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# Holographic phonons by gauge-axion coupling

Xi-Jing Wang1 Wei-Jia Li2 1 Center for Gravitation and Cosmology, College of
Physical Science and Technology, Yangzhou University, Yangzhou 225009, China
2 Institute of Theoretical Physics, School of Physics, Dalian University of
Technology, Dalian 116024, China.

###### Abstract

In this paper, we show that the holographic homogeneous model can realize
spontaneous breaking of translational symmetry by considering a special gauge-
axion higher derivative term. The non-trivial real part and imaginary part of
the stress tensor imply that the dual boundary system is a viscoelastic solid.
By calculating quasi-normal modes and making a comparison with predictions
from the elasticity theory, we verify the existence of (pseudo-)phonons in the
transverse channel. Finally, we discuss how the phonon dynamics affects the
charge transport.

###### Contents

  1. I Introduction
  2. II An elastic black hole
    1. II.1 Spontaneous breaking of translations
    2. II.2 Shear viscoelasticity
  3. III Holographic transverse phonons
    1. III.1 Gapless phonons
    2. III.2 Pinned phonons
  4. IV Electric conductivity
    1. IV.1 Purely SSB pattern
    2. IV.2 Pinned structure
  5. V Conclusions
  6. A Derivation details for analytical Green’s functions
    1. A.1 High temperature limit
    2. A.2 Zero temperature limit
  7. B Perturbation equations and Green’s functions
  8. C DC conductivity and shear viscosity

##  I Introduction

It is widely unveiled that the hydrodynamic formulation, which is based upon a
small amount of conserved quantities as well as the constitutive relations,
can provide a universal description for the low energy collective excitations
in exotic quantum matter Teaney (2010); O’Hara  _et al._ (2002); Levitov and
Falkovich (2016); Bandurin and et al (2016). In relativistic hydrodynamics
Kovtun (2012), the most fundamental conserved quantity is the stress tensor.
Despite the Poincaré symmetry is fundamental for particles in the high energy
region, the spatial translations are ubiquitously broken at low energy scales
in condensed matter physics, especially in solids where there commonly exist
periodic lattice, impurities, defects, etc. If the breaking of translations is
weak, we are still allowed to formulate the so-called quasi-hydrodynamics to
address the problems approximately. However, this framework might break down
when the translational symmetry is strongly violated.

On the contrary, the holographic duality provides a purely geometric approach
to understand universal behaviors in strongly coupled systems, regardless of
whether quantities are conserved or not Hartnoll (2009); Natsuume (2015);
McGreevy ; Hartnoll  _et al._ (2016a). In holography, all the physics of a
many-body system (which is described by some quantum field theories) is
encoded in a dual gravity theory with one extra dimension. Moreover, in the
strong coupling limit of the field theory, the quantum effects on the gravity
side are significantly suppressed. As a result, complicated problems on the
field theory side are simplified into solving classical equations in a curved
spacetime. Consequently, this duality has nowadays become a widely used tool
for exploring strongly coupled many-body systems.

Despite the earlier holographic studies focused more on translation invariant
systems, some recent work attempts to make setups closer to realistic
materials without translations Horowitz  _et al._ (2012a, b); Donos and
Hartnoll (2013); Vegh (2013); Horowitz and Santos (2013); Blake and Tong
(2013); Ling  _et al._ (2013); Donos and Gauntlett (2014a); Andrade and
Withers (2014); Donos and Gauntlett (2014b); Goutéraux (2014); Ling  _et al._
(2015); Davison and Goutéraux (2015a); Donos and Pantelidou (2019). One of the
simplest ways to capture the key features of the symmetry breaking patterns of
translations but allow fast calculations is to consider homogeneous massive
gravity models. Based on the perturbative calculations, it has also been
explicitly shown that gravitons always have a non-zero mass in any holographic
system with an inhomogeneous lattice Blake  _et al._ (2014). For this reason,
massive gravity provides a universal low energy description for strongly
coupled systems with broken translations. A generally and widely used way of
modeling massive gravity is to consider the axions in the bulk Baggioli and
Pujolas (2015); Alberte and Khmelnitsky (2015); Alberte  _et al._ (2016a);
Baggioli and Pujolas (2016); Alberte  _et al._ (2018a). These are massless
scalars enjoying a internal shift symmetry, linearly depending on spatial
coordinates. As a result, the background configuration of the axions
contributes a non-zero mass to gravitons, breaking the translations in the
boundary system but retaining the homogeneity of the background. Better yet,
one can easily realize (spontaneous or explicit) manners of breaking
translations differently by slightly changing the kinetic term of the axions
in the bulk Alberte  _et al._ (2018b); Ammon  _et al._ (2019a, b); Baggioli
and Grieninger (2019). For a comprehensive review on the holographic axion
models, one refers to Baggioli  _et al._ (2021).

In this work, we consider a new way of spontaneously breaking translations by
directly coupling the axions to the $U(1)$ gauge field at finite density. Note
that such types of gauge-axion higher derivative terms have previously
introduced in Goutéraux  _et al._ (2016); Baggioli  _et al._ (2017) in the
holographic studies on the metal-insulator transition (MIT). Here, we consider
the simplest one of them and reveal the fact that this term just introduces a
spontaneous breaking of translations and the pinning effect in presence of
external sources leads to the MIT. Moreover, different from the fluid model
discussed in Li and Wu (2019), the gravity system under consideration is dual
to some solid states on boundary where there exist propagating/pinned modes in
the transverse channel. The paper is organized as follows: In Section II, we
introduce the holographic setup, solve the background solution and analyze the
viscoelasticity of the dual system. In Section III, the spectrum of transverse
phonons is systematically investigated by computing quasi-normal modes (QNMs)
of the black hole. In Section IV, we study how the phononic dynamics impacts
the charge transport. In Section V, we conclude.

##  II An elastic black hole

###  II.1 Spontaneous breaking of translations

In this paper, we consider the following action which can break translational
symmetry of dual field theories (Goutéraux  _et al._ , 2016)

$S=\int d^{4}x\sqrt{-g}\left[R-2\Lambda-\lambda
X-\frac{1}{4}\left(1+\mathcal{K}X\right)F^{2}\right],$ (1)

where $X$ is the kinetic term of the axions $\phi^{I}$ and $F_{\mu\nu}$ is
$U(1)$ gauge field which are described by

$X=\frac{1}{2}\sum_{I=x,y}\partial^{\mu}\phi^{I}\partial_{\mu}\phi^{I},\qquad
F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}.$ (2)

And, $\Lambda$ is the cosmological constant that will be set $\Lambda=-3$ for
simplicity. $\lambda$ and $\mathcal{K}$ are dimensionless couplings. If we set
$\mathcal{K}=0$, this action simply reduces to the simple linear axion model
Andrade and Withers (2014). In our model, we should require that $\lambda\geq
0$ for avoiding the ghost problem and $-1/6\leq\mathcal{K}\leq 0$ for
causality and stability Baggioli and Pujolas (2017); Goutéraux  _et al._
(2016) 111In the paper Goutéraux  _et al._ (2016), the author obtained the
coupling $-1/6\leq\mathcal{K}\leq 1/6$. While in Baggioli and Pujolas (2017),
the author gave the stronger constrain $-1/6\leq\mathcal{K}\leq 0$ to avoid
the problem ghost. In these discussions the effect of magnetic field is not
included. As was pointed in An  _et al._ (2020) that turning on an external
magnetic field will make the constraint on $\mathcal{K}$ even tighter.
However, in this paper, our construction does not involve the magnetism..

Varying the bulk fields, their equations of motion are given by

$\displaystyle
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R-\frac{1}{2}(\lambda+\frac{\mathcal{K}}{4}F_{\sigma\rho}F^{\sigma\rho})\nabla_{\mu}\phi^{I}\nabla_{\nu}\phi^{I}-\frac{1}{2}\left(6-\frac{\lambda}{2}\nabla_{\sigma}\phi^{I}\nabla^{\sigma}\phi^{I}\right)g_{\mu\nu}$
(3)
$\displaystyle=\frac{1}{2}\left(1+\frac{\mathcal{K}}{2}\nabla_{\sigma}\phi^{I}\nabla^{\sigma}\phi^{I}\right)\Big{(}{F_{\mu}}^{\sigma}F_{\nu\sigma}-\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\Big{)},$
$\displaystyle\nabla_{\mu}\left[\left(1+\frac{\mathcal{K}}{2}\nabla_{\sigma}\phi^{I}\nabla^{\sigma}\phi^{I}\right)F^{\mu\nu}\right]=0,$
(4)

and

$\displaystyle\nabla_{\mu}\Big{[}\lambda\nabla^{\mu}\phi^{I}-\frac{1}{4}\left(\mathcal{K}(\nabla^{\mu}\phi^{I})F^{2}\right)\Big{]}=0.$
(5)

Note that in holographic axion models, we have the following isotropic and
homogeneous background metric,

$\displaystyle
ds^{2}=\frac{1}{u^{2}}\left[-f(u)dt^{2}+\frac{1}{f(u)}du^{2}+dx^{2}+dy^{2}\right],$
(6) $\displaystyle A=A_{t}(u)dt,\qquad\phi^{I}=\alpha\delta^{I}_{i}x^{i},$ (7)

where $\alpha$ is a constant characterizing the breaking of translations, and
$x^{i}$ denotes the boundary spatial coordinates. AdS boundary is located at
$u=0$ and horizon is determined by $f(u=u_{h})=0$ in this convention. For our
model, we can solve the black hole solution analytically

$\displaystyle f(u)=1-\frac{1}{2}\alpha^{2}\lambda
u^{2}+\frac{\rho^{2}u^{3}ArcTan\left(\alpha\sqrt{\mathcal{K}}u\right)}{4\alpha\sqrt{\mathcal{K}}}-\frac{\rho^{2}u^{3}ArcTan\left(\alpha\sqrt{\mathcal{K}}u_{h}\right)}{4\alpha\sqrt{\mathcal{K}}}-\frac{u^{3}}{u_{h}^{3}}+\frac{\alpha^{2}\lambda
u^{3}}{2u_{h}},$ (8) $\displaystyle A_{t}(u)=\mu-\frac{\rho
ArcTan\left(\alpha\sqrt{\mathcal{K}}u\right)}{\alpha\sqrt{\mathcal{K}}}.$ (9)

Here, $\mu$ and $\rho$ are two integration constants which are interpreted as
chemical potential and charge density in the dual field theory. The regularity
for gauge field $A_{\mu}$ on the horizon requires

$\mu=\frac{\rho
ArcTan\left(\alpha\sqrt{\mathcal{K}}u_{h}\right)}{\alpha\sqrt{\mathcal{K}}}.$
(10)

According to the holographic dictionary, entropy density, energy density,
momentum susceptibility and temperature are respectively given by

$\displaystyle
s=\frac{4\pi}{u_{h}^{2}},\qquad\epsilon=2\left(\frac{1}{u_{h}^{3}}-\frac{\lambda\alpha^{2}}{2u_{h}}+\frac{\rho^{2}ArcTan\left(\alpha\sqrt{\mathcal{K}}u_{h}\right)}{4\alpha\sqrt{\mathcal{K}}}\right),$
(11) $\displaystyle\chi_{PP}=\epsilon+P=\frac{3}{2}\epsilon,\qquad
T=\frac{3}{4\pi
u_{h}}-\frac{\lambda\alpha^{2}u_{h}}{8\pi}-\frac{\rho^{2}u_{h}^{3}}{16\pi(1+\mathcal{K}\alpha^{2}u_{h}^{2})}.$
(12)

Next, we will explain the two different manners of breaking the translations
by analyzing the asymptotic behavior of scalar field $\phi^{I}$ close to the
AdS boundary $(u\rightarrow 0)$. The story is very similar as in Amoretti  _et
al._ (2017); Alberte  _et al._ (2018b); Amoretti  _et al._ (2018a, b); Li and
Wu (2019); Amoretti  _et al._ (2019a).

  * •

If we set $\lambda\neq 0$ and $\mathcal{K}=0$ in (5), the asymptotic expansion
of $\phi^{I}$ near the UV boundary is

$\phi^{I}=\phi_{(0)}^{I}(t,x^{i})+\phi_{(3)}^{I}(t,x^{i})u^{3}+\cdots.$ (13)

According to the holographic dictionary and standard quantization, the leading
order $\phi_{(0)}^{I}(t,x^{i})=\alpha x^{I}\neq 0$ plays the role of external
source that breaks translations explicitly, relaxing the momentum on the field
theory side. Further properties and applications in holography have been
widely investigated in Andrade and Withers (2014); Davison and Goutéraux
(2015a); Kim  _et al._ (2014, 2015); Davison and Goutéraux (2015b); Blake  _et
al._ (2018); Zhou  _et al._ (2020).

  * •

If we instead set $\lambda=0$ and $\mathcal{K}\neq 0$, the asymptotic
expansion of scalar field will be changed by

$\phi^{I}=\frac{\phi_{(-1)}^{I}(t,x^{i})}{u}+\phi_{(0)}^{I}(t,x^{i})+\cdots.$
(14)

In this case, the background solution of $\phi^{I}$ means that the leading
$\phi_{(-1)}^{I}(t,x^{i})$-term associated to the source should be turned off
while existing a non-zero expectation value $<\mathcal{O}^{I}>$ $\sim$
$\phi_{(0)}^{I}(t,x^{i})=\alpha x^{I}$. This implies the pattern of
spontaneous breaking of translational symmetry.

Note that if the translations are broken spontaneously, there should exist
gapless excitations in the low energy description which are called Goldstone
modes. In solid state physics, these Goldstone modes are called phonons, the
propagating sound modes in elastic media. The dynamics of Goldstone modes has
already been studied in holography Alberte  _et al._ (2018b); Ammon  _et al._
(2019a, b); Baggioli and Grieninger (2019); Amoretti  _et al._ (2019a); Donos
_et al._ (2020); Amoretti  _et al._ (2020a); Baggioli  _et al._ (2020a) and
field theory Leutwyler (1997); Dubovsky  _et al._ (2006); Nicolis  _et al._
(2015); Delacrétaz  _et al._ (2017); Alberte  _et al._ (2019); Musso (2019).

###  II.2 Shear viscoelasticity

One of the most distinctions of a solid and a fluid is that the former is
resistant to shear deformations while the latter one is not.222However, in
high momentum regime, fluids exhibit similar behaviors with solids. For a
comprehensive review, one refers to Baggioli  _et al._ (2020b). To show that
the black hole solution under consideration is dual to a solid system, we
shall compute the Green function of the stress tensor, which in the
hydrodynamic regime can be expressed as

$\mathcal{G}^{R}_{T_{xy}T_{xy}}(\omega,k=0)=G\,-i\,\omega\,\eta+\cdots,$ (15)

where $G$ is the shear elastic modulus and $\eta$ is the shear viscosity. In
the rest of this subsection, we focus on the purely SSB pattern by turning off
the external source, i.e, setting $\lambda=0$ and remaining $\mathcal{K}\neq
0$. One can calculate $G$ and $\eta$ via retarded Green functions of stress
tensor operator $T_{xy}$ numerically and read them by Ammon  _et al._ (2019a)

$G=Re\left[\mathcal{G}_{T_{xy}T_{xy}}^{R}(\omega=k=0)\right],\qquad\eta=-\lim_{\omega\rightarrow
0}\frac{1}{\omega}Im\left[\mathcal{G}_{T_{xy}T_{xy}}^{R}(\omega,k=0)\right].$
(16)

The numeric results have been shown in Fig.1. And we find that the ratio of
shear viscosity $\eta$ to entropy density $s$ always breaks Kovtun-Son-
Starinets (KSS) bound Kovtun  _et al._ (2005) due to the breaking of
translations. Moreover, there is a non-trivial shear modulus which implies the
dual system is in a solid state. To have it positive, i.e avoiding dynamical
instability, we should always require that $\mathcal{K}<0$ in our model.

Figure 1: The dependence of the shear elastic modulus $G$ and the ratio of
$\eta/s$ on $T/\sqrt{\rho}$ for different $\mathcal{K}$. The black dashed line
is KSS bound $1/4\pi$ Kovtun  _et al._ (2005). We have fixed coupling
$\lambda=0$.

Now, we analyze the Green function $\mathcal{G}_{T_{xy}T_{xy}}^{R}$
analytically in the high and low temperature limits. Let us focus on the
former first. The details of the calculation are shown in the Appendix A. In
the high temperature limit, the analytical results of shear modulus and
viscosity are respectively given by

$\displaystyle
G=-\frac{9\mathcal{K}\alpha^{2}T}{128\pi^{3}}\left(\frac{\sqrt{\rho}}{T}\right)^{4}+\cdots,$
(17)
$\displaystyle\frac{\eta}{s}=\frac{1}{4\pi}+\frac{81\mathcal{K}\alpha^{2}}{16384\pi^{7}T^{2}}\left(\frac{\sqrt{\rho}}{T}\right)^{4}+\cdots.$
(18)

Note that the positive shear modulus again requires the coupling constant
$\mathcal{K}<0$.

In the low temperature limit, we only find the analytical result for $\eta$
that is given by

$\frac{\eta}{s}\sim\left(\frac{T}{\sqrt{\rho}}\right)^{2\xi-1},\qquad\xi=\frac{1}{2}\sqrt{1-\frac{8\sqrt{3}\mathcal{K}}{\sqrt{3\mathcal{K}^{2}+1}}}.$
(19)

This result is slightly different from the massive gravity without charge
density Baggioli (2019), which shows that the ratio is proportional to
$T^{2}$. The author of Hartnoll  _et al._ (2016b) have claimed that $T^{2}$
was the fastest decay for the models whose IR geometry has an $AdS_{2}$
factor. In our case, the maximum of exponent is $T^{2}$ that is corresponding
to $\mathcal{K}\rightarrow-\infty$, which satisfies this bound. However, this
claim is not true anymore if one consider more complicated IR fixed points
(see Ling  _et al._ (2016)). Note that for $\mathcal{K}=0$ the $\eta/s$ ratio
is a constant which should be KSS bound.

##  III Holographic transverse phonons

###  III.1 Gapless phonons

The dispersion relation of transverse phonons at low energy is

$\omega=\pm c_{T}k-iD_{T}k^{2}+\cdots,$ (20)

where $c_{T}$ is the propagation velocity of phonons, which is related to the
shear elastic modulus of material given by

$c_{T}^{2}=\frac{G}{\chi_{PP}}.$ (21)

$D_{T}$ is the attenuation constant that is related to the shear viscosity
$\eta$ by $D_{T}=\eta/\chi_{PP}$ Amoretti  _et al._ (2019a). In order to
confirm the presence of transverse phonons, we compute the quasi-normal modes
(QNMs) numerically at zero and finite momenta in the infalling Eddington-
Finkelstein coordinates (74) by using the pseudo-spectral method. It shows
that the Goldstone modes induced by SSB of translations can be confirmed as
transverse phonons in this holographic model (1). One can see more numerical
details in Jansen (2017). 333In the numerical setup throughout the paper, we
fix that $u_{h}=1$ and $\alpha/\sqrt{\rho}=1$.

In Fig.2, we show the real and imaginary parts of the lowest order QNMs for
different temperatures. These Goldstone modes are phonons we expected. The
spectrums of QNMs for different couplings are shown in Fig.3. One can
calculate the sound speed of the transverse phonons in terms of (21) by
holography and compare them to the numeric results extracted from QNMs. We
find that they are in perfect agreement with each other, as is shown in Fig.4.

Figure 2: Real and imaginary parts of the lowest QNMs for different
temperatures. We have fixed coupling $\mathcal{K}=-1/8$ and $\lambda=0$. We
try to use the generalized dispersion relation $\omega=c_{1}k^{a}+ic_{2}k^{b}$
to fit the QNMs data. In our numerical cases, we get $a=1.00003\pm 0.00002$
and $b=2.00003\pm 0.00003$, which satisfies the dispersion relation (20).

Figure 3: Real and imaginary parts of the lowest QNMs for different couplings
$\mathcal{K}$. We have fixed $T/\sqrt{\rho}=0.2$ and $\lambda=0$.  Figure 4:
Comparison of the sound velocity extracted from QNMs (black dots) and the
result from the elasticity formula (21) (solid lines) for various values of
$\mathcal{K}$.

###  III.2 Pinned phonons

In this section, we further introduce slow relaxation on momentum into the SSB
pattern. Physically, this makes the momentum operator no longer exactly
conserved and the would be massless Goldstones get slightly gapped (which is
now called pseudo-Goldstones or pinned phonons). In our model it can be easily
realized by including the canonical kinetic term of the axions as follows,

$\displaystyle\mathcal{L}_{\phi}$ $\displaystyle=-\lambda
X-\frac{1}{4}\mathcal{K}XF^{2}.$ (22)

In order not to change the SSB pattern significantly, we assume that the value
of $\lambda$ is small comparing to $\mathcal{K}$.

In presence of slow momentum relaxation, the relaxation rate $\Gamma$ is
controlled in the manner of Andrade and Withers (2014); Davison (2013)

$\Gamma\propto\lambda\,\alpha^{2}.$ (23)

Therefore we can define a ESB scale which depicts the strength of explicit
breaking of symmetry

$\langle\textsl{ESB}\rangle\equiv\sqrt{\lambda}\,\alpha.$ (24)

Heuristically, we define the SSB scale as follows

$\langle\textsl{SSB}\rangle\equiv\sqrt{\frac{-\mathcal{K}}{4}}\,\alpha.$ (25)

Then, the competition of SSB and ESB should be described by a dimensionless
parameter that is defined by

$\beta\equiv\frac{\langle\textsl{SSB}\rangle}{\langle\textsl{ESB}\rangle}=\sqrt{\frac{-\mathcal{K}}{4\,\lambda}}.$
(26)

That is to say that at $\beta=0$ the breaking of translations is totally
explicit while at $\beta=\infty$ purely spontaneous. Then, the pinned phonons
are expected to appear when $\beta\gg 1$ (pseudo-spontaneous). In this case,
the hydrodynamic formulation predicts the following universal result for zero
momentum Delacrétaz  _et al._ (2017)

$\left(\Omega-i\omega)(\Gamma-i\omega\right)+\omega_{0}^{2}=0,$ (27)

where $\Gamma$ is the momentum relaxation rate (charactering how fast the
total momentum is dissipated), $\Omega$ is the phase relaxation rate
(measuring the lifetime of the the phonons) and $\omega_{0}$ is the so-called
pinning frequency (the mass of the pinned phonons). Solving the equation (27)
we obtain a pair of modes

$\omega_{\pm}=-\frac{i}{2}\left(\Omega+\Gamma\right)\pm\frac{1}{2}\sqrt{4\omega_{0}^{2}-\left(\Gamma-\Omega\right)^{2}}.$
(28)

Note that this formula is valid only when the two modes enter in the
hydrodynamic limit $|\omega_{\pm}|/T\ll 1$.

Next we show the QNMs and see how these two modes move. For this purpose we
fix $\mathcal{K}$ and $\lambda$ and tune the temperature $T/\sqrt{\rho}$. The
results are shown in the left panel of Fig.5. We find that at high
temperatures (i.e, small values of $\langle ESB\rangle/T$) the two modes lie
on the imaginary axes. In this case, we have
$4\omega_{0}^{2}<\left(\Gamma-\Omega\right)^{2}$. When decreasing temperature
$4\omega_{0}^{2}$ grows faster than $(\Gamma-\Omega)^{2}$, the two modes get
closer, collide and finally become off-axis.

Figure 5: Left: The lowest two QNMs at zero momentum for $\mathcal{K}=-1/8$.
We have fixed $\lambda=1\times 10^{-8}$ and $T/\sqrt{\rho}$ $\in$ [0.1,
0.4](green-red). Right: Collision frequency $Im\,\omega_{coll}$ as the
function of $\beta=\langle SSB\rangle/\langle ESB\rangle$. We have fixed
$\mathcal{K}=-1/8$, $\lambda$ $\in$ $[1\times 10^{-8},1\times 10^{-5}]$ and
$\alpha/\sqrt{\rho}=1$. The black line indicates linear scaling.

In condensed matter systems, the presence of the phase relaxation could be
attributed to either the proliferation of dislocations Delacrétaz  _et al._
(2017) or the disorder effects Fukuyama (1978); Huse and Fogler (2000). And in
holographic models, it is associated with the collision mechanism of the two
lowest-lying quasi-normal modes Ammon  _et al._ (2019a). It is worth studying
the relationship between collision frequency $\omega_{coll}$ and
characteristic parameter $\beta$. We can fix $\mathcal{K}$, choose one
specific $\lambda$ and show the QNMs spectrum. Then identify the collision
frequency at the collision point. The final results are shown in the right
figure of Fig.5. For $\beta\gg 1$ the collision happens towards the origin so
that very close to the origin, which can be described effectively by
hydrodynamics. When $\beta=\infty$ (purely SSB case), the two modes are all
located at the origin. As shown in the right panel of Fig.5, the collision
frequency $\omega_{coll}$ shows proportional to the characteristic parameter
$1/\beta$ approximately, that is

$\frac{Im\,\omega_{coll}}{T}\sim\sqrt{\frac{\langle ESB\rangle}{\langle
SSB\rangle}}.$ (29)

Next we check Gell-Mann-Oakes-Renner (GMOR) relation which was originally
found in QCD Gell-Mann  _et al._ (1968) and later was also checked in
holographic studies Kruczenski  _et al._ (2004); Erlich  _et al._ (2005);
Filev  _et al._ (2009); Argurio  _et al._ (2016); Amoretti  _et al._ (2017);
Ammon  _et al._ (2019a); Baggioli and Grieninger (2019). In the pseudo-
spontaneous limit, the momentum relaxation rate can be neglected in (28). By
reading QNMs data, phase relaxation rate $\Omega$ and pinning frequency
$\omega_{0}$ can be identified. We show the dependence of the pinning
frequency $\omega_{0}$ on the ESB and SSB scales in Fig.6. We find that the
pinning frequency $\omega^{2}_{0}$ is proportional to the square root of the
explicit breaking scale (24) and to the half to the third power of the
spontaneous breaking scale (25). That said, in the pseudo-spontaneous limit,
the results indicate the following scaling behavior

$\omega^{2}_{0}\sim\sqrt{\langle ESB\rangle\langle SSB\rangle^{3}},$ (30)

which is different from the original GMOR relation for Pion. Similarly in
Andrade  _et al._ (2018); Andrade and Krikun (2019); Andrade  _et al._ (2021),
they also obtained the results which is different from GMOR relation in
holography.444Note that the derivation of the original GMOR relation was based
on a specific QCD model. To our best knowledge, there is by now no rigorous
proof that this relation should be valid for all cases in the field theory
side. Holographic models suggest a more general form which is
${\omega_{0}^{2}}\sim\langle ESB\rangle^{a}\langle SSB\rangle^{b}$ with two
non-negative exponents $a$ and $b$. For example in holographic model with
helical structure, the exponents are $a=2$ and $b=2$ Andrade  _et al._ (2018).
However in a simple holographic axion model, the authors proved this
conclusion Ammon  _et al._ (2019a).

Figure 6: The dependence of the pinning frequency $\omega_{0}$ on the ESB and
SSB scales. Left: We have fixed $\lambda$ $\in$ $[1\times 10^{-11},1\times
10^{-5}]$, $T/\sqrt{\rho}=0.15$, $\alpha/\sqrt{\rho}=1$ and
$\mathcal{K}=-1/8$. Right: We have fixed $T/\sqrt{\rho}\in[0.06,0.30]$,
$\alpha/\sqrt{\rho}=1$, $\lambda=1\times 10^{-8}$ and $\mathcal{K}=-1/8$. The
black lines indicate linear scalings.

##  IV Electric conductivity

When the charge sector strongly couples with phonons, the combined system will
reach local equilibrium extremally fast. Then, total momentum of the charge-
phonon system should be conserved, resulting an infinite DC
conductivity.555Here, we have assumed that the system has no particle-hole
symmetry. Such a divergency of the conductivity can again be removed by
introducing an ESB of translations which relaxes the momentum. However, unlike
the picture without phonons, the AC conductivity now displays a finite peak at
finite frequency. In this section, we turn to check how (pseudo-)phonons in
our holographic model affect the charge transport in the purely SSB case and
in the presence of weakly and explicitly broken translations, respectively.

###  IV.1 Purely SSB pattern

In the presence of purely SSB case, the AC conductivity at low frequency obeys
the following hydrodynamic formula Hartnoll  _et al._ (2016a)

$\sigma(\omega)=\sigma_{Q}+\frac{\rho^{2}}{\chi_{PP}}\left(\delta(\omega)+\frac{i}{\omega}\right),$
(31)

where $\sigma_{Q}$ is the so-called incoherent conductivity and
$\rho^{2}/\chi_{PP}$ is Drude weight. Note that the DC conductivity is
infinite in the SSB case because of the absence momentum relaxation mechanism.
In this holographic model, the incoherent conductivity can be computed
directly via the membrane paradigm (see Davison  _et al._ (2015) for more
details)

$\sigma_{Q}=\left(1+\mathcal{K}\frac{4\pi\alpha^{2}}{s}\right)\left(\frac{sT}{sT+\mu\rho}\right)^{2}.$
(32)

Here, the first law of thermodynamics $\epsilon+p=sT+\mu\rho$ has been
applied.

We follow the procedure in Andrade and Withers (2014) to calculate AC
conductivity at zero momentum by holographic method. Introduce the time-
dependent perturbations around the background (6)

$\delta g_{tx}=\frac{1}{u^{2}}h_{tx}(u)e^{-i\omega t},\quad\delta
A_{x}=a_{x}(u)e^{-i\omega t},\quad\delta\phi_{x}=\phi_{x}(u)e^{-i\omega t},$
(33)

and choose radial gauge $\delta g_{ux}=0$. The linearized equations of motion
are given by

$\displaystyle 0=2f(\alpha^{2}u^{2}\mathcal{K}+1)^{2}(\rho
u^{3}a_{x}^{\prime}-uh_{tx}^{\prime\prime}+2h_{tx}^{\prime})+\alpha^{2}uh_{tx}(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\rho^{2}u^{4}\mathcal{K})$
$\displaystyle+i\alpha
u\phi_{x}\omega(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\rho^{2}u^{4}\mathcal{K}),$
(34) $\displaystyle 0=2i\rho\omega
a_{x}(\alpha^{2}u^{3}\mathcal{K}+u)^{2}+\alpha
f\phi_{x}^{\prime}(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\rho^{2}u^{4}\mathcal{K})-2i\omega
h_{tx}^{\prime}(\alpha^{2}u^{2}\mathcal{K}+1)^{2},$ (35) $\displaystyle
0=\omega^{2}a_{x}(-(\alpha^{2}u^{2}\mathcal{K}+1))-f(a_{x}^{\prime}(\alpha^{2}u\mathcal{K}(uf^{\prime}+2f)+f^{\prime})+fa_{x}^{\prime\prime}(\alpha^{2}u^{2}\mathcal{K}+1)-\rho
h_{tx}^{\prime}),$ (36) $\displaystyle
0=f(\phi_{x}^{\prime}(2f(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{3}+\rho^{2}u^{4}\mathcal{K}(1-\alpha^{2}u^{2}\mathcal{K}))-uf^{\prime}(\alpha^{2}u^{2}\mathcal{K}+1)(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}$
$\displaystyle-\rho^{2}u^{4}\mathcal{K}))-fu\phi_{x}^{\prime\prime}(\alpha^{2}u^{2}\mathcal{K}+1)(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\rho^{2}u^{4}\mathcal{K}))-u\phi_{x}\omega^{2}(\alpha^{2}u^{2}\mathcal{K}+1)(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}$
$\displaystyle-\rho^{2}u^{4}\mathcal{K})+i\alpha u\omega
h_{tx}(\alpha^{2}u^{2}\mathcal{K}+1)(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\rho^{2}u^{4}\mathcal{K}).$
(37)

It is easy to check there are three independent equations above. One can
reduce them to two equations (see Wang  _et al._ (2019) for analogous
simplification method). The AC conductivity can be computed by solving (34-37)
numerically via Kubo formula Ammon  _et al._ (2019a)

$\sigma(\omega)=\frac{1}{i\omega}\mathcal{G}_{JJ}^{R}(\omega,k=0).$ (38)

We show that the numerical results of AC and incoherent conductivities in
Fig.7. These results imply that the background solution of scalar field
$\phi^{I}=\alpha x^{I}$ should be interpreted as the SSB nature. Moreover, the
incoherent conductivity is also controlled by $\alpha$ and coupling
$\mathcal{K}$, which can be viewed as the effect from phonons. Note that this
can never happen if we neglect the direct coupling the gauge field and the
axions in the bulk theory. It is obvious to see in Fig.7 that our numerical
results from (38) are in agreement with analytical results from (31-32) .

Figure 7: Left: The imaginary parts of AC conductivity for different
temperatures. The inset plot shows the Drude weight $\rho^{2}/\chi_{PP}$. Blue
line is the analytical result obtained from (31) and red dots are the
numerical results from (38). Right: The incoherent conductivity $\sigma_{Q}$
as the function of temperature. Blue line is the analytical result obtained
from (32) and red dots are the numerical results from (38) . We have fixed
$\mathcal{K}=-1/8$ and $\lambda=0$.

###  IV.2 Pinned structure

In section III.2, we study the pinned phonons by turning on small external
source. In this case, the DC conductivity can be computed anatically by the
membrane paradigm Donos and Gauntlett (2014c); Baggioli  _et al._ (2017)

$\sigma_{DC}=1+\mathcal{K}\alpha^{2}u_{h}^{2}+\frac{\rho^{2}u_{h}^{2}}{\alpha^{2}\left(\lambda-\frac{\mathcal{K}\rho^{2}u_{h}^{4}}{2\left(1+\mathcal{K}\alpha^{2}u_{h}^{2}\right)^{2}}\right)}.$
(39)

The structure of the pinned phonons are also captured by the AC conductivity
in the low frequency. AC conductivity (31) from hydrodynamics should be
modified in the presence of both ESB and SSB, which is given by Amoretti  _et
al._ (2020a, 2019b)

$\sigma(\omega)=\sigma_{Q}+\frac{\gamma_{1}^{2}\chi_{PP}^{2}\omega_{0}^{2}(\Gamma-i\omega)+2\gamma_{1}\rho\chi_{PP}\omega_{0}^{2}+\rho^{2}(\Omega-i\omega)}{\chi_{PP}^{2}(\left(\Omega-i\omega)(\Gamma-i\omega\right)+\omega_{0}^{2})}.$
(40)

where $\gamma_{1}$ is the coefficient which is extracted from the correlator
between the electric current and Goldstone operator. When the momentum
relaxation is slow, relaxation rate $\Gamma$ can be ignored. Based on our
calculation, $\omega_{0}^{2}$ is the very small quantity. Thus we can omit the
contribution of the $\gamma_{1}$ term. We identify $\Omega$ and $\omega_{0}$
by extracting QNMs data, which will fix all the parameters (40), to fit the AC
conductivity by holography. The results are shown in Fig.8. We observe that
the numerical AC conductivities are fitted by the hydrodynamic predictions
well. As the authors point out Baggioli and Pujolas (2017); Goutéraux  _et
al._ (2016); An  _et al._ (2020), the pinning effect introduces the metal-
insulator transition. From the perspective of the shear viscosity, there is no
specific behaviour at both sides of the transition point (see Appendix C for
more details).

Figure 8: Comparisons between numerical AC conductivities (black solid lines)
and hydrodynamic predictions (red dots) for different temperatures. The blue
dots are the DC conductivity results from (39). Left: $T/\sqrt{\rho}=0.1$ and
$\omega_{0}^{2}\approx 0.0017$. Here, one should distinguish $\omega_{0}$-the
mass of pinned phonons and $\omega_{\text{peak}}$, the peak of the
conductivity in IR region. Right: $T/\sqrt{\rho}=0.3$. We have fixed
$\lambda=1\times 10^{-7}$ and $\mathcal{K}=-1/8$.

##  V Conclusions

In this work, we consider a holographic model that can break translational
invariance in the dual field theory. We confirm the explicit breaking by
turning on the linear axion term $\lambda X$ and spontaneous breaking by a
gauge-axion coupling $\mathcal{K}XF^{2}$ via the UV analysis.

When turning off the external source, we compute shear modulus and viscosity
by holography. The non-trivial results imply that the dual boundary system is
an elastic solid. By calculating quasi-normal modes and making a comparison
with the hydrodynamics predictions, we verify the existence of the gapless
transverse phonons in the low energy spectrum. Then we turn on the slow
momentum relaxation to see pinned phonons and check GMOR relation. It is found
that the pinning frequency is proportional to the square root of the explicit
breaking scale and to the half to the third power of the spontaneous breaking
scale. This implies that it is different from the “standard” GMOR relation in
our model.

We then study the charge transport. Firstly in the purely SSB pattern, the AC
conductivity is in perfect agreement with the hydrodynamics expectations. This
allows us to claim that the background solution of scalar field should be
interpreted as the SSB contribution. Meanwhile in the presence of weakly
explicit breaking, we show that the numerical AC conductivity can also be
fitted by the hydrodynamic predictions well.

In the small frequency and momentum regime, our model shares the same
hydrodynamic formulas with other holographic axion models Baggioli  _et al._
(2021) or the homogeneous models with more complicated setups Amoretti  _et
al._ (2019b); Andrade  _et al._ (2018) (even though the coefficients and
parameters in the hydrodynamic formulation depend on the setup). This again
verifies the statement that hydrodynamics provides a universal description for
strongly coupled systems.

In this paper we only focus on the transverse channel of the system. The
spectra of the longitudinal channel are also worth analyzing. This channel
contains a pair of sound modes and one crystal diffusion mode in the low
momentum limit, which is more complicated Ammon  _et al._ (2019b); Baggioli
_et al._ (2019); Baggioli and Grieninger (2019). We can study how charge
density affects speed of longitudinal sounds, crystal diffusion and check the
universality of its lower Baggioli and Li (2020); Wu  _et al._ (2021) and
upper Baggioli and Li (2020); Arean  _et al._ (2020); Wu  _et al._ (2021)
bounds proposed previously. In addition, one can also investigate the magnetic
effect with the current model and compare the result with that from the
magneto-hydrodynamics Amoretti  _et al._ (2020b); Delacrétaz  _et al._ (2019)
and other holographic models with magnetic effects Baggioli  _et al._ (2020a);
Amoretti  _et al._ (2020c, 2021); Donos  _et al._ (2021). We will leave these
for future work.

###### Acknowledgements.

We would like to thank Andrea Amoretti, Matteo Baggioli, Guo-Yang Fu, Li Li,
Yi Ling, Jian-Pin Wu and Meng-He Wu for innumerous stimulating discussions and
comments on the topics presented in this paper. We are particularly grateful
to Matteo Baggioli and Li Li for their valuable comments about the manuscript.
XJW is supported by NSFC No.11775036. WJL is supported by NSFC No.11905024 and
No.DUT19LK20.

##  Appendix A Derivation details for analytical Green’s functions

In this appendix, we show the details for the analytical calculations of
Green’s functions $\mathcal{G}_{T_{xy}T_{xy}}^{R}$ at high and zero
temperature limits.

###  A.1 High temperature limit

Let us focus on the high temperature limit first, i.e. $\sqrt{\rho}/T\ll 1$
that is equivalent to small charge density limit. We use Poincaré coordinates
(6) this time and turn on shear perturbation $\delta g_{xy}=e^{-i\omega
t}h(u)/u^{2}$. The perturbation equation is given by

$h^{\prime\prime}+\left(\frac{f^{\prime}}{f}-\frac{2}{u}\right)h^{\prime}+\left(\frac{\omega^{2}}{f^{2}}+\frac{\alpha^{2}\hat{\rho}^{2}T^{4}u^{4}\mathcal{K}}{2f\left(\alpha^{2}u^{2}\mathcal{K}+1\right)^{2}}\right)h=0,$
(41)

where we have introduced a dimensionless parameter which is defined by

$\hat{\rho}=\left(\frac{\sqrt{\rho}}{T}\right)^{2}.$ (42)

First, we show the analytical procedure for the correlator (16). To solve the
equation of motion (41), one should implement the infalling boundary condition
at the horizon and require field $h(u=0)=h_{0}$ at the AdS boundary in zero
frequency limit. The on-shell action can be obtained Alberte  _et al._ (2016b)

$S_{bdy}=\int
d^{2}x\int\frac{d\omega}{2\pi}h_{0}\mathcal{F}(\omega,u)h_{0}\bigg{|}_{u=\epsilon},$
(43)

where $\epsilon$ is the UV cut-off and $h_{0}$ is the value of the bulk field
$h$ at the boundary, which is equivalent to the source $h_{0}\equiv h(u=0).$
As illustrated in Son and Starinets (2002), the retarded Green’s function
$\mathcal{G}^{R}_{T_{ij}{T_{ij}}}$ can be computed by taking two functional
derivative of the above on-shell action with respect to the source $h_{0}$

$\mathcal{G}^{R}_{T_{ij}{T_{ij}}}=-\lim_{\epsilon\rightarrow
0}2\mathcal{F}(\omega,u)\bigg{|}_{u=\epsilon}.$ (44)

The equation (41) can be solved by imposing the follow ansatz

$h(u)=h_{0}\,e^{-\frac{i\omega}{4\pi
T}logf}\left(\Phi_{0}(u)+\frac{i\omega}{4\pi T}\Phi_{1}(u)+\cdots\right),$
(45)

where exponential prefactor represents infalling behaviour and ellipsis
denotes higher frequency contributions which are neglected. The functions
$\Phi_{0}$ and $\Phi_{1}$ are required to be regular. We set

$\Phi_{0}(0)=1,\qquad\Phi_{1}(0)=0,$ (46)

near the boundary.

One can use perturbative method to solve the equation (41) with the ansatz
(45). The equations for $\Phi_{0}$ and $\Phi_{1}$ can be obtained order by
order in the frequency

$\displaystyle
0=\left(\frac{f}{u^{2}}\Phi_{0}^{\prime}(u)\right)^{\prime}+\frac{\alpha^{2}\hat{\rho}^{2}T^{4}u^{2}\mathcal{K}}{2\left(\alpha^{2}u^{2}\mathcal{K}+1\right)^{2}}\Phi_{0}(u),$
$\displaystyle
0=\left(\frac{f}{u^{2}}\Phi_{1}^{\prime}(u)\right)^{\prime}+\frac{\alpha^{2}\hat{\rho}^{2}T^{4}u^{2}\mathcal{K}}{2\left(\alpha^{2}u^{2}\mathcal{K}+1\right)^{2}}\Phi_{1}(u)-\frac{2f^{\prime}}{u^{2}}\Phi_{0}^{\prime}(u)+\left(\frac{2f^{\prime}}{u^{3}}-\frac{f^{\prime\prime}}{u^{2}}\right)\Phi_{0}(u).$
(47)

Since we consider small charge density limit, the emblackening factor $f$ can
be simplified to Schwarzchild-AdS geometry $f(u)=1-u^{3}/u_{h}^{3}$ and the
Hawking temperature is $T=3/(4\pi u_{h})$. Meanwhile, we adopt the following
series expansions for the ansatz (45) in $\hat{\rho}^{2}$

$\Phi_{0}=\sum_{n=0}^{\infty}\hat{\rho}^{2n}\phi_{n},\qquad\Phi_{1}=\sum_{n=0}^{\infty}\hat{\rho}^{2n}\psi_{n},$
(48)

where the requirements (46) for $\Phi_{0}$ and $\Phi_{1}$ make $\phi_{0}=1$
and $\psi_{0}=0$. Substitute the above expansions into (47), and one can
obtain

$\displaystyle\left(\frac{f}{u^{2}}\phi_{1}^{\prime}\right)^{\prime}=-\frac{\alpha^{2}u^{2}\mathcal{K}T^{4}}{2\left(\alpha^{2}u^{2}\mathcal{K}+1\right)^{2}}\approx-\frac{1}{2}\alpha^{2}u^{2}\mathcal{K}T^{4},$
(49)
$\displaystyle\left(\frac{f}{u^{2}}\psi_{1}^{\prime}\right)^{\prime}=2\frac{f^{\prime}}{u^{2}}\phi_{1}^{\prime}.$
(50)

By dimensional analysis, the dimension of $\alpha$ is the same as
$\sqrt{\rho}$. We can expand $\alpha$ to get the leading order contribution,
i.e. approximate term in (49).

The on-shell action is given by Alberte  _et al._ (2016b) 666In this paper
Alberte  _et al._ (2016b), the author re-defined the radial coordinate
$z=u/u_{h}$. We recover the coordinate $u$ here.

$S_{bdy}=-\frac{1}{2}\int
d^{3}x\left(\frac{f}{u^{2}}h(u)h^{\prime}(u)\right)\bigg{|}_{u=\epsilon}^{u=uh},$
(51)

where $\epsilon$ is the UV cut-off. Expand the above formula at leading order
in the charge density and frequency

$S_{bdy}=-\int
d^{3}x\,\frac{h_{0}^{2}}{2}\left[\hat{\rho}^{2}\left(\frac{f}{\epsilon^{2}}\phi_{1}^{\prime}\right)+\frac{i\omega}{4\pi
T}\left(\frac{3}{u_{h}^{3}}+\hat{\rho}^{2}\left(\frac{f}{\epsilon^{2}}\psi_{1}^{\prime}\right)-2\hat{\rho}^{2}log\,f\left(\frac{f}{\epsilon^{2}}\phi_{1}^{\prime}\right)\right)+\cdots\right].$
(52)

The shear elastic modulus and viscosity can be calculated according to (43-44)
and (52)

$\displaystyle G=Re\mathcal{G}_{T_{xy}T_{xy}}^{R}=\lim_{\epsilon\rightarrow
0}\hat{\rho}^{2}\left(\frac{f}{\epsilon^{2}}\phi_{1}^{\prime}\right),$ (53)
$\displaystyle\eta=\lim_{\omega\rightarrow
0}\left[-\frac{1}{\omega}Im\mathcal{G}_{T_{xy}T_{xy}}^{R}(\omega)\right]=\lim_{\epsilon\rightarrow
0}\frac{1}{4\pi
T}\left(\frac{3}{u_{h}^{3}}+\hat{\rho}^{2}\left(\frac{f}{\epsilon^{2}}\psi_{1}^{\prime}\right)\right).$
(54)

Firstly, we compute the shear elastic modulus concretely by using (49)

$\displaystyle G=\hat{\rho}^{2}\lim_{\epsilon\rightarrow
0}\int^{u_{h}}_{\epsilon}\left(-\frac{1}{2}\alpha^{2}u^{2}\mathcal{K}T^{4}\right)$
$\displaystyle\,\,\,\,\,=-\frac{1}{6}\alpha^{2}\hat{\rho}^{2}T^{4}\mathcal{K}u_{h}^{3}.$
(55)

We recover the parameter $\rho$ and replace $u_{h}$ with $T$. Then one can re-
write the above relation

$G=-\frac{9\mathcal{K}\alpha^{2}T}{128\pi^{3}}\left(\frac{\sqrt{\rho}}{T}\right)^{4}+\cdots.$
(56)

Secondly, we show the shear viscosity. In small charge density limit, the
equation (49) for $\phi_{1}$ is solved by

$\frac{f}{u^{2}}\phi_{1}^{\prime}=\frac{1}{6}\alpha^{2}T^{4}\mathcal{K}\left(u^{3}-u_{h}^{3}\right),$
(57)

For (50) and (57),

$\frac{f}{u^{2}}\psi_{1}^{\prime}=\int_{u}^{u_{h}}2\frac{f^{\prime}}{x^{2}}\phi_{1}^{\prime}dx=\int_{u}^{u_{h}}2\frac{f^{\prime}}{f}\left(\frac{f}{x^{2}}\phi_{1}^{\prime}\right)dx=\frac{1}{3}\alpha^{2}T^{4}\mathcal{K}\left(u_{h}^{3}-u^{3}\right).$
(58)

Thus the ratio of shear viscosity $\eta$ to entropy density $s$ is

$\displaystyle\frac{\eta}{s}$ $\displaystyle=\lim_{\epsilon\rightarrow
0}\frac{1}{4\pi
Ts}\left(\frac{3}{u_{h}^{3}}+\hat{\rho}^{2}\left(\frac{f}{\epsilon^{2}}\psi_{1}^{\prime}\right)\right)$
(59)
$\displaystyle=\frac{1}{4\pi}+\frac{81\mathcal{K}\alpha^{2}\rho^{2}}{16384\pi^{7}T^{6}},$
(60)

where we have used simple emblackening factor $f(u)=1-u^{3}/u_{h}^{3}$,
temperature $T=3/(4\pi u_{h})$ and entropy density $s=4\pi/u_{h}^{2}$. Thus
the above formula can be re-written

$\frac{\eta}{s}=\frac{1}{4\pi}+\frac{81\mathcal{K}\alpha^{2}}{16384\pi^{7}T^{2}}\left(\frac{\sqrt{\rho}}{T}\right)^{4}+\cdots.$
(61)

The comparison between the numerical and analytical results for shear elastic
modulus and viscosity is shown in Fig.9.

Figure 9: The comparison between the numerical and analytical results of shear
elastic modulus and viscosity in the high temperature limit for
$\mathcal{K}\in[-1/8,-1/20]$(red-orange). The dots are numerical results
obtained by (16). The gray dashed lines are analytical results obtained by
(56) and (61). We have fixed $\lambda=0$ and $\alpha/\sqrt{\rho}=1$.

###  A.2 Zero temperature limit

Next, we will consider $\eta/s$ ratio at zero temperature limit analytically.
In this case (6), the black hole becomes extreme and the near-horizon geometry
is $AdS_{2}\times R_{2}$ Baggioli (2019). The equation is given by

$h^{\prime\prime}+\left(\frac{f^{\prime}}{f}-\frac{2}{u}\right)h^{\prime}+\left(\frac{\omega^{2}}{f^{2}}+\frac{\alpha^{2}\rho^{2}u^{4}\mathcal{K}}{2f\left(\alpha^{2}u^{2}\mathcal{K}+1\right)^{2}}\right)h=0,$
(62)

where we have recovered the parameter $\rho$. We follow the method in Baggioli
(2019). Assume a power-law ansatz for the field $h$

$h(u)\sim(u-u_{h})^{\nu}.$ (63)

The perturbation equation around the near-horizon geometry is

$\nu\left((\nu-1)u_{h}+2u_{h}\right)f^{\prime\prime}\left(u_{h}\right)\left(\alpha^{2}\mathcal{K}u_{h}^{2}+1\right){}^{2}+\alpha^{2}\rho^{2}\mathcal{K}u_{h}^{5}=0.$
(64)

One can get

$\nu=-\frac{1}{2}\pm\xi,\qquad\xi\equiv\frac{1}{2}\sqrt{1-\frac{4\alpha^{2}\rho^{2}\mathcal{K}u_{h}^{4}}{\left(\alpha^{2}\mathcal{K}u_{h}^{2}+1\right)^{2}f^{\prime\prime}\left(u_{h}\right)}}.$
(65)

Thus we can write the general solution

$h(u)\sim u^{-1/2-\xi}+\mathcal{G}_{T_{xy}T_{xy}}(\omega)u^{-1/2+\xi}.$ (66)

By dimensional analysis, frequency has the dimension of the temperature and
the inverse of the radial coordinate ($\omega\sim T\sim u^{-1}$). This implies

$\omega^{1/2+\xi}\sim\mathcal{G}_{T_{xy}T_{xy}}(\omega)\,\omega^{1/2-\xi},$
(67)

and again

$\mathcal{G}_{T_{xy}T_{xy}}(\omega)\sim\omega^{2\xi}.$ (68)

Therefore

$\eta\sim T^{2\xi-1}.$ (69)

Finally we can obtain

$\frac{\eta}{s}\sim T^{2\xi-1}.$ (70)

Next we analyze $\xi$. In zero temperature limit, one can get

$f^{\prime\prime}\left(u_{h}\right)=-\frac{6}{u_{h}^{2}}-\frac{\alpha^{2}\rho^{2}\mathcal{K}u_{h}^{4}}{2\left(\alpha^{2}\mathcal{K}u_{h}^{2}+1\right){}^{2}}+\frac{3\rho^{2}u_{h}^{2}}{2\left(\alpha^{2}\mathcal{K}u_{h}^{2}+1\right)}.$
(71)

We set $u_{h}=1$ and $\alpha/\sqrt{\rho}=1$. Zero temperature limit gives
another formula

$\rho=2\sqrt{9\mathcal{K}^{2}+3}+6\mathcal{K}.$ (72)

Substitute the above relations into (65), we can finally obtain the $\eta/s$
ratio at zero temperature limit

$\frac{\eta}{s}\sim\left(\frac{T}{\sqrt{\rho}}\right)^{2\xi-1},\qquad\xi=\frac{1}{2}\sqrt{1-\frac{8\sqrt{3}\mathcal{K}}{\sqrt{3\mathcal{K}^{2}+1}}}.$
(73)

We show the comparisons of the ratio obtained analytically and numerically in
Fig.10.

Figure 10: The $\eta/s$ ratio as the function of $T/\sqrt{\rho}$ for
$\mathcal{K}\in[-1/8,-1/20]$(red-orange). The dots are numerical results
obtained by (16). The black dashed lines are analytical results obtained by
(73) in the low temperature limit. The red dashed line is the KSS bound. We
have fixed coupling $\lambda=0$.

##  Appendix B Perturbation equations and Green’s functions

For convenience in numerical calculations of QNMs, we choose Eddington-
Finkelstein (EF) coordinates

$ds^{2}=\frac{1}{u^{2}}\left[-f(u)dt^{2}-2dtdu+dx^{2}+dy^{2}\right].$ (74)

Since the boundary field theory is isotropic, we choose the momentum $k$ to be
parallel to the $y$-axis. Take the following forms of the transverse
perturbations

$\delta g_{tx}=\frac{1}{u^{2}}h_{tx}(u)e^{-i\omega t+iky},\,\delta
g_{xy}=\frac{1}{u^{2}}h_{xy}(u)e^{-i\omega t+iky},\,\delta
A_{x}=a_{x}(u)e^{-i\omega t+iky},\,\delta\phi_{x}=\phi_{x}(u)e^{-i\omega
t+iky},$ (75)

and choose radial gauge $\delta g_{ux}=0$ for simplicity.

The equations are

$\displaystyle
0=u(2ikh_{xy}^{\prime}(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-2h_{tx}^{\prime\prime}(\alpha^{2}u^{2}\mathcal{K}+1)^{2}+\alpha\phi_{x}^{\prime}(\rho^{2}u^{4}\mathcal{K}-2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}))$
$\displaystyle+4h_{tx}^{\prime}(\alpha^{2}u^{2}\mathcal{K}+1)^{2}+2\rho
u^{3}a_{x}^{\prime}(\alpha^{2}u^{2}\mathcal{K}+1)^{2},$ (76) $\displaystyle
0=2i\rho\omega
a_{x}(\alpha^{2}u^{3}\mathcal{K}+u)^{2}+i(\alpha\phi_{x}\omega(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\rho^{2}u^{4}\mathcal{K})-2(\alpha^{2}u^{2}\mathcal{K}+1)^{2}(fkh_{xy}^{\prime}+\omega
h_{tx}^{\prime}))$ $\displaystyle+\alpha
f\phi_{x}^{\prime}(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\rho^{2}u^{4}\mathcal{K})+h_{tx}(2(\alpha^{2}\lambda+k^{2})(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\alpha^{2}\rho^{2}u^{4}\mathcal{K})$
$\displaystyle+2k\omega h_{xy}(\alpha^{2}u^{2}\mathcal{K}+1)^{2},$ (77)
$\displaystyle
0=4ikh_{tx}(\alpha^{2}u^{2}\mathcal{K}+1)^{2}+h_{xy}(\alpha^{2}u(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\rho^{2}u^{4}\mathcal{K})+4i\omega(\alpha^{2}u^{2}\mathcal{K}+1)^{2})$
$\displaystyle-i(\alpha
ku\phi_{x}(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\rho^{2}u^{4}\mathcal{K})+2(\alpha^{2}u^{2}\mathcal{K}+1)^{2}(h_{xy}^{\prime}(-iuf^{\prime}+2if+2u\omega)-ifuh_{xy}^{\prime\prime}+kuh_{tx}^{\prime})),$
(78) $\displaystyle 0=a_{x}^{\prime}(-2\alpha^{2}fu\mathcal{K}-i(2\omega-
if^{\prime})(\alpha^{2}u^{2}\mathcal{K}+1))+a_{x}(k^{2}(\alpha^{2}u^{2}\mathcal{K}+1)-2i\alpha^{2}u\omega\mathcal{K})+\rho
h_{tx}^{\prime}$ $\displaystyle-
fa_{x}^{\prime\prime}(\alpha^{2}u^{2}\mathcal{K}+1),$ (79) $\displaystyle
0=-\phi_{x}(k^{2}u(\alpha^{2}u^{2}\mathcal{K}+1)(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\rho^{2}u^{4}\mathcal{K})+2i\omega(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{3}+\rho^{2}u^{4}\mathcal{K}(1-\alpha^{2}u^{2}\mathcal{K})))$
$\displaystyle+\phi_{x}^{\prime}(-2f(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{3}+\rho^{2}u^{4}\mathcal{K}(1-\alpha^{2}u^{2}\mathcal{K}))+u(f^{\prime}+2i\omega)(\alpha^{2}u^{2}\mathcal{K}+1)(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\rho^{2}u^{4}\mathcal{K}))$
$\displaystyle-2\alpha
h_{tx}(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{3}+\rho^{2}u^{4}\mathcal{K}(1-\alpha^{2}u^{2}\mathcal{K}))-i\alpha
kuh_{xy}(\alpha^{2}u^{2}\mathcal{K}+1)(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\rho^{2}u^{4}\mathcal{K})$
$\displaystyle+fu\phi_{x}^{\prime\prime}(\alpha^{2}u^{2}\mathcal{K}+1)(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\rho^{2}u^{4}\mathcal{K})+\alpha
uh_{tx}^{\prime}(\alpha^{2}u^{2}\mathcal{K}+1)(2\lambda(\alpha^{2}u^{2}\mathcal{K}+1)^{2}-\rho^{2}u^{4}\mathcal{K}).$
(80)

The asymptotic behaviour of the bulk fields near the boundary ($u=0$) are

$\displaystyle h_{tx}=h_{tx(l)}(1+\dots)+h_{tx(s)}u^{3}(1+\dots),$ (81)
$\displaystyle h_{xy}=h_{xy(l)}(1+\dots)+h_{xy(s)}u^{3}(1+\dots),$ (82)
$\displaystyle a_{x}=a_{x(l)}(1+\dots)+a_{x(s)}u(1+\dots),$ (83)

where $l$ and $s$ mark the leading and subleading orders. According to
holographic dictionary and standard quantization, we can write the retarded
Green’s functions as

$\displaystyle\mathcal{G}^{R}_{T_{tx}T_{tx}}=\left(2\Delta-d\right)\frac{h_{tx(s)}}{h_{tx(l)}}=3\frac{h_{tx(s)}}{h_{tx(l)}},$
(84)
$\displaystyle\mathcal{G}^{R}_{T_{xy}T_{xy}}=\left(2\Delta-d\right)\frac{h_{xy(s)}}{h_{xy(l)}}=3\frac{h_{xy(s)}}{h_{xy(l)}},$
(85) $\displaystyle\mathcal{G}^{R}_{JJ}=\frac{a_{x(s)}}{a_{x(l)}}.$ (86)

The physical observables can be extracted from the retarded Green’s functions.

##  Appendix C DC conductivity and shear viscosity

We show the figures of DC conductivity and shear viscosity with temperature in
Fig.11. As shown in the left figure, there exists the metal-insulator
transition near the critical temperature $T/\sqrt{\rho}=0.265$. However there
is no specific behaviour for the shear viscosity at both sides of the
transition point.

Figure 11: DC conductivity and $\eta/s$ as the function of the temperature.
The black dashed line indicates the transition temperature. Left:
$\mathcal{K}=-1/8$ and $\lambda=5\times 10^{-2}$. Right: $\mathcal{K}=-1/8$,
$\lambda=0$ (red solid line) and $\lambda=5\times 10^{-2}$ (blue solid line).

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