)+\frac{L^{2}dr^{2}}{r^{2}f(r)}\left(\frac{Q}{r}+1\right)^{-3/2},$
(109) $\displaystyle
A_{t}=\frac{\sqrt{3Q\tilde{\mu}}}{r+Q}-\frac{\sqrt{3Q\tilde{\mu}^{1/3}}}{L^{2/3}};\qquad\phi=\frac{\sqrt{3}}{2}\ln\left(1+\frac{Q}{r}\right),\quad
f=1-\frac{\tilde{\mu}L^{2}}{(r+Q)^{3}}.$ (110)

In the above equations, $Q$ is related to the black hole charge,
$\tilde{\mu}>0$ is related to the black hole mass and $L$ is the AdS radius.
The black hole horizon is located at $r=(\tilde{\mu}L^{2})^{1/3}-Q\equiv
r_{h}$. Defining the new coordinate $\tilde{r}=r+Q$ so that
$\tilde{r}_{h}=r_{h}+Q$, then

$f=1-\frac{\tilde{\mu}L^{2}}{{\tilde{r}}^{3}}\,;\qquad
A_{t}=-\frac{\sqrt{3Q\tilde{\mu}^{1/3}}}{L^{2/3}}\left[1-\frac{\tilde{r}_{h}}{\tilde{r}}\right]\,,$
(111)

which is similar to that of a Reissner-Nordstrom charged black hole with
chemical potential given by

$\mu=\frac{\sqrt{3Q\tilde{\mu}^{1/3}}}{L^{2/3}}\,.$ (112)

The Hawking temperature, using the $r$ coordinate defined in Eq. (109), is
then given by

$\displaystyle T=\frac{3}{4\pi
L^{2}}\left(\tilde{\mu}L^{2}\right)^{1/6}\sqrt{r_{h}}$ (113)

Then, the imaginary part of the admittance can be written from Eq. (21) as a
function of the temperature as

$\displaystyle\Im\chi(\omega)=\frac{3\alpha^{\prime}}{2\omega
T}\left(\tilde{\mu}L^{2}\right)^{-1/3}=\frac{9\alpha^{\prime}Q}{2\omega
T\mu^{2}L^{2}}\,,$ (114)

while the real part from Eq. (25) is
$\Re\chi\approx-{2\pi\alpha^{\prime}}/{3r_{b}^{3}}\,.$ Note that the real part
of the admittance is small compared with its imaginary part since the former
does not depend on the frequency $\omega$ which is small.

The diffusion coefficient for this model, from Eq.(22), reads

$\displaystyle D=\frac{9Q\alpha^{\prime}}{2\mu^{2}L^{2}}\,,$ (115)

which is temperature independent and decreasing for a growing chemical
potential.

###  8.1 Bosons

The bosonic case is characterized by imposing the condition on the chemical
potential $\mu<0$. Then, for short times $t\ll\beta$, Eq. (74) gives

$\displaystyle s^{2}_{\rm Breg}(t)$ $\displaystyle\approx$
$\displaystyle\frac{2\alpha^{\prime}}{g_{xx}(r_{h})}\text{Li}_{2}\left(e^{\beta\mu}\right)\frac{t^{2}}{\beta^{2}}=\frac{9\alpha^{\prime}Q}{2\pi\mu^{2}L^{2}T}\text{Li}_{2}\left(e^{\beta\mu}\right)\frac{t^{2}}{\beta^{2}}\,,$
(116)

which decreases when $|\mu|$ increases.

On the other side, for long times, $t\gg\beta$, from Eq.(75), one finds

$\displaystyle s^{2}_{\rm Breg}(t)$ $\displaystyle\approx$
$\displaystyle\frac{4\alpha^{\prime}}{g_{xx}(r_{h})}\frac{1}{e^{-\beta\mu}-1}\log\left(\frac{t}{\beta}\right)=\frac{9\alpha^{\prime}Q}{\pi\mu^{2}L^{2}T}\frac{1}{e^{-\beta\mu}-1}\log\left(\frac{t}{\beta}\right)\,.$
(117)

Since $\mu<0$ the diffusion also drops down when $|\mu|$ grows.

###  8.2 Fermions

The fermionic case is described by $\mu>0$. So, for short times, from Eq.(79),
we obtain:

$s^{2}_{\rm
Freg}(t)\approx\frac{\alpha^{\prime}}{g_{xx}(r_{h})}\mu^{2}t^{2}=\frac{9\alpha^{\prime}Q}{2\pi
L^{2}T}t^{2}\,.$ (118)

Note that if one keeps $Q$ fixed, there is no dependence on chemical potential
for this result. On the other hand, for long times we can use the results in
Eq. (80) to obtain:

$s^{2}_{\rm
Freg}(t)\approx\frac{4\alpha^{\prime}}{g_{xx}(r_{h})}\log\left(t\mu\right)=\frac{9\alpha^{\prime}Q}{\pi\mu^{2}L^{2}T}\log\left(t\mu\right),$
(119)

which diminishes for growing $\mu$.

##  9 Conclusions

In this work we presented some results regarding the dynamics of a particle in
a thermal bath at finite density, in the linear regime, using holographic
methods. For a general diagonal metric, we obtained expressions for linear
response function, mean square displacement, correlations functions and
diffusion coefficient, in terms of the metric elements. The mean square
displacement differs for fermions and bosons. Also, we have verified the
fluctuation-dissipation theorem for zero and non-zero chemical potentials for
both statistics. As these results are presented in terms of the metric
components, this is a quite general result.

Since we have a general relation between different metric elements and
physical quantities as the admittance, the regularized mean square
displacement and temperature, it is possible to choose particular backgrounds
in order to produce systems with holographic duals that may have some desired
physical behavior.

For all these results we see a dependence on the IR part of the metric in the
direction of the electrical field on the brane, which acts as an external
force for the system. This is quantified by the presence of $g_{xx}(r_{h})$ in
the general expressions.

For the admittance, we have found the hydrodynamic behavior $\omega^{-1}$ of
its imaginary part, in accordance with the literature deBoer:2008gu ;
Tong:2012nf ; Edalati:2012tc ; Giataganas:2018ekx . However, one finds
different temperature behaviors with respect to different choices of the
metrical components. Note also that the admittance does not depend on the
statistics.

For the regularized mean square displacement for bosons we found, in the late
times limit, that the effect of turning on the chemical potential makes the
diffusion process slower. The usual behavior $s^{2}_{\text{reg}}\sim t$ then
becomes a Sinai-like diffusion, $s^{2}_{\text{reg}}\sim\log t$. On the other
hand, for fermions, we always find $s^{2}_{\text{reg}}\sim\log t$ irrespective
the value of the chemical potential.

In the first application, we studied the Hyperscaling-Lifshitz background
exploring different values for the parameters $z$ and $\theta$ for varied
number of spatial dimensions $d$. It interesting to say that using this
freedom of choice it was possible to obtain some interesting cases without
violating the thermodynamic stability. For instance, the usual Einstein linear
dependence of the temperature in the diffusion constant can be found, for a
certain choice of parameters ($d=\theta$).

For the Guber-Rocha model, Section 8, our second application, we found that
the diffusion coefficient does not depend on the temperature and it is
inversely proportional to the chemical potential. For bosons, the regularized
mean square displacement decreases when the absolute value of chemical
potential increases. Also, this quantity is inversely proportional to the
temperature. As expected, the presence of a non-null chemical potential makes
the diffusion processes becomes slower. The same behavior is found for the
fermionic case, but now $s^{2}_{\text{reg}}\sim\log t$ and it is inversely
proportional to the square of temperature.

We hope that the discussion presented here might help finding new applications
for holographic systems.

## Acknowledgments

N.G.C. is supported by Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES) under finance code 001. H.B.-F. and C.A.D.Z. are partially
supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico
(CNPq) under the grants $\\#$s 311079/2019-9 and 310703/2021-2, respectively.
C.A.D.Z. is also partially supported by Fundação Carlos Chagas Filho de Amparo
à Pesquisa do Estado do Rio de Janeiro (Faperj) under Grant E-26/201.447/2021
(Programa Jovem Cientista do Nosso Estado).

##  Appendix A Solution for the EoM

Here in this Appendix we are going to find the solution for the EoM, Eq. (7).
First we will show how to rewrite the EoM into a convenient form using the
Regge-Wheeler (tortoise) coordinates as a Schrödinger-like equation in
Appendix A.1. We then obtain the solution near the horizon in Appendix A.2.
The solutions for the other regions are discussed in Section 2.1.

###  A.1 Regge-Wheeler coordinate and Schrödinger-like equation

First, we assume that the metric elements $a(r),b(r)$ and $c(r)$, defined in
Eq.(2), are regular functions at $r=r_{h}$ and the function $f(r)$, dubbed the
horizon function, has the following asymptotic values:

$\lim_{r\to r_{h}}f(r)\sim
f^{\prime}(r_{h})(r-r_{h});\qquad\textrm{and}\qquad\lim_{r\to\infty}f(r)=1.$
(120)

The equation of motion, Eq. (7). up to quadratic terms in the Nambu-Goto
action reads

$\frac{\partial}{\partial
r}\left(\frac{g_{xx}\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\frac{\partial\delta
x}{\partial
r}\right)-\frac{g_{xx}\sqrt{g_{rr}}}{\sqrt{g_{tt}}}\frac{\partial^{2}\delta
x}{\partial t^{2}}=0.$ (121)

Substituting the following Fourier decomposition

$\delta x(t,r)=h_{\omega}(r)e^{-i\omega t}$ (122)

into the above equation of motion, one finds

$\frac{\partial}{\partial
r}\left(\frac{g_{xx}\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\frac{\partial
h_{\omega}}{\partial
r}\right)+\omega^{2}\frac{g_{xx}\sqrt{g_{rr}}}{\sqrt{g_{tt}}}h_{\omega}=0.$
(123)

It is convenient to use the Regge-Wheeler coordinate. It naturally comes if we
rewrite the line element as

$ds^{2}=g_{tt}\left(dt^{2}-\frac{g_{rr}dr^{2}}{g_{tt}}\right)+\cdots$ (124)

Then, the Regge-Wheeler coordinate $r_{*}$ is defined by

$dr_{*}^{2}=\frac{g_{rr}dr^{2}}{g_{tt}}\Rightarrow
dr_{*}=\frac{\sqrt{g_{rr}}}{\sqrt{g_{tt}}}dr.$ (125)

The boundary conditions in this new coordinate are $\lim_{r\to
r_{h}}r_{*}(r)\to-\infty$ and $\lim_{r\to\infty}r_{*}(r)\to 0$. So, the
equation of motion (123) can be cast in the form

$\displaystyle\frac{d^{2}h_{\omega}}{dr_{*}^{2}}+\frac{d\log(g_{xx})}{dr_{*}}\frac{dh_{\omega}}{dr_{*}}+\omega^{2}h_{\omega}$
$\displaystyle=0\,.$ (126)

We rewrite the above equation as

$\frac{d^{2}h_{\omega}}{dr_{*}^{2}}+\alpha(r_{*})\frac{dh_{\omega}}{dr_{*}}+\beta(r_{*})h_{\omega}=0$
(127)

which can be transformed into Hill’s standard form Hill

$\frac{d^{2}\psi_{\omega}}{dr_{*}^{2}}+Q(r_{*})\psi_{\omega}=0,$ (128)

by putting

$\displaystyle h_{\omega}(r_{*})$
$\displaystyle=\exp\left[-\frac{A(r_{*})}{2}\right]\psi_{\omega},$ (129)
$\displaystyle\alpha(r_{*})$ $\displaystyle=\frac{dA}{dr_{*}},$ (130)
$\displaystyle Q(r_{*})$
$\displaystyle=-\frac{1}{2}\frac{d\alpha}{dr_{*}}-\frac{1}{4}\alpha^{2}+\beta.$
(131)

Comparing Eq. (126) with (127) we have:

$\alpha(r_{*})=\frac{d\log{g_{xx}}}{dr_{*}}\;\;\;\textrm{e}\;\;\;\beta(r_{*})=\omega^{2}.$
(132)

Equation (130) gives

$\frac{d\log{g_{xx}}}{dr_{*}}=\frac{dA}{dr_{*}}\Rightarrow
A(r_{*})=\log{g_{xx}}.$ (133)

Substituting the above result into Eq. (129) we get

$\displaystyle h_{\omega}(r_{*})$ $\displaystyle=$
$\displaystyle\exp\left[-\frac{A(r_{*})}{2}\right]\psi_{\omega}$ (134)
$\displaystyle=$
$\displaystyle\exp\left[-\frac{\log{g_{xx}}}{2}\right]\psi_{\omega}$ (136)
$\displaystyle=$ $\displaystyle\frac{\psi_{\omega}(r_{*})}{\sqrt{g_{xx}}}$
(138)

and $Q(r)$, Eq. (131), is given by

$\displaystyle Q(r)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{d\alpha}{dr_{*}}-\frac{1}{4}\alpha^{2}+\beta$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{dr}{dr_{*}}\frac{d}{dr}\left(\frac{dr}{dr_{*}}\frac{d\log{g_{xx}}}{dr}\right)-\frac{1}{4}\left(\frac{dr}{dr_{*}}\frac{d\log{g_{xx}}}{dr}\right)^{2}+\omega^{2}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\frac{d}{dr}\left(\frac{\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\frac{d\log{g_{xx}}}{dr}\right)-\frac{1}{4}\frac{g_{tt}}{g_{rr}}\left(\frac{d\log{g_{xx}}}{dr}\right)^{2}+\omega^{2}.$
(140)

Then, Hill’s equation (128), can be written as a Schrödinger-like equation:

$\frac{d^{2}\psi_{\omega}}{dr_{*}^{2}}+(\omega^{2}-V(r))\psi_{\omega}=0,$
(141)

where $\psi_{\omega}$ is defined as
$\psi_{\omega}(r_{*})=\sqrt{g_{xx}}h_{\omega}(r_{*})$, and the potential is
given by

$V(r)=\frac{1}{2}\frac{\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\frac{\partial}{\partial
r}\left(\frac{\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\frac{d\log(g_{xx})}{dr}\right)+\frac{1}{4}\frac{g_{tt}}{g_{rr}}\left(\frac{d\log(g_{xx})}{dr}\right)^{2}.$
(142)

In terms of the metric functions, Eq. (2), this potential reads

$V(r)=\frac{f(r)}{2}\sqrt{\frac{a(r)}{b(r)}}\frac{d}{dr}\left(\sqrt{\frac{a(r)}{b(r)}}\frac{f(r)}{c(r)}\frac{dc(r)}{dr}\right)+\frac{1}{4}\frac{a(r)}{b(r)}\frac{f^{2}(r)}{c^{2}(r)}\left(\frac{dc(r)}{dr}\right)^{2}.$
(143)

Notice that $V(r_{h})=0$ as expected as $f(r_{h})=0$. Furthermore it has units
of squared energy. Now, one can obtain the solution of the Schrödinger-like
equation (141), into two different regions: near by and far from the horizon.

###  A.2 Region A: Deep IR

The deep IR regime is the near horizon region, defined by

$r\sim r_{h},\;\;\;V(r)\ll\omega^{2}.$ (144)

This happens because $V(r)$ is modulated by $f(r)$, and near the horizon this
function becomes arbitrarily close to zero as can be seen in Eq. (143).

In this region, the Schrödinger-like equation (141) is written as

$\frac{d^{2}\psi_{A\omega}}{dr_{*}^{2}}+\omega^{2}\psi_{A\omega}=0\Rightarrow\psi_{A\omega}=A_{1}e^{-i\omega
r_{*}}+A_{2}e^{i\omega r_{*}}.$ (145)

The tortoise coordinate $r_{*}$, Eq. (125), is given by

$\frac{dr_{*}}{dr}=\frac{\sqrt{g_{rr}}}{\sqrt{g_{tt}}}\approx\sqrt{\frac{b(r_{h})}{a(r_{h})}}\frac{1}{f^{\prime}(r_{h})(r-r_{h})}\Rightarrow
r_{*}\approx\frac{1}{f^{\prime}(r_{h})}\sqrt{\frac{b(r_{h})}{a(r_{h})}}\log\left(\frac{r}{r_{h}}-1\right),$
(146)

where Eq. (120) was used. From Eq. (138) and considering only the ingoing
solution ($A_{2}=0$) in Eq. (145), one finds

$h_{A\omega}(r)=\frac{A_{1}e^{-i\omega r_{*}}}{\sqrt{g_{xx}(r_{h})}}.$ (147)

Since one is interested in the long wavelength limit ($\omega\to 0$), using
(146), the above equation reads

$h_{A\omega}(r)=\frac{A_{1}}{\sqrt{g_{xx}(r_{h})}}\left[1-i\frac{\omega}{f^{\prime}(r_{h})}\sqrt{\frac{b(r_{h})}{a(r_{h})}}\log\left(\frac{r}{r_{h}}-1\right)\right].$
(148)

Note also that $Q(r)$, Eq. (131), near the horizon becomes
$Q(r)\approx\omega^{2}$, since $r\approx r_{h}$ and using Eq. (2), one gets

$\frac{\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\approx\sqrt{\frac{a(r_{h})}{b(r_{h})}}\lim_{r\to
r_{h}}f(r)=0.$ (149)

##  Appendix B Neumann boundary condition and Normalization

###  B.1 Neumann boundary condition

The coefficient $B$ from Eq. (28) can be obtained from a Neumann boundary
condition at UV in the position $r=r_{b}$. At zero order in $\omega$, it is
simply

$\displaystyle B=1\,.$ (150)

In general, keeping higher orders in $\omega$ this coefficient will be a pure
phase $e^{i\omega\theta}$ for some $\theta$ real. This will happen because the
general solution can always be written as

$h^{UV}_{\omega}(r)=A\left[g(r)+Bg^{*}(r)\right],$ (151)

that implies

$\displaystyle
B=-\left.\frac{\partial_{r}g(r)}{\partial_{r}g^{*}(r)}\right|_{r=r_{b}}=e^{i\omega\theta}.$
(152)

From a Neumann boundary condition near the horizon the coefficient $B$ can
obtained

$B=-\left.\frac{\partial_{r}h^{in}(r)}{\partial_{r}h^{out}(r)}\right|_{r/r_{h}=1+\epsilon}=\exp\left\\{-2i\frac{\omega}{f^{\prime}(r_{h})}\sqrt{\frac{b(r_{h})}{a(r_{h})}}\log\left(\frac{1}{\epsilon}\right)\right\\}\,,$
(153)

which is a pure phase in $\omega$.

###  B.2 Normalization

Now we proceed to find the coefficient $A$ in Eq. (28). With this purpose, we
write the Klein Gordon inner product, as

$(f,g)=-\frac{i}{2\pi\alpha^{\prime}}\int_{\Sigma}dx\sqrt{|h|}n^{\mu}g_{xx}(f\partial_{\mu}g^{*}-\partial_{\mu}fg^{*}),$
(154)

where $h$ is the induced metric on the Cauchy surface $\Sigma$, $n^{\mu}$ is
the normal vector to this surface, $f$ and $g$ are solutions of the equations
of motion.

As a Cauchy surface we take a constant time slice of our worldsheet. Therefore
we have in the present situation

$\displaystyle n^{\mu}=\left(\frac{1}{\sqrt{g_{tt}}},0\right),$ $\displaystyle
h=g_{rr}.$ (155)

Thus the Klein-Gordon inner product becomes

$\displaystyle(f,g)=-\frac{i}{2\pi\alpha^{\prime}}\int_{\Sigma}dx\sqrt{\frac{g_{rr}}{g_{tt}}}g_{xx}(f\partial_{t}g^{*}-\partial_{t}fg^{*}).$
(156)

In order to obtain a proper normalization, we demand that $(X(t,r),X(t,r))=1$.
Then, for $X(t,r)=e^{-i\omega t}h_{\omega}(r)$ we get

$\displaystyle(X(t,r),X(t,r))$ $\displaystyle=$
$\displaystyle\frac{\omega}{\pi\alpha^{\prime}}\int_{r_{h}}^{r_{b}}dr\sqrt{\frac{g_{rr}}{g_{tt}}}g_{xx}|h_{\omega}(r)|^{2}$
(157) $\displaystyle=$
$\displaystyle\frac{|A|^{2}\omega}{\pi\alpha^{\prime}}\int_{r_{h}}^{r_{b}}dr\sqrt{\frac{b(r)}{a(r)}}\frac{g_{xx}(r)}{f(r)}|h_{\omega}(r)|^{2}$
(158)

For regular functions $b(r)$ and $a(r)$ this integral will be dominated by the
near horizon region because of the zero in $f(r)$. Therefore, we can
approximate this integral by

$\displaystyle(X(t,r),X(t,r))$ $\displaystyle\approx$
$\displaystyle\frac{|A|^{2}\omega}{\pi\alpha^{\prime}}\sqrt{\frac{b(r_{h})}{a(r_{h})}}\frac{g_{xx}(r_{h})}{f^{\prime}(r_{h})}\int_{r_{h}}^{r_{b}}\frac{dr}{r_{h}}\frac{|h_{\omega}(r)|^{2}}{\frac{r}{r_{h}}-1}$
(159) $\displaystyle\approx$
$\displaystyle\frac{4|A|^{2}\omega}{\pi\alpha^{\prime}}\sqrt{\frac{b(r_{h})}{a(r_{h})}}\frac{g_{xx}(r_{h})}{f^{\prime}(r_{h})}\int_{r_{h}}^{r_{b}}\frac{dr}{r_{h}}\frac{1}{\frac{r}{r_{h}}-1},$
(160)

where we used the near horizon expression for $h_{\omega}(r)$ Eq.(26), keeping
only terms smaller than $O(\omega^{2})$. The final result is

$\displaystyle(X(t,r),X(t,r))$ $\displaystyle\approx$
$\displaystyle\frac{4|A|^{2}\omega}{\pi\alpha^{\prime}}\sqrt{\frac{b(r_{h})}{a(r_{h})}}\frac{g_{xx}(r_{h})}{f^{\prime}(r_{h})}\int_{r_{h}}^{r_{b}}\frac{dr}{r_{h}}\frac{1}{\frac{r}{r_{h}}-1}$
(161) $\displaystyle\sim$
$\displaystyle\frac{4|A|^{2}\omega}{\pi\alpha^{\prime}}\sqrt{\frac{b(r_{h})}{a(r_{h})}}\frac{g_{xx}(r_{h})}{f^{\prime}(r_{h})}\log\left(\frac{1}{\epsilon}\right),$
(162)

where the integral was regularized considering in the lower limit of
integration $r_{h}\to r_{h}+\epsilon$. Imposing the normalization condition we
obtain

$\displaystyle A=\sqrt{\frac{\pi\alpha^{\prime}f^{\prime}(r_{h})}{4\omega
g_{xx}(r_{h})\log\left(\frac{1}{\epsilon}\right)}\sqrt{\frac{a(r_{h})}{b(r_{h})}}}\,.$
(163)

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