diffusive mode $\omega=-i\Gamma-iD_{\pi}k^{2}$ for $m/T\in[1,6.5]$ (from black
to light blue). Bottom:  In orange the viscosity to entropy ratio $\eta/s$, in
blue the dimensionless shear diffusion constant $D_{\pi}T$ obtained
numerically and in green the analytic formula (150). The horizontal dashed
value is $1/4\pi$. Figure taken from Baggiolili2005 .

The original discussion of Hartnoll:2014lpa is more general and was initially
focused on the diffusion of energy and charge. In particular, given a generic
diffusion constant $D_{i}$, where $i$ indicates the corresponding operator
associated to the diffusive dynamics, Hartnoll conjectured that an equality of
the type:

$D_{i}\,\geq\,v^{2}\,\tau\,,$ (152)

has to be valid, where $v$ is a characteristic velocity scale and $\tau$ a
minimal relaxation time. When the momentum is relaxed very rapidly (_e.g._ in
the bad metal case of Hartnoll:2014lpa ), $m/T\gg 1$ (we also call it low
temperature limit or incoherent limit.), the dynamics associated with charge
and heat transport become purely diffusive and they decouple:

$\mathcal{D}=\left(\begin{array}[]{ccc}D_{C}&0\\\ 0&D_{T}\\\
\end{array}\right)\,.$ (153)

Then, the matrix of the diffusivities gets completely diagonal. In this case,
the bound (152) should apply. Nevertheless the question is: which are the
characteristic velocity and time scales? The timescale is naturally associated
with the Planckian time $\tau_{pl}=\hbar/k_{B}T$. The discussion about the
velocity is more subtle. This velocity cannot be the Fermi velocity, as it is
in general not sharply defined in the strongly coupled systems without quasi-
particles.

###  V.3 Butterfly velocity and chaos

In the attempt of making Hartnoll’s proposal predictive, M. Blake proposed
that the velocity scale appearing in the diffusivity bounds could be
identified with the butterfly velocity $v_{B}$ Blake:2016wvh ; Blake:2016sud .
$v_{B}$ measures the speed of propagation of information through a system and
it can be generically extracted from the out-of-time-order correlator (OTOC)
1969JETP…28.1200L :

$\displaystyle\Big{\langle}\left[\hat{W}(t,x),\hat{V}(0,t)\right]^{2}\Big{\rangle}_{\beta}\sim
e^{\lambda_{L}(t-t^{*}-|x|/v_{B})}\,,$ (154)
$\displaystyle\,\,\,\,\,\,\,\,\text{for}\,\,\,\,\,\,\,t_{\text{local}}\ll t\ll
t^{*}\,,$

where $\lambda_{L}$ is called the Lyapunov exponent, $t^{*}$ is the scrambling
time and $t_{\text{local}}$ is the timescale that the system reaches local
equilibrium. Here $\hat{W},\hat{V}$ are two generic hermitian operators. In
analogy to the classical chaos (see Fig. 37), this exponential growth can be
viewed as a quantum mechanical definition of it, originating from the non-
trivial commutator of two operators set at different times.

Figure 37: The classical butterfly effect intended as the exponential
sensitivity to the initial boundary conditions. Its quantum generalization can
be formulated as a exponential growth of the OTOC (154).

In holography, the butterfly velocity can be easily calculated by considering
a shock wave solution near the black hole horizon. In full generality, it only
depends on the IR metric and is insensitive to the matter field configurations
in the bulk. Given a metric of the form

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+h(r)dx_{d-1}^{2}\,,$ (155)

the butterfly velocity is given by Blake:2016wvh

$v_{B}^{2}=\frac{f^{\prime}(r_{h})}{d\,h^{\prime}(r_{h})}\,.$ (156)

Exploiting the membrane paradigm Damour:2008ji ; Iqbal:2008by , the transport
coefficients can usually be expressed in terms of horizon data. Then, it is
not hard to find the direct relations between the diffusivities and the
butterfly velocity (see the illustration in Fig. 38):

$D_{C,T}=C_{C,T}\,\frac{v_{B}^{2}}{2\pi T}\,,$ (157)

where $C_{C,T}$ are constants.

Figure 38: The diffusivities and butterfly velocity can be related via the
membrane paradigm.

Note that these relations are only valid in the low temperature limit or
strong momentum relaxation (incoherent) limit Kim:2017dgz , where the charge
and thermal sectors decouple (see Kovtun:2014nsa for a discussion about the
general situation and possible more general bounds). Furthermore, (157) should
be viewed as a low energy IR statement, since both of $C_{C,T}$ are only
determined by the scaling symmetry of the IR fixed point, irrelevant to any UV
parameters, even though they are model-dependent quantities. For the Einstein-
Maxwell-dilaton model, with a hyperscaling violating IR geometry Blake:2016wvh
; Blake:2017qgd , one obtains

$C_{C}=\frac{d-\theta}{\Delta_{\chi}}\,,\qquad C_{T}=\frac{z}{2z-2}\,.$ (158)

Here, $\Delta_{\chi}$ is the scaling dimension of the charge susceptibility,
$z$ is the dynamical critical exponent and $\theta$ the hyperscaling violation
exponent. For anisotropic Q-Lattice models, $C_{T}$ has been computed in
Ahn:2017kvc and it has been shown to obey the lower bound (158), proving that
the latter it is not affected by anisotropy.

###  V.4 Pole-skipping and the complex plane

The chaotic properties of a dynamical system, such as the Lyapunov exponent
($\lambda_{L}$) and the butterfly velocity ($v_{B}$), are encoded in a
specific four point function, out-of-time-order-correlator (OTOC) (154) at
short time scale. Interestingly, it has been observed that $\lambda_{L}$ and
$v_{B}$ can be also detected by a two point function, the retarded Green’s
function of the energy density operator at long time scale. This property has
been dubbed pole-skipping phenomenon Grozdanov:2017ajz ; Blake:2017ris ;
Blake:2018leo and $\lambda_{L}$ and $v_{B}$ have been proven to be related
with the so-called "pole-skipping points". Even though the pole-skipping
phenomenon has been introduced in the context of quantum chaos, its
mathematical concept is more general and physical applications may be wider.
Thus, we start with a general definition of the pole-skipping points.

Pole-skipping points are special points in the complexified momentum (complex
frequency, complex wave number) space. At these points, two point retarded
Green’s functions of given operators are not uniquely defined. The non-
uniqueness of the Green’s function at some value of frequency/momentum is not
very novel and there are prescriptions to define the Green’s function at that
point, making contact with transport properties via Kubo’s formulas. For
example, see (135), where the Green’s functions are not well defined at
$\omega=k=0$ and we specify the order of limit to define them uniquely. It
turns out that the pole-skipping points occur at the integer/half integer
values of $i\omega/(2\pi T)$ for bosonic/fermionic operators.171717However,
see Ahn:2020baf for possibilities for non-integer $i\omega/(2\pi T)$.

Another way to introduce the pole-skipping points is using its literal
meaning: the points where the “pole” is “skipped”. For this purpose, let us
consider the expression for the retarded Green’s function of the operators $A$
and $B$:

$\mathcal{G}_{AB}(\omega,k)\,\sim\,\frac{\mathcal{B}(\omega,k)}{\mathcal{A}(\omega,k)},$
(159)

where we suppress the operator dependence on the RHS not to clutter. The pole
is defined by $\mathcal{A}(\omega,k)=0$, which gives the constraint between
$\omega$ and $k$. If this constraint lives in the pure imaginary $\omega$ and
pure imaginary $k$ space, it defines a curve in the space
($\mathrm{Im}[k],\mathrm{Im}[\omega]$), which corresponds to the red line in
Fig. 39 for example.181818This example is chosen for the pedagogical purpose.
In general, there is no a priori reason that the condition
$\mathcal{A}(\omega,k)=0$ lives in the space
($\mathrm{Im}[k],\mathrm{Im}[\omega]$). For more complete and general
discussion, we refer to Ahn:2020bafxxx .  However, there might be a second
relevant curve coming from the condition $\mathcal{B}(\omega,k)=0$, which is
this time the blue line in Fig. 39.

Figure 39: The pole-skipping points for a scalar field in the hyperbolic
space. The red line indicates the curve on which
$\mathcal{G}_{AB}(\omega,k)=\infty$ and the blue one the curve on which
$\mathcal{G}_{AB}(\omega,k)=0$. The white circle locates the pole-skipping
point $(\omega^{*},k^{*})$. Here, we set $2\pi T=1$. Figure taken from
Ahn:2020bks .

The intersection of these two curves, say $(\omega_{*},k_{*})$, is dubbed the
pole-skipping point because there the would-be pole
($\mathcal{A}(\omega_{*},k_{*})=0$) is skipped
($\mathcal{B}(\omega_{*},k_{*})=0$), _i.e._

$\mathcal{G}_{AB}(\omega_{*},k_{*})\,\sim\,\frac{\mathcal{B}(\omega_{*},k_{*})}{\mathcal{A}(\omega_{*},k_{*})}\,\sim\,\frac{0}{0}\,,$
(160)

where the last expression $0/0$ indicates that the Green’s functions is not
uniquely defined and we need a prescription to define the Green’s function
there. The path dependence prescription was clarified and classified in
Ahn:2020baf . This is nothing but the viewpoint of the first paragraph of this
subsection.

Having this general introduction, let us come back to the original argument
regarding chaos Grozdanov:2017ajz ; Blake:2017ris ; Blake:2018leo . The main
point is that chaotic nature of the OTOC (154) is related with the pole-
skipping points of the retarded Green’s function of energy density
$\mathcal{G}^{R}_{T^{tt}T^{tt}}$. Thus, the first step is to compute the
retarded Green’s function.

Let us consider an Einstein action with the matter Lagrangian
$\mathcal{L}_{M}$

$\mathcal{S}=\int d^{d+1}x\sqrt{-g}\left[{R\over
2}-\Lambda+\mathcal{L}_{M}\right]\,,$ (161)

which includes (47). According to holography, the retarded Green’s function
can be computed by the coupled equations for metric ($\delta g_{\mu\nu}(u)$)
and matter (collectively, $\delta\varphi(u)$) perturbations that are regular
at the horizon in ingoing Eddington-Finkelstein (EF) coordinates: $v=t+u_{*}$
with the tortoise coordinate $u_{*}$. Near the horizon the perturbations can
be written as

$\begin{split}\delta g_{\mu\nu}(u)&=\delta g_{\mu\nu}^{(0)}+\delta
g_{\mu\nu}^{(1)}(u-u_{h})+\cdots,\\\
\delta\varphi(u)&=\delta\varphi^{(0)}+\delta\varphi^{(1)}(u-u_{h})+\cdots\,,\end{split}$
(162)

where we consider metric perturbation near the background (48) in $d+1$
dimension. After plugging in the expansion (162) into Einstein’s equations,
one can organize the equation as an expansion about the horizon. The near
horizon equation including $\delta g_{vv}^{(0)}$ is

$\displaystyle\left(-i\frac{d-1}{u_{h}}\omega+k^{2}\right)\delta
g_{vv}^{(0)}-i(2\pi T+i\omega)\left[\omega\delta g_{x^{i}x^{i}}^{(0)}+2k\delta
g_{vx}^{(0)}\right]$
$\displaystyle=\frac{2}{u_{h}^{2}}\bigg{[}u_{h}^{2}T_{vu}(u_{h})\delta
g_{vv}^{(0)}+\delta T_{vv}(u_{h})\bigg{]}\,,$ (163)

where $T_{\mu\nu}(u_{h})$ is the bulk stress-energy tensor of the background
matter fields and $\delta T_{\mu\nu}(u_{h})$ comes from the matter
perturbations. Thus, the information of the various possible matter content is
explicitly encoded in the last term of (V.4). However, interestingly, it turns
out that this term vanishes identically for a large class of systems such as
Einstein-Maxwell-Dilaton-Axion gravity theories including our models
Blake:2018leo .

What are the consequences of this simplification (vanishing the last term of
(V.4))? In general, (V.4) provides a constraint relating the parameters
$\delta g_{vv}^{(0)},\delta g_{vx}^{(0)}$, and $\delta g_{x^{i}x^{i}}^{(0)}$.
However, if $(\omega,k)=(\omega_{*},k_{*})$, with

$\omega_{*}=2\pi Ti\,,\quad k_{*}=\pm\sqrt{i\frac{d-1}{u_{h}}\omega_{*}}=\pm
i\sqrt{\frac{2\pi T(d-1)}{u_{h}}}\,,$ (164)

we lose such a constraint. It means we have one more degree of freedom than
usual implying the Green’s function can not be uniquely determined. This non-
uniqueness of the Green’s function at a special point in complex momentum
space is precisely the defining property of the pole-skipping point, so (164)
is a pole-skipping point.

This observation is remarkable. This is an alternative holographic way of
understanding the pole-skipping points. In general, computing explicitly the
Green’s function is not easy and usually requires numerics. Finding the points
where the usual constraint at the horizon disappears is on the contrary much
easier. It is not only a technical simplification but also has a conceptual
importance: pole-skipping points are determined by the black hole horizon
property, signaling possible universal properties independent of the UV
details of the theory.

Here comes an example. Note that all information about the matter part of the
action $\mathcal{L}_{m}$ is encoded in the location of the horizon $u_{h}$.
Thus, $\omega_{*}$ is universal regardless of the matter action, while $k_{*}$
is not. Noting that the butterfly velocity from the shock-wave geometry
Blake:2016sud is

$v_{B}=\sqrt{\frac{2\pi Tu_{h}}{d-1}}\,.$ (165)

we find that the pole-skipping point (164) may be related with the chaotic
properties as follows.191919The pole-skipping phenomenon occurs also in a non-
maximally chaotic system where the pole-skipping points capture only the
stress tensor contributions to chaos Choi:2020tdj .

$\omega_{*}=i\lambda_{L}\,,\qquad k_{*}=\pm\,i\frac{\lambda_{L}}{v_{B}}\,.$
(166)

From another perspective, if we knew (166) somehow, we could have computed
(165) just by the pole-skipping points, without calculating nor the OTOC or
the shock wave geometry.

There is another interesting observation related with transport. Let us
consider a system with energy conservation but no momentum conservation, which
corresponds to the case with a very big $\alpha$ in (51). In this case, the
retarded Green’s function of the energy density will have a hydrodynamic
diffusion pole, $\omega=-iD_{T}k^{2}$ – energy diffusion. Extrapolating this
hydrodynamic relation to the pole-skipping point (166) in the non-hydrodynamic
region, we have

$D_{T}=\frac{i\omega_{*}}{k^{2}_{*}}=\frac{v_{B}^{2}}{\lambda_{L}}\,.$ (167)

Interestingly, it turns out that this extrapolation works and indeed (167) is
the universal lower bound for the holographic models with AdS${}_{2}\times
R^{d-1}$ IR geometry Blake:2016jnn . This bound is saturated at the infinite
momentum relaxation limit ($\alpha\to\infty$ in (51)). It can be understood
from an effective field theory perspective Crossley:2015evo . In short, the
pole-skipping point has a potential to predict universal hydrodynamic
properties.

So far we have focused on the specific pole skipping point of the Green’s
function of energy density, which corresponds to the gauge invariant scalar
mode of the metric perturbation. However, the pole skipping points are
ubiquitous in all kinds of Green’s functions. They were observed in other
gauge invariant modes of the metric perturbation and other fields with
different spins such as scalar, vector, and spinor fields Grozdanov:2019uhi ;
Blake:2019otz ; Natsuume:2019xcy ; Ceplak:2019ymw ; Ahn:2020bks ; Ahn:2020baf
; Ahn:2020bafxxx . However, it is still not clear which physical observables
are related with the pole-skipping points in the cases apart from energy
density. This is an interesting and important open question. See
Grozdanov:2020koi ; Ahn:2020bafxxx for this direction.

In general, there are infinite towers of pole-skipping points. The point (166)
from the near horizon equation (V.4) is just one case which is relevant to the
nearest horizon geometry. If we expand the equations to higher orders in
$(u-u_{h})$, we can obtain an infinite set of pole-skipping points whose
imaginary frequencies coincide with the Matsubara values $\omega=-i2\pi Tn$.
For example, one can see part of towers in Fig. 39, where the pole-skipping
points start from $\mathrm{Im}\omega=-1$ in units of $2\pi T=1$, which is the
“highest” pole skipping frequency (on the vertical Im$\omega$ axis). The pole
skipping points continue to appear at every integer frequency smaller than the
highest one. Thus, by coming down on the vertical Im$\omega$ axis, we are
exploring the geometry away from the horizon. The highest pole skipping
frequency is determined by the spin ($\ell$) of the operator Haehl:2019eae ;
Ahn:2020bks :

$\omega_{*}^{\mathrm{highest}}=i(\ell-1)\,.$ (168)

Note that the metric field ($\ell=2$) is the only case with the positive pole-
skipping frequency, signaling an exponentially growing instability related to
the chaotic behaviour. The other operators display only purely relaxational
modes which therefore cannot be related in any clear way to a shock-wave
solution and the onset of quantum chaos. Systematic methods to obtain these
towers of pole skipping points have been developed in Blake:2019otz ;
Natsuume:2019xcy ; Ahn:2020bks ; Ahn:2020baf ; Ahn:2020bafxxx .

The pole-skipping frequency $\omega_{*}$ is universal in the sense it is
independent of the matter action, but $k_{*}$ is not. The pole skipping wave
number $k_{*}$ depends on the momentum relaxation parameter $\alpha$ in (51),
chemical potential $\mu$, and other parameters in the matter action. In
particular, $k_{*}$ can be real number or complex number (see Fig. 40).

Figure 40: Pole-skipping wave number ($k_{*}$) can be real(red),
complex(blue), and pure imaginary(green) for scalar field perturbation in
$\text{AdS}_{4}$. The momentum relaxation parameter $\bar{\alpha}=2,2,1$ for
$\text{Im}[\bar{\omega}]=-1,-2,-3$ (from top to bottom) respectively. Here,
the “bar” variables denote the quantities scaled by $2\pi T$. Figure taken
from Ahn:2020bafxxx .

The pole skipping phenomena have been studied in the SYK models and the
conformal field theory Gu:2016oyy ; Haehl:2018izb ; Haehl:2019eae . The
relations between OTOC and pole skipping points (as well as the comparison
between field theory and gravity analysis) have been investigated in
Ahn:2019rnq ; Haehl:2019eae ; Ahn:2020bks ; Ahn:2020baf where the hyperbolic
space was considered to have an analytic control. However, the final
qualitative results are believed to hold in flat space too. The pole skipping
points were studied also for rotating black holes and topologically massive
gravity Jahnke:2019gxr ; Liu:2020yaf . For more general and detailed pole
skipping analysis for our axion model, we refer to Grozdanov:2019uhi ;
Ahn:2020bafxxx .

Let us conclude with a recent development. The idea of determining a property
coming from a 4-points function using only a 2-points one it is suspicious and
it can hardly be generic. Indeed, the relation discussed in this section is
valid only for maximally chaotic systems (saturating the chaos-bound
$\lambda_{L}\leq 2\pi T$ Maldacena:2015waa ), but it fails in general as
recently shown in Choi:2020tdj . In general, the pole skipping phenomenon
determines only the stress tensor contribution to many-body chaos which
encodes the full chaotic dynamics at maximal chaos but can be decreased in
non-maximally chaotic theories or completely cancelled in integrable systems
Perlmutter:2016pkf .

###  V.5 Bounds on thermal and crystal diffusion

Violations of the bound (152) are reported in the case of the charge
diffusivity $D_{C}.$ More precisely, $C_{C}$ can be made arbitrarily small
when certain gauge-axion couplings are considered Baggioli:2016pia . This is
because the charge transport is controlled by the Maxwell equation while
chaos, and more precisely the butterfly velocity, is controlled by the
Einstein equations. A priori, it is therefore hard to envisage a universal
connection between the two quantities. The case of the energy diffusivity is
much stronger. Indeed, one can verify that

$D_{T}=\frac{f^{\prime}h^{d/2-1}}{(f^{\prime}h^{d/2-1})^{\prime}}\frac{h^{\prime}}{h}\Big{|}_{r_{h}}\frac{v_{B}^{2}}{2\pi
T}\,,$ (169)

with all quantities evaluated at the black hole horizon $r=r_{h}$. Since
$C_{T}$ can always be resolved in terms of the horizon metric, there is no
strong evidence so far to suggest that $C_{T}\sim\mathcal{O}(1)$ can be
violated. We then take the position that there exists a universal lower bound
for thermal diffusion:

$D_{T}\geq C_{T}\frac{v_{B}^{2}}{2\pi T}\,.$ (170)

It is worth noting that this bound was checked recently in experiments
202020It is worth pointing out that the observation of a Planckian timescale
in these experiments can be just the effect of electrons-phonons interactions
as in any common metal mousatov2020phonons . mousatov2020planckian ;
zhang2017anomalous ; xu2020thermal ; zhang2019thermalization ; behnia2019lower
and confirmed by several holographic and field theory computations. Only one
subtle violation has been identified in a class of SYK chains Lucas1608 , and
its meaning is still under debate. Finally, it is interesting to notice that
this bound can be formally derived using a mathematical property of the
hydrodynamic expansion known as univalence Grozdanov:2020koi .

All the diffusion bounds discussed in the previous section attain to the
explicit breaking case. Now, let us move to the SSB case. For simplicity, we
will only focus on the 4-dimensional axions model with $V(X)=X^{N}$ in the
rest of this subsection. All the results can be directly generalized to the
hyperscaling violating case by simply adding a dilaton scalar. As was pointed
in the previous subsection IV.1, in presence of SSBm the diffusive shear mode
become sound-like (propagating) and an additional crystal diffusion mode,
associated with the axions fluctuations in the longitudinal channel, appears.
In this scenario, the only diffusive mode to which the Hartnoll bound (152)
could apply is the latter.

Let us start with the zero charge density case. The crystal diffusive mode is
decoupled from the charge sector. Its diffusion constant is given by
Armas:2019sbe ; Ammon:2020xyv :

$D_{\phi}\,=\,\xi\,\frac{\left(K+G-\mathcal{P}\right)\,\chi_{\pi\pi}}{s^{\prime}\,T^{2}\,v_{L}^{2}}\,.$
(171)

Note that this result perfectly matches the numeric data from the QNMs
Ammon:2020xyv . Using this expression, one can easily check that the
dimensionless quantity $D_{\phi}T$ decreases monotonously upon increasing
$m/T$. On the other hand, there are three distinct velocities in the system,
$v_{L}$, $v_{T}$ and $v_{B}$. The sound’s ones both grow monotonically with
$m/T$ approaching constant values at $m/T\rightarrow\infty$; on the contrary,
the butterfly velocity decreases with $m/T$, exactly as the diffusion constant
$D_{\phi}$. Using all this information, we can check that the dimensionless
ratio $D_{\phi}T/v_{B}^{2}$ obeys a universal bound which is approached at
infinite $m/T$, independently of the value of $N$ (see Fig. 41):

$D_{\phi}\,T\,\geq\,\frac{1}{2\,\pi}\,v_{B}^{2}\,.$ (172)

Figure 41: Top: Various velocities in the holographic systems with SSB of translations. We consider $V(X)=X^{N}$ by fixing $N=3$. In orange the speed of longitudinal sound; in cyan the speed of transverse sound and in green the butterfly velocity. All the velocities are normalized by the conformal value $v_{c}^{2}=1/2$. Bottom: The dimensionless ratio $D_{\phi}T/v_{B}^{2}$ in function of m/T for various $N\in[3,9]$ (from black to blue). The dashed line is the $AdS_{2}$ value $1/2\pi$. Figure taken from Baggiolili2005 . Coefficient | Value
---|---
$\xi$ | $(3-m^{2})^{2}/324$
$K$ | $9/2$
$G$ | $3/2$
$\mathcal{P}$ | $3$
$\chi_{\pi\pi}$ | $3$
$T$ | $(3-m^{2})/4\pi$
$s^{\prime}$ | $8\pi^{2}/9$
$v_{L}$ | $1$
Table 1: Various coefficients in the low temperature limit,
$m/T\rightarrow\infty$, for $V(X)=X^{3}$ where the spatial translations are
broken spontaneously. To achieve the data, we have assumed $u_{h}=1$.

Furthermore, we find that, at least for $N=3$, the linearized field equations
become simple enough to be solved analytically. In the low temperature limit,
the near horizon geometry is $AdS_{2}\times R^{2}$. The butterfly velocity
reads $v_{B}^{2}=\pi T/u_{h}$, going smoothly to be vanishing as
$m/T\rightarrow\infty$. All the other coefficients appearing in Eq. (171) can
be obtained analytically and they are summarized in Table 1. For more details
about the derivation, one refers to Baggiolili2005 . Through a highly non-
trivial cooperation of these coefficients, we finally achieve that

$\frac{D_{\phi}T}{v_{B}^{2}}\rightarrow\frac{1}{2\pi}\equiv\text{AdS}_{2}\,\,\text{bound}$
(173)

as $m/T\rightarrow\infty$, which is the same value as the thermal diffusion in
the EXB case. It would be interesting to understand if this is just a
coincidence or there is some deeper physics behind.

At finite charge density, the crystal diffusion mode couples to the charge
diffusion mode Armas:2020bmo and the model exhibits two diffusive modes in the
longitudinal channel Baggioli:2020edn :

$\omega_{1,2}=-i\,D_{1,2}\,k^{2}+\dots\,,$ (174)

where we denote with subscript “1” the mode related to crystal diffusion and
with subscript “2” the one related to charge diffusion at zero density.
Repeating the numeric calculations, one can check that $D_{1}$ is bounded
below again but $D_{2}$ is not, a property reminiscent of the situation in
presence of gauge-axion couplings or inhomogeneities. Here, we find a novel
approach to violate the proposed bound on charge diffusion, _i.e._ by
introducing the phononic dynamics via the SSB of translational symmetry.

To derive the lower bound above, it was assumed that the IR geometry has an
$AdS_{2}$ sector. However, the conclusions should also be valid for the
Lifshitz or hyperscaling violating case by using the IR scaling argument.
Finally, we conclude that the crystal diffusion is similarly bounded below as
the thermal one.

Figure 42: In relativity, the causality allowed region is bounded by the
lightcone with a slope $c$. Credits: Wikipedia
https://en.wikipedia.org/wiki/Light_cone.

###  V.6 Diffusion bound from causality

Now let us turn to investigate how another fundamental principle – causality –
tha tmay limit diffusive processes by setting a universal upper bound on their
diffusion constants. In a relativistic system, as shown in Fig. 42, any causal
processes must happen in a region enclosed by a lightcone whose slope is set
by the speed of light $c$. In non-relativistic systems, this is not
necessarily true. However, there may also exist some “emergent lightcone” that
limits the growth of operators. In these cases, the speed of operator growth
defines an effective lightcone with speed $v_{lightcone}$, which in general
does not equal to $c$. For instance, in Fermi liquids or graphene, the
lightcone velocity can be identified as the Fermi velocity of the quasi-
particle excitations, and $v_{lightcone}\equiv v_{F}\ll c$ (for graphene $\sim
c/300$).

The transport of any local conserved quantities must happen inside this
effective causal domain. Consider a diffusive process that obeys the Einstein-
Stokes law:

$\langle x^{2}\rangle=D\,t\,.$ (175)

Consider a generic local fluctuation, its position should always be limited by
$x\leq v_{lightcone}t$ for any later time. Combining this and (175), we find
that (see the illustration in Fig. 43):

$\sqrt{D\,t}\leq v_{lightcone}\,t\,.$ (176)

Figure 43: A simple visual derivation of the upper bound on diffusion from
causality. $v$ here is the lightcone speed. The region with grids is diffusion
disallowed.

Here, we should remind the reader of the fact that diffusion sets in only
after the local equilibration. It means that the equation (175) can be applied
only after a timescale $\tau_{eq}$ at which the system reaches local
equilibrium. Suppose that the diffusive process begins at $t=\tau_{eq}$. It is
easy to see that

$D\leq v_{lightcone}^{2}\,\tau_{eq}\,.$ (177)

This is an upper bound on any diffusion constant coming from causality and, in
this respect, it has a more formal and well defined origin that the lower
bound (152). Since the two proposed bounds involve different characteristic
velocities and timescales, some comments are in order:

  * •

The lightcone velocity is a microscopic velocity that in general depends on
the UV parameters of the system. For instance, in a lattice fermionic system,
we may do the identification like $v_{lightcone}\sim v_{F}\sim J\,a/\hbar$,
where $J$ is the interaction scale and $a$ is the lattice spacing. However, in
some relativistic systems, the low energy excitations carrying conserved
charges may also have a microscopic velocity that is not so UV-sensitive. For
example, in relativistic hydrodynamic systems, the lightcone for collective
modes is in general set by the sound speed $v_{s}$ which can be computed in
terms of some macroscopic quantities, say, the pressure, energy density, etc.
On the contrary, the butterfly velocity, $v_{B}$ is a completely IR quantity
in any chaotic systems, characterizing the speed of scrambling the local
perturbations initially set. It is a finite temperature effect, hence strongly
depends on the temperature itself. For example, in a strongly coupled system
with the hyperscaling violating IR fixed point, $v_{B}\sim T^{1-1/z}$ with
$(1-1/z)>0$ at low temperatures Blake:2016wvh ; Roberts:2016wdl . Therefore,
it is in general much slower than the microscopic velocity in the IR region
(see the top panel of Fig. 41).

  * •

The equilibrium timescale $\tau_{eq}$ characterizes how fast a generic system
gets local equilibration. The Planckian time
$\tau_{pl}\sim\frac{\hbar}{k_{B}T}$ sets a universal minimum for $\tau_{eq}$
as $T\rightarrow 0$. In weakly coupled systems with long-lived quasi-
particles, equilibration processes are slow. Then, in general,
$\tau_{eq}\gg\tau_{pl}$. However, for some strongly coupled systems, the
equilibrium timescale may reach its minimum.

A manifest example where the upper bound Eq. (177) is obeyed is linearized
relativistic hydrodynamics with a conserved stress tensor:

$T^{\mu\nu}\,=\,\mathcal{E}\,v^{\mu}\,v^{\nu}\,+\,\mathfrak{p}\,\Delta^{\mu\nu}\,-\,\eta\,\sigma^{\mu\nu}\,\,.$
(178)

Using the hydrodynamics method, one can show that in the transverse channel
there is a diffusive momentum mode with the dispersion relation:

$\omega=-i\frac{\eta}{\mathcal{E}+\mathfrak{p}}\,k^{2}+\dots\,,$ (179)

which is the result of the momentum conservation. This, however, suffers the
problem of superluminality, because its group velocity

$v_{g}\equiv\Big{|}\frac{d\omega}{dk}\Big{|}\sim k\,,$ (180)

can be arbitrarily large if there is no cut-off on the momentum $k$. One well-
known solution to this problem 212121It is actually well-known that a
superluminal group velocity does not imply any violation of causality
doi:10.1119/1.15715 . This is a “problem” only for numerical simulation using
linearized relativistic hydrodynamics as initial conditions. There is
absolutely no fundamental nor physical problem here. is to introduce a
fictitious relaxation time $\tau_{\pi}$ which arises from the constitutive
relation of the stress tensor to higher order. As a result, the diffusive
equation gets modified as

$\omega^{2}\,+\,i\,\omega\,\tau_{\pi}^{-1}\,\,=\,v_{T}^{2}\,k^{2}\,,\quad\quad
v_{T}^{2}\,=\,\frac{\eta}{(\mathcal{E}+\mathfrak{p})\,\tau_{\pi}}\,,$ (181)

which is known as the Israel-Stewart formalism ISRAEL1979341 . Solving this
simple equation, we get

$\omega=\frac{1}{2\tau_{\pi}}\left(-i\pm\sqrt{4v_{T}^{2}\,k^{2}\,\tau_{\pi}^{2}-1}\right)\,.$
(182)

This time, the diffusive dynamics sets only when $k<k_{g}\equiv
1/(2v_{T}\tau_{\pi})$. Here, $k_{g}$ is just the k-gap mentioned previously.
Above the $k_{g}$ momentum, momentum propagates ballistically at the sound
speed $v_{T}\leq c$. Identifying $v_{T}\sim v_{lightcone}$ and
$\tau_{\pi}\sim\tau_{eq}$ and expanding (182), one can observe that
$D_{\pi}=v_{T}^{2}\,\tau_{\pi}\sim v_{lightcone}^{2}\,\tau_{eq}$.
Consequently, the causality requirement

$v_{T}^{2}\,\leq\,c^{2}\,,$ (183)

is equivalent to an upper bound on the diffusion constant

$D_{\pi}\leq c^{2}\tau_{\pi}\,,$ (184)

which takes exactly the form of Eq. (177), with $c$ being the relativistic
lightcone velocity and $\tau_{\pi}$ the equilibration time.

Figure 44: The local equilibrium timescale is extracted from the imaginary
part of the first non-hydrodynamic mode as in Eq. (185). The arrow indicates
its motion as the increase of $m/T$.

In holographic systems, the equilibration process is dominated by the first
non-hydrodynamic modes. 222222The hydrodynamic modes are gapless, hence do not
set any finite timescale for relaxations. Then, we should look at the gapped
modes. Here, the lowest lying gapped QNM is what we term the first non-
hydrodynamic mode. Then, we may identify

$\tau_{eq}\sim\frac{1}{\text{Im}\left[\omega_{\text{QNM}}\right]}\,,$ (185)

where $\omega_{\text{QNM}}$ is the complex frequency of the first non-
hydrodynamic QNM. Back to the holographic axion model, the longitudinal sound
speed plays the role of the lightcone velocity, _i.e._ $v_{L}\sim
v_{lightcone}$. Then, one can directly check the upper bound by computing the
black hole QNMs (see how to extract the local equilibrium timescale
$\tau_{eq}$ in Fig. 44). The results for zero density and charged case have
been shown in Fig. 45, respectively. It is obvious that the upper bound on
diffusion from causality (177) is respected for all the cases.

Figure 45: The upper bound on diffusion.$c$ is the speed of light and $v_{L}$
is the speed of longitudinal sound. Top: The dimensionless ratio at zero
density, where $D_{1}$ is the crystal diffusion and $D_{2}$ the charge
diffusion. Bottom: The dimensionless ratio at finite density, where the blue
points are for $\mu/T=3$ and the red for $\mu/T=5$.

Then, above analysis suggests that the diffusion constants may always be
constrained in an intermediate range:

$v_{B}^{2}\,\tau_{pl}\,\leq\,D\,\leq\,v_{ligthcone}^{2}\,\tau_{eq}\,.$ (186)

In Fig. 46 we show a "sandwich-like" illustration for this feature. Note that
for many cases of holographic systems or quantum critical systems, the
equilibrium time reaches the minimum, $\tau_{eq}\sim\tau_{pl}$ and the
butterfly velocity determines the lightcone speed (assuming that there is no
quasiparticle or well-defined microscopic velocity), for which we may have
that $D\sim v_{B}^{2}\tau_{pl}$, _i.e._ the diffusion allowed region shrinks
to a single point. Recently, Arean:2020eus proposed to define the velocity
scales in the diffusive bound as the ratio between the frequency and the
momentum setting the radius of convergence of the linear hydrodynamic theory.

Figure 46: The diffusion allowed region is bounded by quantum mechanics and
causality.

###  V.7 A bound on stiffness

Now, let us move to check if there is also some constraint on the propagating
speed of the longitudinal sound modes. In some previous studies Hohler:2009tv
; Cherman:2009tw , it has been proposed from the holographic duality that
there exists an upper bound on speed of sound modes, which is set by its
conformal value, $v_{c}\equiv 1/(d-1)$ for $d$ dimensional system. This
proposal has been later related to the physics of neutron stars, very compact
objects which display an extremely stiff equation of state Bedaque:2014sqa .
This bound however has been later checked in various holographic models and
its violation has been observed both at finite charge density case or in
presence of multi-trace deformations Hoyos:2016cob ; Anabalon:2017eri .

Note that all these previous discussions were confined to fluid systems with
no long-range order, where the longitudinal sounds are simply pressure waves,
_i.e._

$v_{L}^{2}=\frac{\partial\mathfrak{p}}{\partial\mathcal{E}}\,.$ (187)

Note that the r.h.s. of Eq. (187) also defines the stiffness of the system

$\kappa\equiv\frac{\partial\mathfrak{p}}{\partial\mathcal{E}}\,.$ (188)

This then means that the bound on the sounds is simultaneously a bound on the
stiffness:

$\kappa\leq\kappa_{c},\quad\kappa_{c}\equiv\frac{1}{d-1}\,.$ (189)

In the holographic axion model, the SSB of translations results in a non-zero
shear elasticity $G$ and the dual systems on boundary is a solid (see also Eq.
(111)). The presence of a finite $G$ speeds up the longitudinal sounds, which
becomes Esposito:2017qpj :

$v_{L}^{2}=v_{c}^{2}+v_{T}^{2}\geq v_{c}^{2}\,.$ (190)

The first equivalence is a direct consequence of conformal invariance, _i.e._
$\langle T^{\mu}_{\mu}\rangle=0$. Again, this is a new way to break the
proposed bound on the sound speed. Nevertheless, we will verify that the
result above does not imply a violation of the stiffness bound. That is to say
the bounds conjectured in Hohler:2009tv ; Cherman:2009tw has to be viewed as a
limit to the stiffness instead of a bound on the speed of sound.

Figure 47: The stiffness $\kappa$ for $N\in[3,9]$ (from black to yellow) in
function of $m/T$. Figure taken from Baggiolili2005 .

As was revealed earlier, in presence of the finite strain pressure
$\mathcal{P}$, the total pressure $\langle T^{xx}\rangle$ is no longer equal
to the thermodynamic pressure $\mathfrak{p}$. For the conformal solid, the
traceless stress tensor implies that

$\langle
T^{xx}\rangle=\mathfrak{p}+\mathcal{P}=\frac{1}{d-1}\mathcal{E},\,\,\,\mathcal{P}>0\,.$
(191)

Applying the definition of the stiffness, we obtain

$\kappa\equiv\frac{\partial\mathfrak{p}}{\partial\mathcal{E}}=\frac{1}{d-1}-\frac{\partial\mathcal{P}}{\partial\mathcal{E}}.$
(192)

For the holographic model $V(X)=X^{N}$ with $N>3/2$ and $d=3$, one can easily
show that the second term in the last step is always positive by using

$\displaystyle\mathcal{P}\,=\,\frac{m^{2}\,N\,u_{h}^{2N}}{(2\,N-3)\,u_{h}^{3}}\,,\quad\mathcal{E}\,=\,\frac{1}{u_{h}^{3}}-\frac{m^{2}\,u_{h}^{2N}}{(3-2N)\,u_{h}^{3}}\,.$
(193)

To make it clearer, we plot the final result of (192) in Fig. 47. It confirms
our statement that

$\kappa\leq\kappa_{c}\,,$ (194)

independently of $N$. For $N<3/2$, the phonons are destroyed by the external
source that breaks spatial translations, and there is no low energy modes
propagating at the speed set by the stiffness. Note also that when $N<3/2$,
both of $G$, $\mathcal{P}$ become negative, implying the existence of a
dynamical instability. Hence, we will not view this as a violation of the
stiffness bound.

##  VI Holographic pinned structures

###  VI.1 Pseudo-Goldstone modes

So far, we have focused on two very distinct symmetry patterns: (A) the
explicit breaking of translations, giving rise to the physics of momentum
dissipation and the Drude model, and (B) the spontaneous symmetry breaking of
translations, related to the physics of elasticity and the dynamics of
phonons. In the previous sections, we have explained how to technically
achieve these two limits using the holographic axion model.

Now, we want to make a step forward and combine the two in what is called the
pseudo-spontaneous regime. As we will see, the holographic axion model is rich
enough to encompass also this different situation. From the physical point of
view, this regime is realized in QCD for the Pions Burgess:1998ku , where
chiral symmetry is both broken spontaneously and explicitly, giving a small
mass to the corresponding pseudo-Goldstones and in the so-called pinned charge
density waves RevModPhys.60.1129 , where impurities produce a pinning
frequency for the corresponding phason modes.

Indicating with $\langle EXB\rangle$ the explicit breaking scale, and with
$\langle SSB\rangle$ the spontaneous one, we want to work in the limit of:

$\langle EXB\rangle/T\,\ll\,1\,\,,\quad\frac{\langle EXB\rangle}{\langle
SSB\rangle}\,\ll\,1\,,$ (195)

such that the corresponding charge (momentum in this case) is “approximately”
conserved (corresponding to a weak explicit breaking mechanism) and a pseudo-
Goldstone mode can be still defined. Let us first address the question of how
to realize this limit within holography and later describe in detail which are
the phenomenological consequences. First, let us give an intuitive argument
taken from Alberte:2017cch . What is the substantial difference between the
explicit breaking and the spontaneous one? The explicit breaking appears at
the level of the fundamental action of the theory and it relates to the
presence of an operator whose source breaks the specific symmetry. The
spontaneous breaking does not relate to the action of the system but rather to
its ground state, the preferred solution in which the system wants to sit. In
a sense, the EXB can be thought as an ultraviolet (UV) breaking, while the SSB
as an infrared (IR) breaking. This idea is beautifully encoded in the
holographic picture by considering that the UV dynamics is localized close to
the AdS boundary, meanwhile the IR dynamics nearby the black hole horizon. One
could think, for example, about the famous holographic model for
superconductivity Hartnoll:2008vx where the mass of the gauge field
(determining the breaking of the $U(1)$ symmetry) is localized close to the
horizon and it is zero at the boundary (_i.e._ no explicit breaking).

The idea here is exactly analogous but this time, since we are dealing with
translations, is the mass of the graviton that has to be considered 232323See
Amoretti:2016bxs for a detailed analysis of the Ward’s identities in the
pseudo-spontaneous regime.. In particular, one would naively connect the
explicit breaking of translations with the presence of a finite UV mass, and
the spontaneous breaking with the presence of a growing mass peaked in the IR.
Therefore, in this picture, the pseudo-spontaneous regime should be
accomplished by considering a bulk configuration where the graviton mass in
the UV is small and it rapidly grows in the IR (see Fig. 48).

Figure 48: The intuitive picture relating the spontaneous and explicit
breaking of translations with the profile of the graviton mass in the
holographic bulk. Picture taken from Alberte:2017cch .

In order to have such a situation, one could consider a potential of the form:

$V(X)\,=\,X\,+\,\beta\,X^{N}\,,\quad\beta\gg\,1\,,$ (196)

since the graviton mass is proportional to $m_{g}^{2}(u)\,\sim\,V_{X}(X)$ and
the argument $X\sim u^{2}$ vanishes at the boundary and grows towards the
black hole horizon.

This intuition turns out to be correct and it can be formalized better using
the language of the previous sections. Given a potential $V(X)=X^{N}$, we know
now that the breaking of translations is explicit if $N<5/2$ and spontaneous
if $N>5/2$. Therefore, to reach the pseudo-spontaneous regime, we need to
utilize a potential of the type:

$V(X)\,=\,X^{N}\,+\,\beta\,X^{M}\,,\qquad N<5/2,\,M>5/2\,.$ (197)

This is exactly what has been done in Alberte:2017cch which first 242424The
same day, a work Andrade:2017cnc studying this regime in the context of
holographic helical lattices was posted in arXiv. studied the dynamics of
pinning phonons in holography. Curiously 252525It is also funny to notice that
historically this regime was achieved before the fully spontaneous one
Alberte:2017oqx , which for technical reasons has been the most difficult to
construct., this model was considered long time before, in Baggioli:2014roa ,
but the connection with the pseudo-spontaneous breaking of translations has
not been given therein. A full understanding of this regime has appeared later
in Ammon:2019wci . A slightly different way to realize this regime can also be
found in Li:2018vrz .

Let us analyze in detail what one would expect in this regime. Following the
hydrodynamic description of Delacretaz:2017zxd , in presence of both explicit
and spontaneous breaking of translations, the low-energy hydrodynamic spectrum
at zero momentum is described by the solutions of the equation:

$\left(\Omega\,-\,i\,\omega\right)\,\left(\Gamma\,-\,i\,\omega\right)\,+\omega_{0}^{2}\,=\,0\,,$
(198)

where $\Gamma$ is the momentum relaxation rate (how fast momentum is
dissipated), $\Omega$ the phase relaxation rate (measuring the lifetime of the
Goldstones) and $\omega_{0}$ the so-called pinning frequency – the mass of the
(not anymore) Goldstone modes. The two modes involved in the quadratic
equation (198) are the transverse momentum $\pi^{\perp}$ and the transverse
component of the Goldstone field. Solving Eq. (198) we obtain a pair of
excitations whose frequencies are given by

$\omega_{\pm}\,=\,-\,\frac{i}{2}\,\left(\Gamma+\omega\right)\,\pm\,\frac{1}{2}\sqrt{4\,\omega_{0}^{2}\,-\,\left(\Gamma-\Omega\right)^{2}}\,.$
(199)

For $4\,\omega_{0}^{2}-(\Gamma-\Omega)^{2}<0$, the two modes lie along the
imaginary axes; they are purely decaying modes. Exactly at
$4\,\omega_{0}^{2}-(\Gamma-\Omega)^{2}=0$, the two modes collide on the
imaginary axes and for $4\,\omega_{0}^{2}-(\Gamma-\Omega)^{2}>0$ they move
onto the complex plane acquiring a finite and growing real part. This dynamics
is perfectly obeyed by the holographic models, see, for example, Fig. 49.

Figure 49: The dynamics of the low-energy hydrodynamic modes in the pseudo-
spontaneous regime. The dance of the modes is perfectly described by Eq.
(198). Figure adapted from Ammon:2019wci .

Interestingly, this collision is not an exclusive feature of the pseudo-
spontaneous regime. Indeed, this collision is the same giving rise to the
incoherent-coherent transition in the models with pure explicit breaking
Davison:2014lua . Rather, a peculiar characteristic of the pseudo-spontaneous
regime is the fact that such collision happens at low frequency, within the
so-called hydrodynamic regime. Increasing the amount of spontaneous breaking,
both $\Gamma$ and $\Omega$ become smaller and the collision happens close to
the origin $\omega=0$. This tendency is shown explicitly in Fig. 50. Notice
that in the purely spontaneous regime, both modes would collapse at the origin
and form the propagating shear sound.

Figure 50: The collision between the two modes by moving towards the pseudo-
spontaneous regime. Figure adapted from Ammon:2019wci . Notice how this
dynamics was already present in Baggioli:2014roa .

Finally, one can perform also the computation at finite momentum and verify
the dispersion relation of the various modes. As shown in Fig. 51, the pseudo-
phonon mode shows a characteristic dispersion:

$\mathrm{Re}\left[\omega\right]=\,\sqrt{\omega_{0}^{2}\,+\,v^{2}\,k^{2}}\,,$
(200)

typical of massive modes (cf. Pions).

Figure 51: The dispersion relation of the pseudo-phonon. Figure adapted from
Alberte:2017cch .

Before moving to the next subsection, it is interesting to analyze how the
parameters entering in Eq. (198) depend on the explicit and spontaneous
breaking scales, $\langle EXB\rangle,\langle SSB\rangle$. Combining
hydrodynamic and holographic arguments, we can derive a simple general
formula:

$\displaystyle\Gamma\,+\,\frac{\omega_{0}^{2}}{\Omega}\,=\,m^{2}\,\frac{V_{X}}{2\,\pi\,T}\,+\,\mathcal{O}(m^{4})\,.$
(201)

Assuming the benchmark potential for the pseudo-spontaneous regime
$V(X)=X+\beta X^{N}$ we can also obtain that

$\displaystyle\Gamma\,=\,\frac{m^{2}}{2\,\pi\,T}\,\sim\,\langle
EXB\rangle^{2}+\dots\,,$ (202)
$\displaystyle\frac{\omega_{0}^{2}}{\Omega}\,=\,\frac{m^{2}\,\beta\,N}{2\,\pi\,T}\sim\langle
SSB\rangle^{2}+\dots\,.$ (203)

Moreover, using the numerical data (see Fig. 52), we can prove robustly that

$\omega_{0}^{2}\,=\,\langle EXB\rangle\,\langle SSB\rangle\,,$ (204)

as imposed by the Gell-Mann-Oakes-Renner (GMOR) relation PhysRev.175.2195 (cf.
Pions).

Figure 52: The numerical confirmation of the GMOR relation (204). Figure
adapted from Ammon:2019wci .

Notice that combining (203) together with the GMOR relation (204) we find
immediately that

$\Omega\,\sim\,\frac{\langle EXB\rangle}{\langle SSB\rangle}\,\,!$ (205)

This is surprising for various reasons and it will be the topic of the next
subsection.

###  VI.2 Phase relaxation and universality

The fact that the phase relaxation rate is proportional to the explicit
breaking scale is per se very surprising. In general, phase relaxation is
induced by the presence of elastic defects such as dislocations and
disclinations and it has nothing to do with the explicit breaking of momentum.
It comes from the fact that the displacement vectors are not anymore single-
valued and a non trivial Burgers vector appears.

First, let us notice how we have already encountered part of the scalings in
Eq. (205) in the previous section, when we were discussing the collision of
the modes. As also shown in Fig. 49, the collision moves towards the origin
increasing the strength of the SSB and one of the reason is exactly that the
second pole, approximately located at $\omega=-i\Omega$, moves upwards. This
is in perfect agreement with the fact that $\Omega$ is inversely proportional
to such strength and it becomes smaller by increasing the amount of SSB.
Second, it is important to emphasize that the relation (205) has been
confirmed numerically in a large class of holographic axion models
Ammon:2019wci ; Baggioli:2019abx (and in similar models Donos:2019txg ;
Amoretti:2018tzw ; Amoretti:2019cef ; Andrade:2018gqk ; Andrade:2020hpu ).
This confirms the universal character of this scaling.

The story is even more fascinating, since the authors of Amoretti:2018tzw
motivated and proposed an even more universal relation:

$\Omega\,=\,\mathrm{M}^{2}\,\xi\,=\,\frac{\omega_{0}^{2}\,\chi_{\pi\pi}}{G}\,\xi\,,$
(206)

where $\mathrm{M}$ is the mass of the pseudo-Goldstone mode and $\xi$ the
dissipative parameter determining the diffusion constant of the Goldstone mode
in the un-relaxed theory. Using the fact that at leading order the Goldstone
diffusion is given by $D_{\phi}=G\,\xi$ and that $v_{T}^{2}=G/\chi_{\pi\pi}$,
we can re-write Eq (206) as:

$\Omega\,=\,\frac{\omega_{0}^{2}\,D_{\phi}}{v_{T}^{2}}\,.$ (207)

Notice how Eq. (206) is compatible with the scalings discussed and derived in
the previous paragraphs:

$\underbrace{\frac{\langle EXB\rangle}{\langle
SSB\rangle}}_{\Omega}\,\sim\,\underbrace{\langle EXB\rangle\langle
SSB\rangle}_{\omega_{0}^{2}}\,\underbrace{\frac{1}{\langle
SSB\rangle^{2}}}_{\xi/G}\,.$ (208)

Figure 53: Numerical verification of the universal relation (206). Figure
taken from Ammon:2019wci .

In the context of axions models, the universal relation (206) has been
confirmed numerically for various choices of the potential $V$ in
Ammon:2019wci ; Baggioli:2019abx (see Fig. 53).

Importantly, the dissipative coefficient $\xi$ can be derived analytically in
terms of horizon data and it generally reads:

$\frac{\xi}{G}\,=\,\frac{4\,\pi\,s\,T^{2}}{2\,m^{2}\,\chi_{\pi\pi}^{2}\,V_{X}}\,.$
(209)

Combining this result, with the previous equations (201) and (203), one can
immediately prove that for the benchmark model $V(X)=X+\beta X^{N}$:

$\frac{\Omega\,G}{\omega_{0}^{2}\,\xi\,\chi_{\pi\pi}}\,=\,\frac{1\,+\,N\,\beta}{N\,\beta}\,\underbrace{\rightarrow}_{\beta\rightarrow\infty}\,1\,,$
(210)

where the pseudo-spontaneous limit $\beta\rightarrow\infty$ has been taken. In
summary, the universal relation (206) can be proven explicitly, in agreement
with the numerical data (see Fig. 53).

Finally, the universal relation (206) has been formally derived in the context
of dissipative EFT using Keldysh-Scwhinger techniques in Baggioli:2020haa (see
also Baggioli:2020nay ). It turns out that this universal relation arises
directly from the intertwined symmetry breaking pattern between spacetime
translations and internal shifts. From the holographic point of view, this
intertwined dynamics has been also discussed in similar models in
Amoretti:2019kuf ; Donos:2019hpp .

###  VI.3 Optical conductivity and pinning

Very importantly, in presence of a finite charge density, the physics just
described couples to the dynamics of the electric current and induces
important effects in the electric conductivity. In particular, the low
frequency behaviour of the electric conductivity is given by

$\sigma(\omega)\,=\,\sigma_{0}\,+\,\frac{\frac{\rho^{2}}{\chi_{\pi\pi}}\,(\Omega-i\omega)-\omega_{0}^{2}\,\gamma\,[2\,\rho+\gamma\chi_{\pi\pi}(\Gamma-i\omega)]}{(\Gamma-i\omega)\,(\Omega-i\omega)\,+\,\omega_{0}^{2}}\,,$
(211)

where $\sigma_{0}$ is the incoherent conductivity Davison:2015taa (the one
coming from the incoherent current and not sensitive to the dynamics of
momentum), $\rho$ the charge density and $\gamma$ a dissipative parameter
coming from the coupling between the electrical current and the Goldstone
mode. Beyond the complicated expression for the optical conductivity, the most
important feature arising from Eq. (211) is the spectral weight transfer from
zero frequency to an intermediate IR frequency which is governed by
$\omega_{0}$, the mass of the Goldstone mode. More specifically, the real part
of the optical conductivity displays a peak at a certain real frequency
$\omega^{*}=\sqrt{4\omega_{0}^{2}-(\Gamma-\Omega)^{2}}$. In other words, the
Drude peak moves towards higher frequency and, because of the sum-rule, the DC
component of the conductivity decreases at the same time.

Figure 54: Top:  The shift of the Drude peak to finite intermediate
frequencies as a consequence of the pseudo-spontaneous dynamics. Bottom:  The
temperature dependence of the conductivity peak. Figures taken from
Ammon:2019wci .

This dynamics is shown in the top panel of Fig. 54. The position of this peak
plays an important role in the proposal that bad metals can be understood as
strongly coupled material with fluctuating charge density waves relics. More
precisely, in Delacretaz:2016ivq , it has been proved that in order for this
theory to describe the optical properties of strange metals the peak must move
to larger frequencies increasing the temperature. Not only that, but it has
been proposed that the energy scale controlling such peak is the same as the
one controlling the linear in $T$ resistivity of those materials, the
Planckian time $\tau=\hbar/k_{B}T$. Accordingly to their analysis of the
experimental data, the position of the peak in the optical conductivity should
increase linearly with the temperature.

Unfortunately, this feature is not recovered in the holographic axion model
Ammon:2019wci . It was found that the optical conductivity follows a more
standard and natural transition towards an insulating state. Increasing the
temperature, the system tends to become more metallic, the DC conductivity
wants to grow and the optical conductivity wants to return to its original
Drude shape (see the bottom panel of Fig. 54). This drawback can be cured by a
fine-tuned generalization, involving a dilatonic coupling, and displaying this
behaviour in a very small range of temperatures (see the inset of Fig. 3 in
Amoretti:2018tzw ).

##  VII Phenomenology

###  VII.1 Metal-Insulator transitions

The metal-insulator transition (MIT) is one of the oldest as well as the
fundamentally least understood problems in condensed matter. Although many
theories have been proposed, the mechanisms toward the metal-insulator
transition remain controversial and somewhat incomplete (see Imada:1998zz ;
2011arXiv1112.6166D for reviews). It is obvious that a good metal and a good
insulator are very different physical systems, characterized by quite
different elementary excitations. In particular, in the intermediate regime of
the transition, different types of excitations coexist and simple theoretical
tools prove of little help. Since the discovery of high temperature
superconductivity, the study of metal-insulator transition came to the strong
correlation era, for which physical pictures based on weak-coupling approaches
prove insufficient or even misleading.

Holography provides a new approach to tackle states of quantum matter without
quasiparticle excitations, for which the transport properties deviate strongly
from conventional approach described, in particular, by Fermi liquid theory.
From an EFT point of view, the holographic axion model concerns broken
translation invariance and implements non-perturbative renormalization group
flows. To realise a holographic metal-insulator transition, one should first
overcome the obstruction found in Grozdanov:2015qia where it was proven that
in some simple holographic theories with arbitrary spatial inhomogeneity
(disorder) the electrical conductivity is bounded from below by a universal
minimal conductance. Therefore, one can not obtain an insulating phase where
the electric DC conductivity at zero temperature is very small or eventually
zero. It was then found in Baggioli:2016oqk that it is possible to introduce
additional couplings between the charge and translation breaking sectors
allowed by the symmetries (see also Gouteraux:2016wxj ; Baggioli:2016pia ).
Then a clear disorder-driven metal-insulator transition was observed
Baggioli:2016oqk .

The minimal holographic model of a metal-insulator transition is described by
(47) with $Z=0$

$\mathbf{\mathcal{S}}=\int
d^{4}x\sqrt{-g}\left[\frac{R}{2}-\Lambda-\frac{Y(X)}{4e^{2}}F_{\mu\nu}F^{\mu\nu}-m^{2}V(X)\right]\,.$
(212)

The consistency of a theory imposes some constraints on the couplings that
appear in the Lagrangian. For the theory (212), it was shown that $V(X)$ and
$Y(X)$ should satisfy the following constraints Baggioli:2016oqk :

$\displaystyle V^{\prime}(X)>0,\quad Y(X)>0,\quad Y^{\prime}(X)<0\,,$ (213)

in which $Y^{\prime}(X)<0$ plays a key role in triggering a metal-insulator
transition.262626See _e.g._ Donos:2012js ; Mefford:2014gia ; Rangamani:2015hka
; Baggioli:2016oju ; Ling:2014saa ; Cremonini:2017qwq ; Andrade:2017cnc for
other holographic realizations of metal-insulator transitions driven via other
mechanisms. More recently, a much stringent constraint was found for $Y$ in
An:2020tkn . Without loss of generality, one can parametrize the couplings $Y$
and $V$ in the following expansion as $X\rightarrow 0$ (weak momentum
dissipation):

$Y(X)=1-\gamma\,X+\mathcal{O}(X^{2}),\quad
V(X)=\frac{1}{2m^{2}}X+\mathcal{O}(X^{2})\,,$ (214)

with $\gamma$ a constant. By requiring a positive definite longitudinal
conductivity in the presence of charge density and magnetic field restricts
the allowed parameter space of theory parameters.

$0\leqslant\gamma\leqslant 1/6\quad\Rightarrow-1/6\leqslant
Y^{\prime}(0)\leqslant 0\,.$ (215)

The working and phenomenological definition of a metal versus an insulator
behavior is given by

$\text{metal:}\quad\frac{d\,R_{xx}}{d\,T}>0,\quad\text{insulator:}\quad\frac{d\,R_{xx}}{d\,T}<0\,,$
(216)

where $R_{xx}$ is the longitudinal DC resistivity. Some generic features
without being concerned with details of the holographic theory was uncovered
in An:2020tkn . In particular, the temperature dependence of resistivity is
found to be well scaled with a single parameter $T_{0}$, which approaches zero
at some critical charge density $\rho_{c}$, and increases as a power law
$T_{0}\sim|\rho-\rho_{c}|^{1/2}$ both in metallic $(\rho>\rho_{c})$ and
insulating $(\rho<\rho_{c})$ regions in the vicinity of the transition.
Similar features also happen by changing the disorder strength $\alpha$ as
well as magnetic field (see Fig. 55). It was also found that the metallic and
insulating curves are mirror symmetry in the high temperature regime:

$R_{xx}(\rho-\rho_{c},T)=1/R_{xx}(\rho_{c}-\rho,T)\,.$ (217)

These results suggest that the mechanism responsible for the temperature
dependence of conductivity on both insulating and metallic sides of the
transition would be the same, or it would originate with some fundamental
feature that is common to both.

Figure 55: Temperature dependence of the resistivity versus disorder strength
at zero magnetic field for the model with
$Y=1-\frac{X}{6},V=\frac{X}{2m^{2}}$. Top: The metal-insulator transition
driven by the disorder strength $\alpha$. Bottom: Scaling of resistivity with
scaled temperature $T/T_{0}$. The collapse of data into two separated curves
both in the metallic and insulating sides is manifest. Other parameters are
chosen by $e=\rho=1$. Figure taken from An:2020tkn .

The holographic results are reminiscent of the scaling behaviors for
resistivity near the transition point reported in some two dimensional samples
and materials, which shows the collapse of data into two separated curves and
displays remarkable mirror symmetry over a broad interval of temperatures
1995PhRvB..51.7038K ; 1996PhRvL..77.4938K ; 1997PhRvB..5612764C ;
1998PhRvL..80.1292S . This observation has been interpreted as evidence that
the transition region is dominated by strong coupling effects characterizing
the insulating phase 1997PhRvL..79..455D . Nevertheless, the dependence of
$T_{0}$ near $\rho_{c}$ is a power law with the power that is different from
holographic result $1/2$, suggesting that holographic theory (212) falls into
a different universality class from those materials. A natural extension is to
consider a holographic setup that is asymptotically Lifshitz with a dynamical
exponent $z$ which parametrizes the relative scaling of space and time, making
the model compatible with experimental data.

The phase diagram gets richer and incorporates several phases of matter
depending on the parameters. As shown in Fig. 56, there are as many as four
different phases in the temperature-disorder phase diagram: good metal (a),
bad or incoherent metal (b), bad insulator (c) and good insulator (d). Some
representative examples of the AC electric conductivity for each phase are
shown in Fig. 57. A coherent metallic phase with a sharp Drude peak is
obtained for small $\alpha$. As $\alpha$ increases, the Drude peak is
suppressed and one arrives at an incoherent metallic phase where there is no
clear and dominant localized long lived excitation, see green curves of Fig.
57. When the disorder is strong enough, the spectral weight transfers to the
mid-infrared, resulting in an insulating behavior. As a consequence, the DC
conductivity keeps decreasing, and then triggers the transition from bad
insulator to good insulator. Therefore, there is a clear disorder-driven
transition from a coherent metal with a sharp Drude peak to a good insulator
with a tiny or vanishing DC conductivity at zero temperature.

Figure 56: Phases diagram for the model with
$Y=1-\frac{X}{6},V=\frac{X}{2m^{2}}$ in the absence of magnetic field. Four
regions are denoted by (a) good metal, (b) incoherent metal, (c) bad insulator
and (d) good insulator, respectively. Other parameters are chosen by
$e=\rho=1$. The figure is updated from An:2020tkn .

Figure 57: Representative examples of the AC electric conductivity
$\sigma_{xx}$ with unitary charge density $\rho=1$. There are four phases: (a)
good metal (red) with $(\alpha=0.6,T=0.5)$, (b) incoherent metal (green) with
$(\alpha=1.5,T=0.5)$, (c) bad insulator (blue) with $(\alpha=4.5,T=0.3)$ and
(d) good insulator (purple) with $(\alpha=7.8,T=0.05)$. Figures taken from
An:2020tkn .

The investigation of transport in this minimal holographic setup of a metal-
insulator transition uncovered some interesting features, shedding light on
this interesting transition and the physical mechanism that drives it. There
are still many interesting questions. There are as many as four different
phases in the temperature-disorder phase diagram of Fig. 56, but all phases
share the same symmetries of the underlying theory, and thus beyond a simple
Ginzburg-Landau description. Some observables were examined in order to
characterise different phases, but failed An:2020tkn . It is still an open
question to find a good probe to the metal-insulator transitions. The scaling
exponent of $T_{0}$ from the holographic setup is different from the
experiments which yielded scaling exponents between 1.25 and 1.6
1995PhRvB..51.7038K ; 1996PhRvL..77.4938K ; 1997PhRvB..5612764C ;
1998PhRvL..80.1292S . It is interesting to generalise the model to obtain an
exponent that is compatible with experimental observation. It is also worth
studying the thermal response and the mechanical response.

###  VII.2 The scalings of strange metals

The anomalous metallic transport in the high-temperature superconducting
cuprates is one of the most remarkable puzzles in condensed matter physics.
The so-called strange metal phase is characterized by universal temperature
scalings which are robust across widely different systems, and are believed to
be controlled by an underlying strongly interacting quantum critical sector.
Its anomalous features include a linear temperature dependence for the
resistivity $R_{xx}\sim T$ and the scaling of the Hall angle
$\cot(\Theta_{H})\sim T^{2}$ (see Fig. 58). Realizing the anomalous
temperature dependence of both $R_{xx}$ and $\cot(\Theta_{H})$ at once within
a holographic model has proven to be a challenge. Although many efforts were
made, standard Einstein-Maxwell-dilaton (EMD) theories have thus far been
unable to give the expected anomalous scalings of the strange metal (see
_e.g._ Blake:2014yla ; Amoretti:2015gna ; Amoretti:2016cad ). As we are
dealing with strongly correlated electron matter, it may be crucial to take
into account the nontrivial dynamics between the charge degrees of freedom to
reliably capture transport in these phases Hartnoll:2009ns ; Kim:2010zq ;
Karch:2014mba .

Figure 58: Experimental observation of the cuprates scalings for the in-plane
resistivity $R_{xx}\sim T$ and inverse Hall angle $\cot(\Theta_{H})\sim
T^{2}$. Figures taken from 1997PhyC..282.1185T .

The first consistent holographic realization of the strange metal scalings of
the resistivity and Hall angle was given in Blauvelt:2017koq by working with a
string-theory-motivated gravitational model encoded by the Dirac-Born-Infeld
(DBI) action.272727Note that the strangle metal scalings have also been
realized in the Einstein-Maxwell-dilaton-axion model with a hyperscaling IR
geometry only for a special value of $B$ Zhou:2015dha . The motivation is to
describe a strongly coupled quantum theory containing a sector of dilute
charge carriers that interact amongst themselves as well as with a quantum
critical bath. The charge degrees of freedom is treated as a probe when
compared to the larger set of neutral quantum critical degrees of freedom. The
gravitational theory takes the generic form

$\mathcal{S}=\int
d^{4}x\sqrt{-g}[\mathcal{L}_{\text{bath}}+\mathcal{L}_{\text{charge}}]\,.$
(218)

The bath sector $\mathcal{L}_{\text{bath}}$ is supported, for example, by a
neutral scalar and axionic scalars. The charge sector
$\mathcal{L}_{\text{charge}}$ describes the dynamics of a $U(1)$ gauge field,
taking into account non-linear interactions between the charged degrees of
freedom. In Blauvelt:2017koq the bath geometry is nonrelativistic and
hyperscaling-violating supported by a neutral scalar field and two axions
$\phi^{I}$. In the so-called probe limit for which the backreaction of the DBI
interactions on the geometry can be safely neglected, the nonlinear dynamics
of the gauge field sector allows a clean scaling regime for the cuprate
strange metals:

$R_{xx}\sim T,\quad\cot(\Theta_{H})=\frac{R_{xx}}{R_{yx}}\sim T^{2}\,,$ (219)

with $R_{yx}$ the Hall resisitivity.

Because of its richness, the DBI theory can support a wide spectrum of
temperature scalings. Therefore, one can use similar construction to realize
other scaling behaviors observed in strange metals. Recently, novel strange
metal behavior was observed in the pnictides 2016NatPh..12..916H , for which
the magnetic field $B$ plays the same role as the temperature $T$ (see also
the observation of linear-in-field resistivity in cuprates 2018Sci…361..479G
). The measurements imply that the in-plane resistivity behaves as
2016NatPh..12..916H

$R_{xx}=\sqrt{\gamma\,T^{2}+\eta\,B^{2}}\,,$ (220)

with $\gamma$ and $\eta$ two constants. This striking behavior was realized
from holography in Kiritsis:2016cpm . The holographic theory also predicts a
Hall resistivity in the same temperature regime that is linear in the magnetic
field and approximately temperature independent. This idea has been
generalised to include a generic nonlinear gauge field sector in
Cremonini:2018kla . A particularly simple nonlinear model whose structure is
natural from the point of view of the DBI action was found to be able to
realise the temperature scalings of the entropy $\sim T$, resistivity $\sim
T$, Hall angle $\sim T^{2}$ and weak-field magneto-resistance $\sim T^{-4}$
observed in cuprates.

The underlying mechanism for above holographic construction relies on having a
quantum critical IR fixed point and on the nonlinear structure of the
interactions between the charges. It would be desirable to understand the
underlying dynamics and to match the holographic description with the expected
interactions of electrons in real materials, building intuition for the
mechanisms underlying the unconventional behavior of strange metals. The first
step towards connecting to phases and critical points of Hubbard model
Sachdev:2010uz was addressed Sachdev:2010um . The realisation of strange metal
scalings using holography beyond the probe approximation is still an open
question. It is fair to say that a consensus about the strange metals
phenomenology is far from being reached both from the holographic point of
view khveshchenko2020die and the condensed matter one singh2020leading .

###  VII.3 Superconductivity

The holographic superconductors model Hartnoll:2008vx ; Hartnoll:2008kx is one
of the first and most popular frameworks in Applied Holography (see
Cai:2015cya for a recent review). Its simplicity comes nevertheless with a
price. In particular, because of the translational invariance of the
background, both the normal state and the broken SC phase exhibit an infinite
conductivity:

$\sigma(\omega)\,=\,\sigma_{0}\,+\,\frac{i}{\omega}\,\left(\frac{\rho_{s}}{\mu}\,+\,\frac{\rho_{n}^{2}}{\mu\rho_{n}+sT}\right)\,,$
(221)

with $\rho_{n}$ and $\rho_{s}$ respectively the normal and superfluid
densities. The first infinity comes from the superfluid flow while the second
from translational invariance. This drawback makes it impossible to
distinguish the two phases at the level of electric transport without a very
careful analysis. To avoid this problem, the power of the holographic axion
model was originally used to dissipate momentum and have a normal state with
finite DC conductivity. The holographic superconductors model was endowed with
the axion fields in Andrade:2014xca ; Kim:2015dna ; Baggioli:2015zoa and
further explored in Baggioli:2015dwa ; Kim:2016hzi ; Ling:2016lis ;
Gouteraux:2019kuy .

Once momentum dissipation is introduced, via the axion scalars, the expression
(221) gets modified as:

$\sigma(\omega)\,=\,\sigma_{0}\,+\,\frac{i}{\omega}\,\frac{\rho_{s}}{\mu}\,+\,\frac{\rho_{n}^{2}}{\mu\rho_{n}+sT}\,\frac{i}{\omega\,+\,i\,\Gamma}\,,$
(222)

where $\Gamma$ is the usual momentum relaxation rate. In the normal phase,
$\rho_{s}=0$, the DC conductivity is finite and it becomes infinite only for
$T<T_{c}$. The corresponding AC finite frequency conductivity is shown in Fig.
59. Above the critical temperature, there is no $1/\omega$ pole in the
imaginary part of conductivity thanks to a finite $\alpha$. Below the critical
temperature, a simple pole (corresponding to the second-sound in
superconductors) appears in the imaginary part of the conductivity, yielding -
through Kramers-Kroenig relations - the appearance of a delta function at zero
frequency in the real part. In other words, the DC conductivity goes to
infinity, as we would expect for a superconductor.

Figure 59:  $\alpha/\mu=1$, $T/T_{c}=3.2,1,0.89,0.66,0.27$ (dotted, red,
orange, green, blue). Figures adapted from Kim:2015dna

Interestingly, the presence of momentum dissipation has two important effects
(see Fig. 60): (I) it shrinks the area in the phase diagram where the
superconducting instability can appear, (II) it decreases the value of the
superconducting condensate.

Figure 60: The effects of momentum dissipation on the phase diagram of the
holographic superconductors model and on the SC condensate. $\Delta$ is the
conformal dimension of the complex bulk scalar and $q$ its charge. Figures
adapted from Kim:2015dna .

There is an interesting open problem regarding holographic superconductors. In
high-temperature superconductors and some conventional superconductors, there
is a universal property called Homes’ law Homes_2004 . It relates three
quantities: the superfluid density at zero temperature ($\rho_{s}$), the phase
transition temperature ($T_{c}$), and the DC conductivity in the normal phase
close to $T_{c}$ ($\sigma_{\mathrm{DC}}$) with a material independent
universal number ($C$). i.e.

$\rho_{s}(T=0)=C\sigma_{\mathrm{DC}}(T_{c})\,T_{c}\,.$ (223)

Here, the DC conductivity is involved, so momentum relaxation is necessary to
study the Homes’ law and the axion model is obviously the simplest to
consider. In the context of the axion model, the universality of $C$ in (223)
means $C$ is independent of $\alpha$, the strength of momentum relaxation,
which can be interpreted as a parameter effectively specifying the microscopic
material properties. However, it turns out the Homes’ law does not work in the
holographic superconductor-axion model Kim:2016hzi . The Homes’ law have been
studied also in other models, _e.g._ helical lattices Erdmenger:2015qqa and
Q-lattices Kim:2016jjk . In these cases, the Homes’ law hold for a window of
momentum relaxation parameters, but there is still not a good understanding of
this mechanism from the holographic (geometric) viewpoint. To remedy this
situation, it would be advantageous to have a holographic model with robust
linear-$T$-resistivity up to high temperature Jeong:2018tua ; Ahn:2019lrh
because in this case the factor $T_{c}$ in in (223) would cancel. It seems
that the strong momentum relaxation is also a necessary ingredient Homeswip .

Finally, let us emphasize that all the homogeneous models do not display any
commensurability effects Andrade:2015iyf . In fact, because of their
homogeneous nature, and contrarily to standard periodic lattices, they do not
select any preferred wave-vector. In order to introduce such effects in the
holographic framework, one should consider more complicated and fully
inhomogeneous models Andrade:2017leb which go far beyond the scope of this
review.

###  VII.4 Conductivities at finite magnetic field

As explained in subsection VII.2, conductivities at finite magnetic field such
as the Hall conductivity, the Nernst effect, and the Hall angle, play
important roles in understanding strongly correlated electron systems such as
cuprates. Indeed, transport in strongly correlated material has been one of
the leading themes of the early AdS/CMT era Hartnoll:2007ai ; Hartnoll:2007ih
; Hartnoll:2007ip . Here, a constant magnetic field $B$ is introduced by the
following background gauge potential

$A_{\mu}\mathrm{d}x^{\mu}=\frac{B}{2}(x\,\mathrm{d}y-y\,\mathrm{d}x)\,.$ (224)

However, in these pioneering works, momentum relaxation was lacking or not
treated in a full manner. To remedy it, the axion model was employed
Amoretti:2015gna ; Blake:2015ina ; Lucas:2015pxa ; Kim:2015wba , where the
electric, thermoelectric, and thermal conductivity at finite magnetic field
have been computed. For a general class of Einstein-Maxwell-Dilaton-Axion
theories all DC conductivities were expressed in terms of the black hole
horizon data. In particular, for the dyonic black hole modified by axions, the
background solution was analytically obtained and the AC electric,
thermoelectric, and thermal conductivity were numerically computed.

For the dyonic black hole, the Hall angle (219) is computed as BlakeHall ;
Kim:2015wba

$\cot({\Theta_{H}})=\frac{\alpha^{2}}{u_{h}\mu
B}\frac{u_{h}^{2}B^{2}+(\mu^{2}+\alpha^{2})}{u_{h}^{2}B^{2}+(\mu^{2}+2\alpha^{2})}\,,$
(225)

where $\mu$ is chemical potential, $u_{h}$ is horizon location, and $\alpha$
is the momentum relaxation parameter. Because $u_{h}$ is a complicated
function of $T,B,\mu,\alpha$ it is not easy to figure out the $T$ dependence
of the Hall angle. By numerical analysis it was found that the $T$ dependence
of the Hall angle ranges between $T^{0}$ and $T^{1}$. In the large $T$ regime,
$u_{H}\sim 1/T$ so the Hall angle always scales as $T$. Therefore, this
standard but simple model does not exhibit the characteristic Hall angle
behavior $\sim T^{2}$. For an improved version displaying a $T^{2}$-Hall
angle, see (218) and discussions therein.

There is another important phenomenon to consider, where both the magnetic
field and momentum relaxation are essential. It is the Nernst effect. In the
presence of a magnetic field, a transverse (say, $x$ direction) electric field
can be generated by a longitudinal (say, $y$ direction) or transverse thermal
gradient. The former is the Nernst effect and the latter is the Seebeck
effect. The Nernst effect is characterized by the the Nernst signal $e_{N}$

$e_{N}=-{(\sigma^{-1}\cdot\mathscr{A})_{y}}^{x}\,,$ (226)

where $\sigma$ is electric conductivity and $\mathscr{A}$ is the
thermoelectric conductivity ($2\times 2$) matrix. Note that it is zero if
there is no momentum relaxation ($\alpha=0$), because the electric
conductivity becomes infinite. Thus, a finite $\alpha$ is essential for the
holographic model of the Nernst effect. For the dyonic black hole, the Nernst
signal yields Kim:2015wba

$e_{N}=\frac{4\pi\alpha^{2}B}{u_{h}^{2}\mu^{2}B^{2}+(\mu^{2}+\alpha^{2})^{2}}\,.$
(227)

We want to see the $B$ dependence of the Nernst signal $e_{N}$ since it
displays a different behaviour in cuprates with respect to conventional
metals. For example, in the normal state of a cuprate it is bell-shaped as a
function of $B$, while in conventional metals it is linear in $B$ Wang_2006 .
Because $u_{h}$ in (227) is a complicated function of $T,B,\mu,\alpha$ it is
not easy to figure out the $B$ dependence of the Nernst signal. Thus, we make
a plot of $e_{N}$ as a function of $B$ at a fixed $\alpha$ in Fig. 61.

Figure 61: Nernst signal for the dyonic black hole. $\alpha/T=0.5,1,4$(red,
green, blue). For a large $\alpha$ it is linear to magnetic field
(conventional-metal-like) and for a small $\alpha$ it is bell-shaped (cuprate-
like). Figure taken from Kim:2015wba .

Interestingly, by looking at the Nerst signal, this model shows the transition
from a conventional metal (blue line) to a cuprate-like state (green and red)
as $\alpha$ decreases.

In general, the AC conductivities with non vanishing $B$ display a peak at the
finite $\omega$. This pick is related to a pole of the conductivity, in
complex $\omega$ plane, dubbed the cyclotron resonance pole Hartnoll:2007ih ;
Hartnoll:2007ip .

$\omega_{*}\equiv\omega_{c}-i\gamma\,,$ (228)

where the “cyclotron frequency” $\omega_{c}$ is the relativistic hydrodynamic
analog of the free particle case, $\omega_{f}=eB/mc$, even though here it
should be understood as coming from a collective fluid motion. A damping
$\gamma$ could be due to interactions between the positively charged current
and the negatively charged current of the fluid, which are counter-
circulating. Momentum relaxation $\alpha$ shifts both $\omega_{c}$ and
$\gamma$. For small $B$ these shifts scale as $\sim\alpha^{2}B$ and
$\sim\alpha^{2}$ Kim:2015wba . It is natural that $\alpha$ increases the
damping effect. We refer to Kim:2015wba for more detailed analysis of AC
conductivities and the effect of $\alpha$ on them.

###  VII.5 Magnetophonons

In presence of an external magnetic field $B$, together with the SSB of
translations, the dynamics of the low-energy Goldstone modes become richer. In
particular, the transverse and longitudinal phonon modes mix together and give
rise to the so-called magnetophonons and magnetoplasmons Chen2005QuantumSO ;
PhysRevB.18.6245 ; PhysRevB.46.3920 , with dispersion relations:

$\begin{split}&\mathrm{Re}\,[\omega_{-}]\,=\,\frac{v_{\perp}\,v_{\parallel}}{\omega_{c}}\,k^{2}\,+\,\dots\,,\\\
&\mathrm{Re}\,[\omega_{+}]\,=\,\omega_{c}\,+\,\frac{(v_{\parallel}^{2}+v_{\perp}^{2})}{2\,\omega_{c}}\,k^{2}\,+\,\dots\,,\end{split}$
(229)

with $\omega_{c}$ being the cyclotron frequency. The fundamental reason is
that the Poincaré algebra is now modified into:

$\left[P_{i}\,,\,P_{j}\right]\,=\,-\,i\,\epsilon_{ij}\,B\,\mathcal{Q}\,,$
(230)

where $\mathcal{Q}$ is the electric charge operator. This implies that
translations do not commute anymore with each other and the effective low
energy description for the Goldstone fluctuations $\pi^{i}$ associated with
translations can contain a new term:

$\mathcal{L}\,=\,\epsilon^{ij}\,\pi_{i}\,\partial_{t}\pi_{j}\,+\,\dots\,.$
(231)

In accordance with the Watanabe-Brauner formalism Watanabe:2014fva , the
system will display the presence of a type-B Goldstone mode – the
magnetophonon.

This mechanism was successfully verified within the holographic axion model in
Baggioli:2020edn . See Fig. 62 for the dispersion relations of the modes just
mentioned. Interestingly, it was observed that the imaginary part of the
magnetophonon is compatible with a quadratic diffusive behavior which is not
envisaged from EFT methods Hayata:2014yga . Actually, field theory approaches
suggest a $\sim k^{4}$ behavior of imaginary part for quadratic type-B
Goldstone modes manifesting the quasiparticle nature of excitation. In
contrast, a quasiparticle excitation in the holographic axion model
Baggioli:2020edn is not guaranteed. This is a good example to show that
holography is able to describe strongly coupled quantum matter without
quasiparticle excitations. It was verified explicitly that the number of
type-B phonons and the number of gapped partners sum up to the number of
broken generators. For the holographic model the broken generators are the two
momenta. One has two linear phonon modes at zero magnetic field. At finite
magnetic field, there are one massless type-B magnetophonon and its gapped
partner – the magnetoplasmon.

Figure 62: The dispersion relations of the magnetophonon and magnetoplasmon in
the holographic axion model with an external magnetic field. The arrow
indicates the direction of growth of $B$. Figures adapted from
Baggioli:2020edn .

The situation becomes even more interesting when a small amount of disorder –
EXB of translations – is introduced in the system. In this case, the
magnetophonon gets pinned producing a characteristic peak feature in the
optical transport. Despite several theoretical frameworks PhysRevB.59.2120 ;
PhysRevB.62.7553 ; PhysRevB.65.035312 ; Delacretaz:2019wzh , a concrete
understanding of this phenomenon and in particular of the $B$ dependence of
the pinning frequency is still lacking. The dependence of the magnetophonon
peak $\omega_{pk}$ as a function of magnetic field is easy to measure
accurately and can give useful information on the feature of disorder in the
material. In more detail, the dependence is sensitive to whether the material
is in a classical or quantum regime. In the classical regime the classical
treatment of the pinning mechanism predicts $\omega_{pk}\sim 1/B$, while in
the quantum regime, the results can be quite different and the peak can
increase with the magnetic field.

In the holographic axion model, it was found Baggioli:2020edn that the pinning
frequency increases with the magnetic field $B$ (in contrast to what discussed
in Delacretaz:2019wzh ) and at large magnetic field it scales like $\sim
B^{1/2}$ (see Fig. 63). Interestingly, this scaling is consistent with the
experimental measurement in certain two-dimensional materials
Chen2005QuantumSO . This does not imply that the holographic model describes
any specific material, but rather that the scaling found from holography is
consistent with realistic observation, while at odds with the discussion in
Delacretaz:2019wzh . It would be interesting to understand what these results
tell us about the nature of the “disorder” implemented by the holographic
axion model.

Figure 63: The optical conductivity moving the magnetic field $B$ and the
scaling of the peak. The position of the peak increases monotonically with the
magnetic field. Figures adapted from Baggioli:2020edn .

###  VII.6 Non-linear elasticity and rheology

A basic aspect of matter is to understand and characterize the response of
materials under mechanical deformations. So far, we have considered the
elastic properties of the holographic axion model only in the linear regime,
where the external strain is small, and the stress-strain relation can be
linearized as:

$\sigma^{ij}\,=\,C^{ijkl}\,\epsilon_{kl}\,.$ (232)

More in general, one could consider an arbitrarily large external strain, such
that the stress-strain relation becomes highly non-linear282828Here, $\sigma$
has not to be confused with the electric conductivity.:

$\sigma(\epsilon)\,.$ (233)

This scenario indicates the onset of non-linear elasticity, in which the
higher order corrections:

$\sigma\sim\epsilon\,+\,\underbrace{\epsilon^{2}+\epsilon^{3}+\epsilon^{4}+\dots}_{\text{higher-
order}}\,,$ (234)

cannot be neglected anymore.

The non-linear elastic features of the various solids can vary a lot and they
can be very useful to characterize them. For example, metals and rubbers are
very different in this respect (see Fig. 64).

Figure 64: The different nonlinear elastic behaviour between metals and
rubbers.

In particular, in one case (the metal) the stress-strain curve exponent
becomes sub-linear at large strain. This phenomenon is called strain-softening
and it indicates that the material becomes softer by increasing the
deformation strain. In the second case (the rubber), the non-linear behaviour
is faster than linear; the material becomes more rigid at finite deformations
– strain hardening.

From the field theory point of view, the non-linear elasticity theory has been
recently implemented in PhysRevD.100.065015 . In order to follow the same
logic from the holographic perspective, a few ingredients must be changed.
Since the deformation strain is now an $\mathcal{O}(1)$ external field, it
must be endowed in the background configuration and, in particular, the
scalars profile must be modified into:

$\phi^{I}\,=\,O^{I}_{j}\,x^{j}\,,$ (235)

with

$O^{I}_{j}\,=\,\alpha\begin{pmatrix}\sqrt{1+\varepsilon^{2}/4}&\varepsilon/2\\\
\varepsilon/2&\sqrt{1+\varepsilon^{2}/4}\end{pmatrix}\,.$ (236)

For $\alpha=1$, we have $\mathrm{Det}\,O^{I}_{j}=1$, which means the
deformation does not change the volume of the system, it is a pure-shear
deformation, parametrized by the parameter $\varepsilon$. On the contrary, the
$\alpha$ parameter accounts for the changes of volume: it is a bulk-strain
deformation. Notice that the background configuration (236) breaks explicitly
the isotropy of the system.

Figure 65: Top panel: Shear stress strain curve for various potentials and
relative (dashed) large strain scaling. Bottom panel: Shear stress strain
curves for different temperatures and comparison with the analytic formula
(238) (dashed lines). As expected for $T/m\gg 1$ the formula gives a very good
approximation. Figures taken from Baggioli:2020qdg .

In order to find a background solution with the scalar configuration of Eq.
(236), the metric ansatz must be modified into:

$ds^{2}\,=\,\frac{1}{u^{2}}\left(-f(u)\,e^{-\chi(u)}\,dt^{2}+\frac{du^{2}}{f(u)}+\gamma_{ij}(u)\,dx^{i}dx^{j}\right)\,,$
(237)

where $\gamma_{ij}$ is a two dimensional spatial metric with unitary
determinant. This time, the background has to be found numerically by solving
a simple set of ordinary differential equations in the radial coordinate $u$.
The stress-tensor $T^{ij}$ can be then obtained using standard methods
Balasubramanian:1999re and it is a non-linear function of the background
strains $\alpha$ and $\varepsilon$. In particular, for a generic potential
$V=(X,Z)$, an analytic formula can be derived:

$\sigma(\varepsilon)=\frac{1}{2}m^{2}\alpha^{2}\varepsilon\sqrt{4+\varepsilon^{2}}\int_{0}^{u_{h}}\frac{V_{X}\left(\bar{X},\bar{Z}\right)}{\zeta^{2}}\,d\zeta\,,$
(238)

which is valid at small graviton mass, $m/T\ll 1$. Here, we have defined
$(\bar{X},\bar{Z})=(\alpha^{2}\frac{1}{2}(2+\varepsilon^{2})\zeta^{2},\zeta^{4}\alpha^{4})$
and $\sigma\equiv T^{x}_{y}$. The convergence of the integral in (238) is
equivalent to the positivity of the linear bulk modulus. In order to make some
more quantitative predictions, we will consider the benchmark potential:

$V(X,Z)\,=\,X^{\mathfrak{a}}\,Z^{\frac{\mathfrak{b}-\mathfrak{a}}{2}}\,,$
(239)

The non-linear stress-strain curves for different powers are shown in Fig. 65
together with the comparison with the perturbative expression Eq. (238). At
intermediate strain, a power law scaling $\sigma\sim\varepsilon^{\nu_{1}^{S}}$
appears, with

$\nu_{1}^{S}=2\mathfrak{a}\,.$ (240)

Additionally, at much larger values of the strain, a secondary scaling appears
$\sigma\sim\varepsilon^{\nu_{2}^{S}}$ with a different exponent

$\nu_{2}^{S}\,=\,3\,\frac{\mathfrak{a}}{\mathfrak{b}}\,.$ (241)

The presence of two scaling regimes happens only at high enough temperature.
At low temperature, the stress-strain curve directly interpolates from the
linear regime to the $\nu_{2}^{S}$ scaling. A similar power law behaviour
appears in the bulk stress-strain curve Baggioli:2020qdg , where there is a
universal scaling

$\sigma_{L}\,\propto\,\kappa^{3}\,,$ (242)

with $\kappa\equiv\partial\cdot\phi$ and
$\sigma_{L}=T_{xx}(\kappa)-T^{eq}_{xx}$.

Until now, we have discussed the realm of non-linear elasticity only in the
context of static deformations. Nevertheless, the biggest interest in the
field of rheology deals with time dependent deformations and the corresponding
reaction of the system. In particular, a typical experiment – oscillatory
shear test – consists in an external shear strain taking a simple sinusoidal
form

$\gamma(t)\,=\,\gamma_{0}\,\sin(2\pi\omega t)\,,$ (243)

where $\gamma_{0}$ is the strain amplitude and $\omega$ the characteristic
frequency.

Particularly challenging is the regime where the amplitude of the external
strain is large, $\gamma_{0}=\mathcal{O}(1)$, which takes the name of large
amplitude oscillatory shear regime (LAOS). In the LAOS regime, linear
viscoelasticity is not applicable anymore and the response is fully nonlinear
and very little is known pipkin1986lectures ; HYUN20111697 ;
4f5dcea3a806435da7b4b4cbadb06d63 . The LAOS regime has been recently studied
in the holographic axion model in Baggioli:2019mck and the non-linear regime
has been directly observed with several methods (see Fig. 66).

Figure 66: The onset of nonlinear elasticity by increasing the strain
amplitude. Top panel: the real time dynamics and the Lissajiou’s figures.
Bottom panel:  the non-linear complex modulus. Figures updated from
Baggioli:2019mck .

Moreover, it has been found that at least for the potentials considered, the
holographic model displays a well-defined strain-hardening mechanism. More
generally, depending on the potential chosen in Eq. (239) the holographic
axion model can exhibit either strain-softening or strain hardening. See Fig.
67 for a map of the two situations depending on the powers in Eq. (239).

Figure 67: The phase diagram of the holographic axion model with benchmark
potential (239) according to the non-linear elastic properties.

###  VII.7 Plasmons

An interesting phenomenological direction which has been recently investigated
is the dynamics of plasmon modes in strongly coupled materials and therefore
holographic models. The seminal work can be found in Aronsson:2017dgf and it
was motivated by the recent surprising experimental results of
mitrano2018anomalous ; husain2019crossover (see also Krikun:2018agd ;
Andrade:2019bky for discussions around this point). The main idea is to modify
the boundary conditions of the gauge fields fluctuation to impose the Maxwell
equations in the boundary dual field theory. This can be achieved by fixing:

$\omega^{2}\,\delta A_{x}\,+\,\lambda\,\delta A^{\prime}_{x}\,=\,0\,,$ (244)

which is a mixed boundary condition and makes the gauge field dynamical at the
boundary. The parameter $\lambda$ measures the strength of the emergent
Coulomb force in the boundary theory. Doing so, a nice plasmon mode
$\mathrm{Re}\left[\omega\right]=\sqrt{\omega_{p}^{2}\,+\,k^{2}}$ is obtained
at finite charge (see Fig. 68).

Figure 68: The appearance of plasmons in the Reissner-Nordstrom background as
a consequence of imposing the mixed boundary conditions (244). Figure from
Aronsson:2017dgf .

The effects of the explicit and spontaneous breaking of translations, using
the holographic axion model, have been studied in a series of follow-up works
Baggioli:2019aqf ; Baggioli:2019sio . The first observation is that the
lifetime of the plasmon mode obeys a inverse Matthiessen’ rule (see Fig. 69):

$\tau^{-1}\,=\,\tau_{EM}^{-1}\,+\,\tau_{M}^{-1}\,,$ (245)

where $\tau_{EM}$ is the contribution for the Coulomb interactions and
$\tau_{M}$ is the contribution coming from momentum dissipation and equal to
the inverse of the momentum relaxation rate $\Gamma$.

Figure 69: The numerical confirmation of the inverse Matthiessen’ rule for the
plasmon lifetime. The dashed lines are Eq.(245) and the colored dots the
numerical data for various $\lambda$. Figure taken from Baggioli:2019aqf .

Additionally, in the transverse spectrum, momentum dissipation induces a
peculiar modes repulsion dynamics which was discussed in Baggioli:2019sio .

Finally, Baggioli:2019aqf analyzed also the plasmons dynamics in presence of
elasticity – SSB of translations. An interesting transition appears at the
point in which the shear viscosity becomes comparable with the shear modulus
(see Fig. 70). At that value, the fluid becomes to behave more like a solid
and the plasma frequency – the mass of the plasmon – starts to rapidly decay
as shown in Fig. 70.

The physics of holographic plasmons is still highly unexplored and more work,
specially in connection with a possible hydrodynamic description, is needed.

Figure 70: The fluid to solid crossover and the depletion of the plasma
frequency $\omega_{p}$. Figure taken from Baggioli:2019aqf .

##  VIII Additional topics

###  VIII.1 SYKology

The Sachdev-Ye-Kitaev (SYK) model is a many body quantum system which has
become very popular in the physics community because it is strongly coupled,
exactly solvable, chaotic and nearly conformal invariant Rosenhaus:2018dtp .
Moreover, it bears several interesting relations with AdS2 gravity, black
holes physics and strange metals Trunin:2020vwy .

The relation between the SYK model and axions-like holographic model was put
forward in Davison:2016ngz with special emphasis on the thermodynamic,
transport and quantum chaos properties. More specifically several connections
between the two frameworks were successfully established and analyzed later in
PhysRevX.5.041025 .

Recently, several works Choi:2020tdj ; Arean:2020eus computed exactly the
energy-energy correlator in the SYK model and they compared it with that
extracted from the linear axion model of Andrade:2013gsa . Again, in the
strong coupling limit, the results obtained from the two frameworks were found
to be very similar. These findings contribute to the question about which is
the gravity dual of the disordered SYK model and if that has to do with our
holographic axion model with momentum dissipation.

###  VIII.2 Quantum information

There are increasing evidences for the existence of a deep connection between
quantum information in the boundary field theory and the spacetime geometry in
bulk. This link was initially triggered by the definition of entanglement
entropy and its holographic dual – the Ryu-Takayanagi formula Ryu_20061 ;
Ryu_20062 . See Rangamani_2017 for a review. However, there are many other
quantum informational quantities, which capture several aspects of quantum
information different from the entanglement entropy. According to holography,
these informational concepts "must" have their own dual geometric objects. For
example, quantum information probes for mixed states have been proposed: the
entanglement of purification, the logarithmic negativity, the odd entanglement
entropy and the reflected entropy. Their holographic dual is related to the
so-called entanglement wedge cross section.292929For more description of the
concepts we refer to Jeong:2019xdr and references therein. Quantum complexity
is another important concept, because it is conjectured to explore the inside
of the black hole horizon, while entanglement entropy can not Susskind:2018pmk
. This line of research played a key role to achieve a possible resolution of
the black hole information paradox Almheiri:2020cfm .

It is natural to consider the axion model to study various quantum
informational quantity, because momentum relaxation is ubiquitous and plays an
important role in real quantum systems. The momentum relaxation effect on the
holographic entanglement entropy Mozaffara:2016iwm and the complexity in the
complexity-action conjecture Babaei-Aghbolagh:2020vsz have already been
studied. Under thermal quench, holographic entanglement entropy Li_2020 ,
subregion complexity in the complexity-volume conjecture Zhou:2019xzc have
been investigated. Towards the entanglement measure for mixed states,
holographic entanglement entropy, mutual information, and entanglement of
purification have been considered in Huang:2019zph .

###  VIII.3 Fermionic response

The fermionic spectral function is a very important observable, specially in
strongly correlated materials, which can be directly probed experimentally
with Angle Resolved Photoemission Spectroscopy (ARPES) or Scanning Tunneling
Microscopy (STM). The fermionic spectral function has been considered in the
realm of Holography in several pioneering works Lee:2008xf ; Liu:2009dm ;
Cubrovic:2009ye ; Faulkner:2009wj in relation to possible non-Fermi liquids
signatures.

The holographic spectral function can be computed by solving the bulk Dirac
equation. A class of fermion bulk action with the mass and the dipole coupling
is given by

$\displaystyle\mathcal{S}_{\text{spinor}}$ $\displaystyle=$ $\displaystyle
i\int
d^{4}x\sqrt{-g}\bar{\psi}\left(\Gamma^{M}D_{M}-m-\,\frac{ip}{2}\Gamma^{MN}F_{MN}\right)\psi\,,$
$\displaystyle\Gamma^{MN}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\Gamma^{M},\Gamma^{N}]\,,$ (246)

with $\Gamma^{M}$ and $D_{M}$ the Gamma matrices and the covariant derivative
in a curved spacetime, respectively. The dipole interaction drives the
dynamical formation of a Mott-like gap in the absence of continuous symmetry
breaking Edalati:2010ww . The study of holographic fermions in terms of
fermion mass ($m$), dipole coupling ($p$) and the strength of momentum
relaxation ($\alpha$) has been conducted in Jeong:2019zab .

The holographic spectral function with momentum relaxation in two linear axion
models was systematically investigated in Jeong:2019zab , where the momentum
relaxation strength $\alpha$ is introduced via the bulk profile
$\phi^{I}=\alpha\delta^{I}_{i}x^{i}$. By classifying the shape of spectral
functions, the complete phase diagrams in ($m,p,\alpha$) space were
constructed (see Fig. 71). Although the DC electric transport of two models
are very different, the effects of momentum relaxation on the spectral
function are similar. This may be due to the fact that holographic fermion
does not back-react to geometry.

Some common features were highlighted as follows Jeong:2019zab . First, it was
found that for a given dipole coupling and momentum relaxation, the spectral
functions tend to become sharper by increasing the mass of the bulk fermion.
Second, a new peak at finite frequency can be generated as the dipole coupling
increases. Third, in general the spectral function becomes more suppressed and
broader as the strength of momentum relaxation is increased. Interestingly,
the suppression of spectral weight and the gradual disappearance of Fermi
surface along the symmetry breaking direction were also observed in the
inhomogeneous holographic models by increasing the lattice strength
Cremonini:2018xgj ; Cremonini:2019fzz ; Balm:2019dxk ; Iliasov:2019pav ;
Mukhopadhyay:2020tky . The homogeneous holographic lattices can simulate the
effects of translational symmetry breaking while retaining the homogeneity of
the spacetime geometry. However, homogeneous lattices are unable to capture
the physics of Umklapp, motivating the need to work with periodic lattices
Bagrov:2016cnr .

Figure 71: Phase diagram in ($m,p,\bar{\alpha}$) space for the Einstein-
Maxwell-linear axion model of Jeong:2019zab . Depending on the shape of
spectral functions, one can classify different phases, such as fermi liquid
like (FL), bad metal prime (BM’), bad metal (BM), pseudogap (PG), gapped (G)
and so on. See Jeong:2019zab for more details. Figure taken from Jeong:2019zab
.

###  VIII.4 Modeling graphene

Experimental measurements have uncovered evidence of the strongly coupled
nature of the graphene. As a matter of fact, the Wiedemann-Franz law (the
ratio of heat and electric conductivities, $L=\kappa/T\sigma$) is violated by
up to a factor of 20 near the charge neutral point in extremely clean graphene
2016Sci…351.1058C . It has been argued that graphene near charge neutrality
forms a strongly coupled Dirac fluid without well-defined quasiparticle
excitations. A fundamental reason for the appearance of the strong interaction
in graphene is due to the smallness of the Fermi sea: electron-hole pair
creation near the Dirac cone is insufficient to screen the Coulomb
interaction, thus there should be a regime where electrons are strongly
correlated. A hydrodynamics description with disorder and a single conserved
$U(1)$ current was adopted to explain experimental observations
Hartnoll:2007ih ; Lucas:2015sya , but still left room for improvement.

In contrast to the one current model, there are a few motivations including an
extra current in the graphene Seo:2016vks . The first one is from the effect
of imbalance between the electrons and holes due to the kinematic constraints
of the Dirac cone, which also suggests the two conserved charges can be
proportional. Other candidates include spin charge separation, valley
currents, phonons and so on. The linear axion model with two distinct
conserved $U(1)$ currents was proposed in Seo:2016vks to describe the
experimental data. The electric, thermo-electric and thermal conductivities
can be computed analytically. Then, under the assumption that the two
conserved charges are proportional to each other, the holographic results for
the density dependence of the electric and heat conductivities have a
significantly improved match to the experimental data than the models with
only one current (see Fig. 72). The holographic model also suggested an
additive structure in the transport coefficients: the additivity of
dissipative part of the inverse heat conductivity.

$D[1/\kappa]=\sum_{i}D[1/\kappa_{i}],\quad\bar{D}[1/\sigma]=\sum_{i}\bar{D}[1/\sigma_{i}]\,,$
(247)

where $\kappa_{i}$ and $\sigma_{i}$ are the heat conductivity and electric
conductivity for the $i$-th current. $D[f]$ denotes the dissipative part of
$f$ and $\bar{D}[f]=f-D[f]$.

Figure 72: Comparison with experimental data. Density plot of electric
conductivity $\sigma$ and of thermal conductivity $\kappa$. Red circles are
for data used in 2016Sci…351.1058C and black curves are for the holographic
model with two currents. The regime marked in blue is for the Fermi Liquid
(FL) that is far from the holographic theory. Figure taken from Seo:2016vks .

Quantum criticality has been argued to be crucial for interpreting a wide
variety of experiments. A large class of quantum critical points can be
characterized by two scaling exponents, known as the dynamical critical
exponent $z$ and the hyperscaling violation exponent $\theta$. The case for
the holographic model of Seo:2016vks corresponds to $(z=1,\theta=0)$. A
different set of dynamical exponents was considered in Song:2020ojc where it
was found that the graphene data can be fit much more naturally by considering
$(z=3/2,\theta=1)$. Furthermore, the Seebeck coefficient can also be fit using
$(z=3/2,\theta=1)$ (see Fig. 73). In contrast, the previous model with
$(z=1,\theta=0)$ fails to describe features of the experimental data at large
density. The fact that this model does not fit with the experimental data for
large temperature was argued to be due to the absence of phonon effect that is
important for large temperature Song:2020ojc .

Figure 73: Seebeck coefficient as a function of charge density $Q$. Circles
are for experimental data used in 2016PhRvL.116m6802G and dashed line for
hydrodynamics result. Seebeck coefficient at low temperature fits well with
experiment for the holographic two currents model with $(z=3/2,\theta=1)$.
Figure updated from Song:2020ojc .

###  VIII.5 Topological effects

The bulk Chern-Simons terms play an important role in holography since it
contributes to various new effects and new physics. For example, the Chern-
Simons term $A\wedge F\wedge F$ can yield a charge density wave instability
Nakamura:2009tf ; Donos:2013gda , a metal-insulator transition Donos:2012js ,
and the presence of a non-trivial chiral magnetic conductivity Yee:2009vw ;
Kim:2010pu .

On the contrary, what if we want to study the boundary Chern-Simons term,
$A\wedge F$? The boundary $A\wedge F$ term is important because it may be
interpreted as the spin-orbit coupling. The spin-orbit coupling in $2+1$
dimension is relevant to interesting phenomena in topological insulators and
Weyl semi-metals Qi_2011 .303030The spin-orbit interaction involves fermions
but after integrating out fermions we may effectively deal with the Chern-
Simons term $A\wedge F$ PhysRevD.31.2137 ; PhysRevD.32.1020 . The boundary
$A\wedge F$ term can be holographically lifted to $F\wedge F$ in bulk
Seo:2015pug . To have a non-trivial dynamical effect we may couple $F\wedge F$
with some scalar operators.

Related with the axion model, one possible coupling is

$q_{\chi}XF\wedge F\,,$ (248)

where $X$ is the kinetic term of the axion fields and $q_{\chi}$ is introduced
to quantify the strength of this interaction. For the model (46) with
$Y=1,V=X$ together with the interaction (248), thermodynamic properties of the
system and electric and thermal conductivities have been computed Seo:2015pug
; Seo:2017oyh ; Song:2019rnf .

From the structure of (248), which is schematically $\sim
q_{\chi}\alpha^{2}\rho B$, we see that there can be a magnetization ($\sim
q_{\chi}\alpha^{2}\rho$) even without explicit magnetic field. In this sense,
the axion charge $\alpha$ may be interpreted as a magnetic impurity. This
magnetic impurity induces a Hall current without an external magnetic field,
so it may explain the presence of an anomalous Hall effect, which is ten times
larger than the one observed in non-magnetic materials.

Regarding the AC electric conductivity, the interaction (248) induces a new
quasi-particle pole Song:2019rnf . This excitation is attributed to a new
coupling between two gauge field fluctuations $a_{x}$ and $a_{y}$ by (248):
$q_{\chi}\alpha^{2}\partial_{u}a_{y}\partial_{t}a_{x}$. This quasi-particle
pole may be considered as a kind of cyclotron pole (228) induced this time by
a magnetic impurity, not by an external magnetic field. Note that, for a
cyclotron pole, $a_{x}$ and $a_{y}$ are connected indirectly by a metric
fluctuation.

As another interaction, we may consider Kim:2019lxb ; Kim:2020ozm

$\varphi F\wedge F\,,$ (249)

where $\varphi$ is a real scalar in the holographic superconductor model
Hartnoll:2008kx with axion Kim:2016hzi . The basic idea is to use spontaneous
$\mathbb{Z}_{2}$ symmetry breaking to induce spontaneous magnetization.
Because $\varphi$ is a real scalar (the symmetry is not $U(1)$ but
$\mathbb{Z}_{2}$) the system is not superconducting and the conductivity is
finite below the symmetry breaking transition Hartnoll:2008kx . The essential
structure of (249) is $\sim\langle\mathcal{O}\rangle\rho B$, where
$\langle\mathcal{O}\rangle$ is the (spontaneous) condensate of the operator
dual to $\varphi$. Thus, even without $B$ the magnetization can be finite due
to spontaneous $\mathbb{Z}_{2}$ symmetry breaking Kim:2019lxb . If $\alpha$
increases, the magnetization increases so $\alpha$ can be interpreted as
magnetic impurity. This model exhibits magnetic hysteresis as well.
Interestingly this model finds its applications in topological insulators,
bao123 ; bao1234 , where it is observed that the magnetoconductance starts
showing hysteresis behavior similar to magnetization as magnetic doping
increases. This is called hysteric magnetoconductance phase. The holographic
model with the new coupling (249) qualitatively reproduces this phase as
$\alpha$, which is identified with magnetic impurity strength, increases.

###  VIII.6 Non-equilibrium physics and thermalization

This review so far has mainly focused on the nature of holographic quantum
matter at equilibrium and on the consequence of perturbing states very near
equilibrium, for which the linear response theory applies. The non-equilibrium
physics of strongly coupled quantum matter is an important but largely
unexplored frontier. Non-equilibrium phenomena are general problems in ultra-
relativistic heavy-ion collisions, cold atom systems, condensed matter physics
and so on, for which there are almost no techniques from standard field theory
to apply. By mapping the physics of quantum matter into a dual gravitational
theory, one is able to study difficult non-equilibrium process by solving
tractable non-stationary general relativity problems Chesler:2013lia .

A simple way to drive a system far from equilibrium is through a quantum
quench by turning on a time dependent source $s(t)$. Two typical examples are
as follows. Typically, $s(t)$ can either interpolate between an initial and a
final state $s(-\infty)=0$ and $s(\infty)=1$ or alternatively oscillate
between them. The far from equilibrium dynamics will tend to drive quantum
matter to a finite temperature states. This thermal effect is natural from
bulk perspective as the energy will be absorbed by the black hole.

Transport properties of strongly coupled systems from holographic duality have
been the subject of much recent interest. The nonlinear response of a finite
charge density system resulting from an electric field quench in a simple
Einstein-Maxwell-axion model was investigated in Withers:2016lft ;
Bagrov:2017tqn . For the finite-time pulsed quench, the electric field is
smoothly turned on, held for some time and then turned off again, see the red
dashed curve in top panel of Fig. 74. As can be seen from Fig. 74, the system
returns to equilibrium after turning the electric field off. It was found in
Withers:2016lft that the system returns to equilibrium with the approach
governed by the longest lived QNMs of the final black brane whose spectrum
depends on the strength of momentum relaxation $k$ via
$\phi^{I}=\alpha\delta^{I}_{i}x^{i}$. By dialing $\alpha$, one can see a
qualitative change in the relaxation of currents, due to the pole collision
and the presence of off-axis mode. In the small $\alpha$ coherent regime, the
relevant QNMs are purely decaying, while in a large $\alpha$ incoherent
regime, the heat current acquires enhanced contributions from a branch of QNMs
that oscillate and decay.

Figure 74: Evolution for the case with top hat electric field. Top panel: The
solid curve denotes the electric field $J$. The red dashed curve shows the
quenched electric field $E(t)$ and the blue dashed curve gives the
approximation to the electric conductivity. Middle panel: The energy current.
Bottom panel: The evolution of event horizon and apparent horizon. The bulk
distribution of the axion field with the linear $x$-dependence subtracted is
illustrated in color. The charge density has been set to one. Figure updated
from Withers:2016lft .

The nonlinear thermoelectric response induced by holding the electric field
constant was also discussed in Withers:2016lft . For small electric field,
there is a steady state described by DC linear response, due to a balance
between the driving electric field and the momentum sink. When the electric
field is large, Joule heating will introduce significant time dependence on
the bulk geometry. Nevertheless, in a regime where the rate of temperature
increase is small, the nonlinear electric conductivity can be well
approximated by a DC linear response calculation, once an appropriate
effective temperature $T_{E}$ is taken into account (see Fig. 75). In
contrast, the linear response result for the thermoelectric DC conductivity
$\bar{\alpha}$ does not give good agreement over the same timescales, which
means $\bar{\alpha}$ should have an explicit dependence on the electric field.

Figure 75: Nonlinear electric current response as a function of effective
temperature $T_{E}$. Curves from left to right represent runs of different
initial temperatures $T_{i}=i0^{-2},10^{-1},1/4,1/2,3/4,1$, respectively. The
black dashed line shows the DC linear response conductivity after the
equilibrium temperature is promoted to $T_{E}$. The charge density has been
set to one. Figure updated from Withers:2016lft .

Another interesting case is driving a system with a very short and intense
coherent electromagnetic pulse, after which the time evolution of the system
is monitored by a linear response probe. The study of this pump-probe
experiment from holography was presented in Bagrov:2017tqn . A holographic
state at finite density with mildly broken translation invariance through
linear axions was excited by oscillating electric field pulse. The
thermalization process was numerically investigated by varying the pulse
frequency $\omega_{P}$. It was found that in all circumstances the
thermalization continues to be instantaneous for pump pulses devoid of a zero
frequency component. For pump electric field with a significant DC component,
the full time evolution is governed by a single thermalization time which is
precisely half of the equilibrium momentum relaxation time at the final
temperature. This feature can be understood from the fact that the metric
component corresponding to momentum appears squared in the computation of the
time-dependent conductivity. It was conjectured in Bagrov:2017tqn that large
class of systems with a holographic dual will exhibit the phenomenon of
instantaneous thermalization. These holographic findings can be tested by
experiments which are in principle feasible in condensed matter laboratories.

The holographic axion model has been also used to study out-of-equilibrium
dynamics in presence of anomaly Fernandez-Pendas:2019rkh ; Morales-
Tejera:2020xuv . In this last context, it has been proven before
Copetti:2017ywz that the chiral magnetic and chiral vortical conductivities
are completely independent of the momentum relaxation rate introduced via the
axions.

##  IX Outlook

###  IX.1 Open questions

Before concluding, we find useful to collect the main open questions related
to the holographic axion model discussed in this review. We will take a very
direct attitude and list them one by one.

  1. 1.

The viscosity of holographic solids. Why does the viscosity decrease by
increasing the amount of SSB of translations? Going from a fluid to a solid
the viscosity rapidly increases in Nature, but it is not the case here. Is
this connected to the fact that in most of the holographic systems (see
Cremonini:2012ny for a counterexample) the viscosity grows with temperature?
This is again not the case in liquids, but only in gases.

  2. 2.

Holographic fluids. We always sell the idea that AdS Schwarzchild is the dual
of a relativistic fluids at strong coupling. That cannot be since fluids
clearly have a finite electric conductivity. This suggests that a realistic
holographic fluid must be encoded in the axions model with $V(Z)$
Alberte:2015isw ; Baggioli:2019abx , in agreement also with the effective
theories expectations Dubovsky:2011sj ; deBoer:2015ija . What is really the
difference between these two setups? Is this really dual to a fluid or a gas?

  3. 3.

The cost of homogeneity. How do these homogeneous models compare with more
realistic inhomogeneous setups? It seems that at low energy, _i.e._ low
frequency and low momentum, there is absolutely no difference. We did not
learn anything about DC transport coefficients from these very complicated
inhomogeneous models. We have recently proved that even the low energy
spectrum is the same Andrade:2020hpu . Where can we find differences? It is
clear that one has to go to more microscopic features, related to higher
momenta. One case is the property of commensurability Andrade:2015iyf .
Anything else? Does the numerical effort really pay back?

  4. 4.

Phase relaxation vs. pseudo-spontaneous breaking. We now know that the pseudo-
spontaneous breaking of translations produces an effective phase relaxation
term which is fundamentally different from the one usually considered and
coming from the presence of elastic defects such as dislocations. Why
nevertheless do we not see any Drude peak in the frequency dependent
viscosity? Can we understand why the $\Omega$ pole is somehow hidden by the
presence of explicit breaking?

  5. 5.

Holographic dislocations. How can we introduce in a simple way elastic defects
in the holographic picture? And is their phenomenology what we do expect?

  6. 6.

Holographic Glasses. We have discussed in details fluids and solids. What
about glasses? Have the axions model anything in common with glasses? Is there
a Boson peak in the spirit of Baggioli:2018qwu ? Similar ideas appeared
already in Anninos:2013mfa ; Facoetti:2019rab .

  7. 7.

Phonons Hydrodynamics. There has been recent effort in linking the idea of
electron hydrodynamics Lucas:2017idv with holography Erdmenger:2018svl . What
about phonons hydrodynamics Cepellotti2015 ? How to implement such limit?

  8. 8.

Physical nature of holographic axions. Despite a lot of work on the axions
model, the physical nature of the dual system is still not well understood. To
what extent can this simple homogeneous setup be trusted? And which phase of
matter are we actually describing? These questions remain unanswered.

  9. 9.

Negative energy and possible instability. A common feature of many simple
linear axion models is that the energy density $\mathcal{E}$ becomes negative
at large $\alpha$. It was argued that there might be some instability
signalled by the appearance of negative energy density, but such instability
was never found and all the linearized excitations are well behaved. How to
understand this negative energy issue? Does it have any important consequence
or limitation for some of the simplest holographic quantum matters?

  10. 10.

The residual entropy density. Another common feature of linear axion models is
the residual entropy density at zero temperature, corresponding to the
$AdS_{2}$ IR extreme geometry. What’s the nature of this residual entropy? Is
there any possible relation to localization or glasses? Recent developments in
the SYK model taught us that such feature does not come from a degeneracy of
the ground state but from the piling up of a very close excited states
PhysRevX.5.041025 . Can we understand it from the gravitational picture? Is it
again related to the highly unstable character of the $AdS_{2}$ geometry?

  11. 11.

Make it useful. It is a truth acknowledged by several researchers in (and
specially outside) the community, that AdS-CMT has not produced yet a strong
smoking-gun result able to justify its usefulness for realistic condensed
matter systems. Can we push the framework further, connect it to experiments,
predict new phenomena and explain unresolved ones? This seems the only way for
the tool to survive without becoming a niche product for a small group of
enthusiasts.

###  IX.2 Conclusions

In conclusion, we hope to have convinced the Reader that, despite the apparent
simplicity, the holographic axion model displays an incredibly rich set of
features and applications which go far beyond the idea of dissipating momentum
and make the DC electric conductivity finite.

In any case, whether you want to use them just to avoid annoying infinities or
if you want to dig deeper in the physics of solids and fluids at strong
coupling, this review is made for you. At the cost of resulting rather
lengthy, we have made the effort of being as comprehensive as possible and
discuss all the different faces of the model. We hope that any of you, in one
way or another, will benefit from this read and will learn something new you
were not aware of before. We also wish to have inspired new thoughts on the
topic and the incentive for future developments in the field.

## Acknowledgments

We are grateful to the uncountable number of colleagues which participated
with us in the process of understanding all the secrets of the holographic
axion model. We thank Teng Ji, Giorgio Frangi, Hyun-Sik Jeong, Xi-Jing Wang
and Yongjun Ahn for useful comments and helping proof-reading an early version
of this manuscript.

M.B. acknowledges the support of the Shanghai Municipal Science and Technology
Major Project (Grant No.2019SHZDZX01) and of the Spanish MINECO “Centro de
Excelencia Severo Ochoa” Programme under grant SEV-2012-0249. K.K. was
supported by Basic Science Research Program through the National Research
Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future
Planning (NRF-2017R1A2B4004810) and the GIST Research Institute(GRI) grant
funded by the GIST in 2020. L.L. is supported in part by the National Natural
Science Foundation of China (NSFC) Grants No.12075298, No.11991052 and
No.12047503. W.J.L. is supported in part by the NSFC under grant No.11905024
and DUT under grant No.DUT19LK20.

##  Appendix A Notations and conventions

In order to avoid confusion, in this appendix we describe in detail all the
symbols and notations used in this review.

Greek letters $\mu,\nu,...$ run over spacetime indices, while Latin letters
$i,j,...$ denote spatial ones. The axion flavor indices $I,J,...$ run over the
number of broken translations. In this review, we also omit the summation
symbol over the axion flavor indices, and use the Einstein convention for them
too.

We always utilize a mostly plus metric $(-1,1,1,1)$ and we define the Fourier
transform of a field $\Psi$ using the plain-wave $e^{-i\omega t+ikx}$.
Finally, we indicate in Table I all the symbols used.

Symbol | Meaning
---|---
$\eta$ | shear viscosity
$G$ | shear modulus
$\zeta$ | bulk viscosity
$K$ | bulk modulus
$E$ | electric field
$B$ | magnetic field
$\tau$ | relaxation time
$\Gamma$ | momentum dissipation rate
$\chi_{AB}$ | susceptibility
$\chi_{\pi\pi}\,\text{or}\,\chi_{pp}$ | momentum susceptibility
$\sigma_{ij}\,\text{or}\,T_{ij}$ | stress
$\varepsilon_{ij}$ | strain
$\varrho$ | mass density
$v_{T}\,\text{or}\,v_{\perp}$ | shear sound speed
$v_{L}\,\text{or}\,v_{\parallel}$ | longitudinal sound speed
$u_{i}$ | displacements
$\omega_{0}$ | pinning frequency
$\Omega$ | phase relaxation rate
$\mathcal{E}$ | energy density
$\Gamma_{L}\,\text{or}\,\Gamma_{\parallel}$ | longitudinal sound attenuation
$\Gamma_{T}\,\text{or}\,\Gamma_{\perp}$ | transverse sound attenuation
$D_{\phi}$ | crystal diffusion constant
$\xi$ | Goldstone diffusion parameter
$\mathfrak{p}$ | pressure
$\mathcal{P}$ | crystal pressure
$m_{g}$ | graviton mass
$v_{B}$ | butterfly velocity
$u$ or $r$ | radial coordinate
$u_{h}$ | horizon radius
$L$ | AdS radius
$\Lambda$ | cosmological constant
$T$ | temperature
$s$ | entropy density
$\rho$ | charge density
$\mu$ | chemical potential
$k_{g}$ | k-gap
$z$ | Lifshitz exponent
$\theta$ | hyperscaling parameter
$\langle EXB\rangle$ | explicit breaking scale
$\langle SSB\rangle$ | spontaneous breaking scale
$D_{\pi}$ | momentum diffusion constant
$\omega$ | frequency
$k$ | momentum in Fourier space or wave-number
$H$ | Hamiltonian
$\mathcal{G}_{AB}$ | Green’s function
$\omega_{p}$ | plasma frequency
Table 2: Notations and symbols.

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