additions aainstitutetext: Wilczek Quantum Center, School of Physics and
Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang
District, Shanghai 200240, China. bbinstitutetext: Shanghai Research Center
for Quantum Sciences, Shanghai 201315.

# Holographic supersolids

Matteo Baggioli a , Giorgio Frangi<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A supersolid is a system that presents long-range order and shear rigidity as
a solid but which also supports a non-dissipative superflow as a superfluid.
From an effective perspective, supersolids are identified with phases of
matter that break spontaneously translational invariance together with a
global $U(1)$ symmetry. By using this symmetry prescription, we build a
holographic bottom-up model for supersolids and we start the investigation of
its thermodynamic and mechanical properties. More precisely, we analyze the
behaviour of the critical temperature, the condensate, the shear modulus and
the viscosity across all the phase diagram. Finally, we successfully compare
our results with a simple Ginzburg-Landau model for supersolids deriving some
universal physical correlations between the observables mentioned above.

##  1 Introduction

Crystalline solids111We will ignore in this manuscript the possibility of
having solids lacking long-range order – amorphous solids (e.g. glasses). are
characterized by the presence of long-range order and they exhibit macroscopic
shear rigidity chaikin_lubensky_1995 . From a more fundamental perspective,
solids correspond to ground states which spontaneously break translational
invariance (usually down to a discrete group defined by the lattice vectors)
and whose low-energy dynamics is controlled by the corresponding Goldstone
modes – the phonons Leutwyler:1996er . In a similar fashion, superfluids are
identified with ground states which support a non-dissipative flow with zero
viscosity – the superflow RevModPhys.71.S318 . Superfluid states break
spontaneously a $U(1)$ global symmetry Son:2002zn ; Nicolis:2011cs  and their
low-energy physics is dominated by a Goldstone mode as well, accounting for
the propagation of second sound doi:10.1063/1.3248499 .222More precisely,
second sound arises as a combination of the $U(1)$ Goldstone mode and the
original charge diffusion mode. The same pattern holds for a solid in which
the phonons are not exactly the Goldstone excitations but a combination of
them with momentum and energy fluctuations.

A supersolid is a phase of matter that displays simultaneously the distinctive
features of solids and superfluids RevModPhys.84.759 ; Balibar2010 . How can a
rigid body with "infinite" viscosity flow freely as a superfluid? The
questions of whether such a system could exist, how the superflow could appear
in a solid and where a supersolid could be experimentally observed have
occupied the minds of several physicists in the past seventy years
PhysRev.106.161 ; GROSS195857 ; RevModPhys.34.694 ; PhysRevA.2.256 ;
PhysRevLett.25.1543  since the original observation by O. Penrose and Onsager
PhysRev.104.576 . Despite the tremendous theoretical progress, supersolid
systems have eluded clear detection for the longest time, and it was in recent
times only that they have finally been observed in laboratories: historically,
supersolid behavior has been first looked for in solid helium
doi:10.1126/science.1196409 ; kim2004probable ; kim2004observation ;
rittner2006observation ; PhysRevLett.109.155301 ; kim2014upper , while in the
last few years advancements in the field of ultracold atom gases provided some
clear signatures of it doi:10.1126/science.aba4309 ; Guo2021 ; Norcia2021 .
Nonetheless, a complete understanding of this phenomenon is still lacking.

From an abstract point of view, a supersolid is a state which spontaneously
breaks translational (and rotational) invariance together with a $U(1)$ global
symmetry. This will be our operative, and indeed agnostic in a microscopic
sense, definition of a supersolid. In this work, we will limit ourselves to
analyzing the possible low energy dynamics and physics of such a hypothetical
system PhysRevLett.104.075302 ; doi:10.1063/1.2883895 ; Syshchenko_2009 ;
Day2007 ; prokof2007makes ; lin2009heat ; anderson2008bose . In the past
decades, a lot of effort has been devoted to this direction, especially into
finding the correct low-energy effective description and hydrodynamic
framework for supersolids PhysRevB.53.5670 ; Saslow2012 ;
hofmann2021hydrodynamics ; PhysRevLett.97.125302 ; PhysRevLett.96.055301 ;
PhysRevLett.94.175301 ; RevModPhys.46.705 . Recently, supersolids have been
also considered as possible ground states in low-energy effective field
theories with spontaneously broken Lorentz invariance Nicolis:2013lma ;
Delacretaz:2014jka ; LVEFT3 ; Krichevsky:2020ury ; Celoria:2017bbh , which
will be closer in spirit to our approach.

Inspired by these ideas and questions, we will reformulate the problem of
supersolids and their low-energy description in terms of holographic methods
zaanen2015holographic ; Natsuume:2014sfa ; Hartnoll:2016apf ; Baggioli:2019rrs
. Superfluid states have been successfully incorporated in the holographic
framework more than ten years ago, using a simple abelian Higgs model
Hartnoll:2008vx ; Hartnoll_2008  (for a review of this model see Cai:2015cya )
and the corresponding low energy spectrum has been successfully matched to the
predictions of relativistic superfluid hydrodynamics Herzog:2011ec ;
Schmitt:2014eka  in Arean:2021tks .333The hydrodynamics description and the
agreement with the holographic model has been also recently extended to the
case of softly broken $U(1)$ symmetry Ammon:2021slb ; Donos:2021pkk .
Holographic solids, i.e. holographic systems spontaneously breaking
translations, have attracted a lot of interest recently as possible toy-models
to understand the complex dynamics of strongly correlated materials such as
strange metals. A very convenient scenario is obtained by imposing large scale
homogeneity and avoiding any spatial dependence in the stress tensor and
related quantities. Models of this sort are usually referred to as holographic
homogeneous solids baggioli2015electron ; Baggioli:2021xuv ; Donos:2013eha ;
Grozdanov:2018ewh ; Nakamura:2009tf ; Amoretti:2017frz  and have represented a
very efficient platform to perform computations in a controllable way. The
simplest among the homogeneous setups is the holographic axion model
Baggioli:2021xuv , which presents a well-defined elastic response together
with the presence of propagating phonon modes Alberte:2016xja ;
Alberte:2017oqx ; Baggioli:2019abx . As in the case of superfluids, its low
energy dynamics is well described by hydrodynamics with broken translations
Armas:2019sbe ; Ammon:2020xyv ; Armas:2020bmo ; Baggioli:2020edn .

In this work, we combine the holographic superfluid model of Hartnoll:2008vx
with the holographic solid model of baggioli2015electron  to build a
holographic supersolid – a holographic system which breaks spontaneously
translations together with a global $U(1)$ symmetry444See Kiritsis:2015oxa
for a previous attempt of describing supersolids using holographic superfluid
models with momentum relaxation.. The combination of superfluidity with broken
translations has been already discussed in the holographic literature
Baggioli:2015zoa ; Baggioli:2015dwa ; Zeng:2014uoa ; Ling:2014laa ;
Andrade:2014xca ; Erdmenger:2015qqa ; Kim:2015dna ; Jeong:2021wiu ;
Ling:2016lis ; Kim:2016jjk ; KimHomes ; Kiritsis:2015hoa  but always in the
context of momentum dissipation, i.e. explicit breaking of translations. A
crucial novel ingredient is therefore to allow for translations to be broken
spontaneously and give rise to the rigidity of our holographic system.

The manuscript is organized as follows. In Section 2 we describe a simple
Ginzburg-Landau (GL) theory for supersolids which will serve as our
theoretical background; in Section 3 we introduce the holographic model for
supersolids used in this work; in Section 4 we present our main results and
characterize in detail the thermodynamic and mechanical properties of the
holographic model, comparing it with the GL theory mentioned above; finally,
in Section 5 we conclude and provide some thoughts for the future. Appendices
A-B are left for further details about the theory of nonlinear elasticity and
for technicalities regarding the holographic model; Appendix C provides some
additional results that are not discussed in the main text.

#### Note added

This work is a direct continuation of the Master thesis of Giorgio Frangi "A
holographic study of a two-dimensional superconductor under strain
deformations" [2021, Università degli studi di Milano].

##  2 Ginzburg-Landau theory of supersolids

In this Section, we review the simple effective description of supersolids
presented in PhysRevLett.96.055301  and its main features. For a more complete
description of the low-energy physics and hydrodynamics of supersolids we
refer to PhysRevB.53.5670 ; Saslow2012 ; hofmann2021hydrodynamics ;
PhysRevLett.97.125302 ; PhysRevLett.96.055301 ; PhysRevLett.94.175301 ;
RevModPhys.46.705 .

The phenomenology of an isotropic $s$-wave superfluid near its transition can
be captured via the Ginzburg-Landau (GL) formalism by writing the following
free energy functional Schmitt:2014eka :

$F_{s}[\Psi(x)]=\int
d^{d}x\left[\frac{1}{2}\,|\partial\Psi|^{2}+a_{s}(T)|\Psi|^{2}+b_{s}(T)|\Psi|^{4}\,+\,\dots\right],$
(1)

where $\Psi$ is the (complex) scalar order parameter and $d$ is the number of
spatial dimensions of a flat space. This expression, usually referred to as
the Mexican hat potential (see Fig.1), has to be taken as a low-energy
effective description valid close to the phase transition, where
$\langle\Psi\rangle\ll 1$. The ellipsis refers to possible higher order
corrections, both in $\Psi$ and its derivatives. The functional in Eq.(1) is
clearly invariant under a global $U(1)$ transformation $\Psi\rightarrow
e^{i\varphi}\,\Psi$.

Whenever $\langle\Psi\rangle\neq 0$, the ground state of the system breaks
spontaneously the original $U(1)$ global symmetry and the system is in a
superfluid phase. This state is often called the broken phase, as opposed to
the normal phase, $\langle\Psi\rangle=0$, in which the $U(1)$ symmetry is
intact. The transition from one phase to the other is introduced
phenomenologically by imposing that the coefficient $a_{s}(T)$ changes sign at
some critical temperature $T_{c}^{(0)}$, becoming negative below it and
shifting the minimum of the free energy away from the unbroken ground state
$\langle\Psi\rangle=0$. Here, $T_{c}^{(0)}$ is used to indicate the value of
the critical temperature in the purely superfluid state, where translations
are not broken in any way. The additional requirement $b_{s}(T)>0$ ensures
that the argument of the functional in Eq.(1) is bounded from below and
consequently the system is stable.

The $a_{s}(T)$ parameter in the GL functional (1) vanishes linearly at the
critical temperature:

$a_{s}(T)=a_{0}\,\left(T-T_{c}^{(0)}\right)\,+\dots\,,$ (2)

below which, if the condition $a_{0}>0$ is fulfilled, it acquires a negative
value, causing the condensation.

In a similar way, it is possible to write down the free energy of an elastic
medium deformed away from its equilibrium configuration as
chaikin_lubensky_1995 :

$F_{e}[u]=\int
d^{d}x\,\frac{1}{2}\,C_{\alpha\beta\gamma\delta}\,u_{\alpha\beta}\,u_{\delta\gamma}\,+\,\dots\,,$
(3)

with $u$ being the linear strain tensor, i.e. the symmetrized derivative of
the displacement vector $u_{\alpha}=x^{\prime}_{\alpha}-x_{\alpha}$, and $C$
the elastic tensor. Here, $x_{\alpha}$ is the position of the atoms in the
initial configuration and $x^{\prime}_{\alpha}$ the one after the mechanical
deformation. Indices refer only to the spatial directions. For isotropic
materials, $C$ can be decomposed as:

$C_{\alpha\beta\gamma\delta}=\mathcal{K}\,\delta_{\alpha\beta}\delta_{\gamma\delta}+\mathcal{G}\left(\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma}-\frac{2}{d}\,\delta_{\alpha\beta}\delta_{\gamma\delta}\right),$
(4)

with $d$ the number of spatial dimensions. $\mathcal{G}$ and $\mathcal{K}$ are
respectively called shear and bulk modulus, since they are related to volume-
preserving and shape-preserving deformations.

Given this premise, the simplest way to think of a supersolid in its low-
energy regime is to combine the superfluid free energy (1) with the solid one
(3), by making the phenomenological coefficients of the former analytic
functions of the linear strain tensor $u_{\alpha\beta}$. In this picture, the
low-energy degrees of freedom are given by the phonons, together with the
superfluid Goldstone mode. At quadratic order in $u_{\alpha\beta}$, the
coupling between the two types of degrees of freedom can be reconstructed via
the following prescription:

$\begin{cases}a_{s}(T)\longrightarrow
a(T,u)=a_{s}(T)+{a_{1}(T)}_{\alpha\beta}u_{\alpha\beta}+\frac{1}{2}{a_{2}(T)}_{\alpha\beta\gamma\delta}u_{\alpha\beta}u_{\gamma\delta}+O(u^{3})\\\
b_{s}(T)\longrightarrow
b(T,u)=b_{s}(T)+{b_{1}(T)}_{\alpha\beta}u_{\alpha\beta}+\frac{1}{2}{b_{2}(T)}_{\alpha\beta\gamma\delta}u_{\alpha\beta}u_{\gamma\delta}+O(u^{3})\end{cases}\,.$
(5)

Notice that, because the background metric is flat, at this point the position
of the indices is not important. All in all, the total free energy of a
supersolid (ss) is given by:

$\begin{split}F_{ss}[\Psi,u]=F_{s}[\Psi]+F_{e}[u]+\int
d^{d}x\,&\left(a_{1}(T)|\Psi|^{2}+b_{1}(T)|\Psi|^{4}\right)_{\alpha\beta}u_{\alpha\beta}+\\\
&+\frac{1}{2}\left(a_{2}(T)|\Psi|^{2}+b_{2}(T)|\Psi|^{4}\right)_{\alpha\beta\gamma\delta}u_{\alpha\beta}u_{\gamma\delta}\,.\end{split}$
(6)

Before analyzing in detail the physics hidden in Eq. (6), let us notice that,
due to isotropy, the tensorial coefficients of the expansion (5) can be
decomposed in a way analogous to the one reported in (4):

$\displaystyle a_{1}(T)_{\alpha\beta}=a_{1}(T)\,\delta_{\alpha\beta}\,,$ (7)
$\displaystyle
a_{2}(T)_{\alpha\beta\gamma\delta}=a_{2}^{(\mathcal{K})}(T)\delta_{\alpha\beta}\delta_{\gamma\delta}+a_{2}^{(\mathcal{G})}(T)\left(\delta_{\alpha\beta}\delta_{\gamma\delta}+\delta_{\alpha\gamma}\delta_{\beta\delta}-\frac{2}{d}\delta_{\alpha\delta}\delta_{\beta\gamma}\right).$
(8)

A similar decomposition holds for $b$ coefficients as well.

Figure 1: Effect of small, homogeneous deformations $(\delta=\pm
0.1,\,\epsilon=0)$ on a slice of the Mexican hat potential appearing in (6),
at a given temperature $T$. We start with assuming $a_{s}(T)=-2$ and
$b_{s}(T)=1.5$ (black, dashed curve), then apply (10) and its analogous
expression for $b_{s}(T)$, neglecting $\delta^{2}$ terms and taking
$a_{1}(T)=b_{1}(T)=1$. For this choice of coefficients, a compression (red,
$\delta<0$) enhances the condensate, whilst a dilatation (blue, $\delta>0$)
weakens it.

In the remainder of this work, we restrict our attention to homogeneous
deformations of isotropic media with two spatial dimensions ($d=2$). In
Appendix A, we show with rigorous symmetry arguments that homogeneity allows
us to parameterize our deformations of interest with the following strain
tensor:

$u_{lin}=\frac{1}{2}\begin{pmatrix}\delta&\epsilon\\\
\epsilon&\delta\end{pmatrix}.$ (9)

The subscript emphasizes that the strain tensor in the equation above is the
linear one which is valid only for small deformations, i.e
$\delta,\,\epsilon\ll 1$.555In the following, we will distinguish it from the
non-linear Lagrangian strain tensor which will be indicated with $E_{ij}$.
This justifies the expansions in $\delta$, $\epsilon$ performed in the next
subsections.

If $\epsilon=0$, $\delta$ describes a purely volumetric deformation with
$\Delta V/V=\delta$666This definition will be kept also at the nonlinear level
in the following., while in the opposite case ($\delta=0$), $\epsilon$ gives
rise to a purely deviatoric term which modifies the shape and the angles in
the medium without changing its volume.777At linear level $\epsilon/2$ is both
the off-diagonal component of the strain tensor and of the deformation
gradient. At nonlinear level, the two definitions will no longer coincide. We
will keep defining $\epsilon$ as the off-diagonal component of the deformation
gradient. Homogeneity implies that the superfluid condensate $\Psi$ is
constant in space as well.

We now review the main effects that the coupling of elasticity and
superfluidity produces in this framework.

#### Critical temperature

The first important consequence of coupling the solid and superfluid sectors
into the supersolid effective action (6) is that the value of the original
critical temperature $T_{c}^{(0)}$ is modified by the presence of a mechanical
deformation. In particular, using (7) and (9), one finds that the corrected
coefficient of the quadratic term $|\Psi|^{2}$ is now given by:

$a(T,u)=a_{s}(T)+a_{1}(T)\,\delta+a_{2}^{(\mathcal{K})}(T)\,\delta^{2}+a_{2}^{(\mathcal{G})}(T)\,\epsilon^{2}\,,$
(10)

so that it depends explicitly on the mechanical deformation parameters
$\delta,\,\epsilon$. It is then straightforward to obtain the corrections to
the superfluid critical temperature $T_{c}^{(0)}$ induced by the presence of a
solid component in the medium. By identifying the new critical temperature
$T_{c}$ as the zero of the corrected quadratic coefficient $a(T,u)$ in Eq.
(10), we find (respectively, for a purely deviatoric and a purely volumetric
deformation):

$\displaystyle\delta
T_{c}^{(\epsilon)}=T_{c}-T_{c}^{(0)}=-\frac{a_{2}^{(\mathcal{G})}\left(T_{c}^{(0)}\right)}{a_{0}\left(T_{c}^{(0)}\right)}\,\epsilon^{2}\,,$
(11) $\displaystyle\delta
T_{c}^{(\delta)}=T_{c}-T_{c}^{(0)}=-\frac{a_{1}\left(T_{c}^{(0)}\right)}{a_{0}\left(T_{c}^{(0)}\right)}\,\delta-\frac{a_{2}^{(\mathcal{K})}\left(T_{c}^{(0)}\right)}{a_{0}\left(T_{c}^{(0)}\right)}\,\delta^{2}\,.$
(12)

Importantly, we already observe that the correction to the critical
temperature is quadratic in the shear parameter $\epsilon$ but linear in the
volumetric deformation $\delta$. This is a direct consequence of isotropy.

#### Elastic moduli

A second and important consequence which arises from the supersolid effective
action (6) is that the elastic moduli of the original solid are corrected by
the presence of a superfluid condensate $\langle\Psi\rangle\neq 0$. More
precisely, the linear elastic tensor becomes:

$\hat{C}_{\alpha\beta\gamma\delta}=C_{\alpha\beta\gamma\delta}+a_{2}(T)_{\alpha\beta\gamma\delta}|\Psi|^{2}+b_{2}(T)_{\alpha\beta\gamma\delta}|\Psi|^{4}.$
(13)

In the isotropic case, using (7), this simply reduces to:

$\begin{cases}\hat{\mathcal{K}}=\mathcal{K}+a_{2}^{(\mathcal{K})}(T)|\Psi|^{2}+b_{2}^{(\mathcal{K})}(T)|\Psi|^{4}\\\
\hat{\mathcal{G}}=\mathcal{G}+a_{2}^{(\mathcal{G})}(T)|\Psi|^{2}+b_{2}^{(\mathcal{G})}(T)|\Psi|^{4}\end{cases}\,.$
(14)

Let us elaborate more about the correction to the shear modulus. The ground
state in the broken homogeneous phase which arises from the effective action
in Eq.(6) is defined as:

$|\Psi|^{2}(T)=-\frac{a(T)}{2\,b(T)}\,,$ (15)

as it follows from free energy’s minimization. Substituting this expression in
(14), and going in the vicinity of the superconducting transition (so that the
quartic-in-the-condensate term can be neglected), one obtains:

$\delta\mathcal{G}=\hat{\mathcal{G}}-\mathcal{G}=a_{2}^{(\mathcal{G})}|\Psi|^{2}=a_{2}^{(\mathcal{G})}\,\frac{a_{0}}{2\,b}\,\left|T-T_{c}^{(0)}\right|\,,$
(16)

where all the parameters $a_{2}^{(\mathcal{G})},\,a_{0},\,b$ are evaluated at
the original critical temperature $T_{c}^{(0)}$. An analogous formula holds
for the bulk case. Most interestingly, though, we notice that, since the
fraction in (16) is positive, it follows that the corrections (11) and (16)
must have opposite sign:

$\delta\mathcal{G}\cdot\delta T_{c}^{(\epsilon)}<0.$ (17)

As a conclusive remark, we would like to stress that the GL model presented in
this Section does not provide any of the phenomenological coefficients, which
should be derived by other means. In the remainder of this work, we will see
how the adoption of holographic methods, instead, gives us access to those
coefficients, thus completing in a sense the GL model.

##  3 The holographic model

We consider the following four dimensional gravitational action (in natural
units $\hbar=c=1$):

$\mathcal{S}_{grav}=\int
d^{4}x\sqrt{-g}\left\\{\frac{1}{2\kappa^{2}}\left(R-2\Lambda-2m^{2}V(X)\right)-\frac{1}{4e_{0}^{2}}F^{2}-|D\psi|^{2}-M^{2}|\psi|^{2}\right\\},$
(18)

which combines the original holographic superconductor model of Hartnoll_2008
with the generalized axions model of baggioli2015electron . Here, $\Lambda$ is
the AdS4 cosmological constant (so that $\Lambda=-3/L^{2}$ with $L$ being the
AdS length), $F=dA$ is the field strength of a $U(1)$ bulk gauge field
$A_{\mu}$, $\psi$ is a complex scalar field charged under the $U(1)$ symmetry
and $D_{\mu}=\partial_{\mu}-iqA_{\mu}$ is the standard covariant derivative.
The equations of motion of the system allow to set the phase of the complex
scalar $\psi$ to zero, at least at the level of the (homogeneous) background.
Following dimensional analysis, $V(X)$ is a generic, dimensionless scalar
potential defined in terms of the scalar variable
$X\equiv\frac{1}{2}\partial_{\mu}\phi^{I}\partial^{\mu}\phi^{I}$ with
$\phi^{I}$, $I\in\\{x,y\\}$, being a set of two massless real scalar fields
(the axion fields). The function $V(X)$ is not entirely arbitrary as it must
obey specific consistency requirements, derived in baggioli2015electron .
Notice that by setting $m=0$, one recovers the action of Hartnoll_2008  and
the standard holographic superfluid model, which we can then consider as a
limiting case of our model. We adopt Poincaré coordinates:
$\left\\{t,x,y\right\\}$ for the boundary temporal and spatial coordinates,
and $u$ for the radial bulk one. The UV boundary is placed at $u=0$ while the
event horizon of the black brane solution at $u=u_{h}$.

The action (18) is invariant under global shifts in the axion fields:
$\phi^{I}\rightarrow\phi^{I}+a^{I}$, with $a^{I}$ a constant two-dimensional
vector. Because of this property, their equation of motion:

$\partial_{\mu}\left(\sqrt{-g}\,\frac{\partial V}{\partial
X}\,g^{\mu\nu}\partial_{\nu}\phi^{I}\right)=0,$ (19)

is trivially solved by a background configuration which is linear in the
spatial coordinates of the boundary $x^{i}$ and taken for convenience as:

$\phi^{I}={\Theta^{I}}_{j}\,\frac{x^{j}}{L_{0}},$ (20)

where ${\Theta^{I}}_{j}$ is a constant, dimensionless, rank 2 matrix, which we
will identify, in the following Subsection 3.1, with the inverse of the
deformation gradient tensor (about the physical significance of this quantity,
see Appendix A), and $L_{0}$ is a characteristic lengthscale necessary to have
the matrix $\Theta$ dimensionless. All in all, the axion fields $\phi^{I}$ are
dimensionless, as expected.

We will later identify the reference configuration with
${\Theta^{I}}_{j}\equiv{\delta^{I}}_{j}$. Therefore, $L_{0}$ can be thought to
be related to the distance between the atomic positions in the reference
configuration. In order for this holographic model to be dual to a field
theory exhibiting spontaneous translational symmetry breaking, we will need to
require the background solution $\phi_{ref}^{I}=x^{I}/L_{0}$ to correspond to
the expectation value of the scalar operators dual to the bulk axion fields.
This is obtained by placing further constraints on the $V(X)$ function, as
shown later (see Baggioli:2021xuv  for more details).

Following baggioli2020black , we parameterize the constant matrix appearing in
(20) as:

$\Theta=\alpha\begin{pmatrix}\cosh{\left(\Omega/2\right)}&\sinh{\left(\Omega/2\right)}\\\
\sinh{\left(\Omega/2\right)}&\cosh{\left(\Omega/2\right)}\end{pmatrix}\,.$
(21)

We also take the following ansatz for the rest of the bulk fields:

$\begin{gathered}ds^{2}=\frac{L^{2}}{u^{2}}\left(-f(u)e^{-\chi(u)}dt^{2}+\frac{du^{2}}{f(u)}+\gamma_{ij}(u)dx^{i}dx^{j}\right),\\\\[1.0pt]
\gamma(u)=\begin{pmatrix}\cosh h(u)&\sinh h(u)\\\ \sinh h(u)&\cosh
h(u)\end{pmatrix},\qquad A=A_{t}(u)dt,\qquad\psi=\psi(u),\end{gathered}$ (22)

with $\gamma(u)$ obeying the property $\det(\gamma)=1$. We restrict to black-
brane solutions where $f(u)=0$ at $u=u_{h}$. The bulk equations of motion
which follow are presented in Appendix B together with a more detailed
analysis of the symmetries of our system and their meaning. Such analysis
allows us to set the AdS length $L$ to unity ($L=1$); we then make a choice
for the other parameters appearing in the action (18), namely:
$\kappa/L=1/\sqrt{2}$, $e_{0}=1$, $mL=1$, $q=3$ and $(ML)^{2}=-2$. This choice
sets the UV asymptotics of the charged scalar field $\psi$ to be:

$\psi\simeq\psi^{(1)}u+\psi^{(2)}u^{2}+\dots\,.$ (23)

Assuming a standard quantization scheme, the leading term in this expansion,
$\psi^{(1)}$, corresponds to the source of the dual operator $\mathcal{O}$ and
the subleading one, $\psi^{(2)}$, to its expectation value
$\langle\mathcal{O}\rangle$. Background solutions in the broken phase
($T<T_{c}$) are found by solving numerically the bulk equations of motions.
Details are reported in Appendix B. As a benchmark model, we will consider a
potential of the form:

$V(X)\,=\,X^{N}\,,\qquad\text{with}\qquad N>5/2\,.$ (24)

The choice of the exponent guarantees that (in the standard quantization
scheme) the operators dual to the axion fields $\phi^{I}$ break the
translational invariance of the dual field theory spontaneously, making such a
system a solid. For a complete description and understanding of this mechanism
we refer to Alberte:2017oqx ; Ammon:2019wci ; Baggioli:2021xuv . For the
remainder of this work, we set $N=3$. In making such a choice, we focus on a
field theory that has no canonical kinetic term. This may raise a question on
whether the time evolution of the system is non-unitary; however, we stress
that the theory at hand does not have any higher derivative kinetic term,
rather a higher power of the usual two-derivative one. As such, no unitarity
problem arises; instead, the unusual feature of theories of this form is that
they make sense only around non-trivial vacuums (and not around the trivial
one $\phi^{I}=0$ where the model becomes strongly coupled). We refer to
Alberte:2017oqx  for a more detailed discussion as for why that is the case
and which are the physical implications.

Finally we can define the the temperature $T$, the charge density $\rho$ and
the chemical potential $\mu$ of the dual field theory using the standard AdS-
CFT dictionary zaanen2015holographic ; Baggioli:2019rrs :

$\displaystyle
T\,=\,-\frac{f^{\prime}(u_{h})}{4\pi}\,e^{-\chi(u_{h})/2}\,,\qquad\mu\,=\,A_{t}(0)\,,\qquad\rho\,=\,-A_{t}^{\prime}(0)\,,$
(25)

where primes indicate differentiation with respect to the radial coordinate
$u$.

To summarize, any state of the boundary theory can be then parameterized by
the value of the three following dimensionless parameters:

$\\{TL_{\alpha},\,\rho L_{\alpha}^{2},\,\Omega\\},$ (26)

where we have defined $L_{\alpha}=L_{0}/\alpha$. Unsheared configurations
corresponds to $\Omega=0$.

###  3.1 On the definition of mechanical deformations in holography

Before proceeding to the results of our work, let us briefly describe how the
holographic model at hand naturally includes mechanical deformations. Our
notations follow closely the ones adopted in the EFT framework of
Alberte:2018doe ; Pan:2021cux . Importantly, we will assume the deformations
to be finite, or even large, and we will therefore consider the full framework
of nonlinear elasticity. For a brief summary of nonlinear elasticity theory we
refer to Appendix A; a more in depth treatment can be found in
ZAMM:ZAMM19850650903 .

The starting point of our analysis is the definition of the Lagrangian strain
tensor (see Armas:2019sbe ). Let us consider a medium in a space equipped with
a background flat metric $g_{\mu\nu}=\eta_{\mu\nu}$. The distance between
points in such a medium can be computed by using the crystal metric defined
as:

$ds^{2}_{cr}=h_{IJ}\left(e^{I}_{\mu}\,dx^{\mu}\right)\left(e^{I}_{\nu}\,dx^{\nu}\right),$
(27)

where $\mu$, $\nu$ are indices running on spacetime coordinates, and
$e^{I}_{\mu}(x)$ are a set of one-form fields, with $I\in\\{1,...,d\\}$.
Deformations are then defined with respect to an initial reference
configuration:

$ds^{2}_{ref}=h^{ref}_{IJ}\left(e^{I}_{\mu}dx^{\mu}\right)\left(e^{I}_{\nu}dx^{\nu}\right)\,.$
(28)

For practical purposes, we will later make the convenient choice
$h^{ref}_{IJ}=\delta_{IJ}$.

Once a reference configuration is selected, it is straightforward to define
the Lagrangian strain tensor $E_{IJ}$ chaikin_lubensky_1995  as:

$ds^{2}_{cr}-ds^{2}_{ref}=2E_{IJ}\left(e^{I}_{\mu}dx^{\mu}\right)\left(e^{I}_{\nu}dx^{\nu}\right)\quad\Longrightarrow\quad
E_{IJ}=\frac{1}{2}\left(h_{IJ}-h^{ref}_{IJ}\right).$ (29)

In absence of defects, the medium can be smoothly parameterized by mean of the
crystal fields $\varphi^{I}$ which serve as the material coordinates (see
Appendix A for more details). Then, the one forms $e^{I}_{\mu}(x)$ can be
written as:

$e^{I}_{\mu}=\partial_{\mu}\varphi^{I}\,.$ (30)

The material coordinates $\varphi^{I}$ are related to the axion fields
$\phi^{I}$ appearing in our holographic model via the simple identification:

$\varphi^{I}=L_{0}\,\phi^{I}.$ (31)

Making such a combination is necessary to take into account the fact that the
axion fields $\phi^{I}$ are dimensionless Goldstone phases, and cannot be
directly identified with spatial coordinates. As a direct consequence of
Eq.(31), the reference configuration888Within the holographic picture, this
configuration does not minimize the free energy and possesses a residual
background stress labelled crystal pressure in Armas:2019sbe . Nevertheless,
one can still consider linearized perturbations around such a state, and show
that their dynamics is perfectly consistent with the hydrodynamics
expectations Ammon:2020xyv ; Baggioli:2020edn . is then given by:

$\varphi_{ref}^{I}={\delta^{I}}_{j}\,x^{j}.$ (32)

Using the previous results, the inverse of the crystal metric can easily be
written in terms of the crystal fields as:

$h^{IJ}=\left(h_{IJ}\right)^{-1}=g^{\mu\nu}\partial_{\mu}\varphi^{I}\partial_{\nu}\varphi^{J}.$
(33)

This provides us with everything that we need to compute the strain tensor
(29) in our holographic model. By using (31) along with (20) in the above
expression, we find:

$h^{IJ}=\delta^{ij}{\Theta^{I}}_{i}{\Theta^{J}}_{j}\quad\Longleftrightarrow\quad
h^{-1}=\Theta^{T}\Theta\,.$ (34)

Using (29), we then obtain:

$E=\frac{1}{2}\left((\Theta^{-1})^{T}\Theta^{-1}-\mathbb{I}\right)\,.$ (35)

By comparing this expression with the standard definition of the Lagrangian
strain tensor in nonlinear elasticity theory, Eq.(72), we find the important
result:

$\Xi=\Theta^{-1},$ (36)

where $\Xi$ is the deformation gradient tensor. As expected, the inverse of
the deformation gradient tensor is given by the Jacobian matrix of the
material coordinates with respect to the spacetime ones.

In the rest of the manuscript, we will consider two separate cases: (I) a pure
volumetric, shape-preserving deformation, and (II) a purely deviatoric strain,
which preserves the volume of any subregion of the medium throughout the
deformation process.

The first scenario is realized by setting $\Omega=0$ and identifying:

$\delta=\frac{\Delta
V}{V}\,=\,\mathrm{Tr}\left[E_{ij}\right]\,=\,\mathrm{det}\,\Xi\,=\left(\frac{1}{\alpha^{2}}-1\right)\,.$
(37)

From this expression, a dilatation, $\delta>0$, corresponds to the choice
$\alpha<1$ and a compression with $\alpha>1$. This is consistent with the fact
that $L_{\alpha}$ is related to the lattice size of the system; the case
$\alpha=1$ corresponds to absence of mechanical deformations (with respect to
the reference configuration (32)). Finally, a linear isotropic deformation
corresponds to an infinitesimal shift $\alpha\rightarrow\alpha+\Delta\alpha$,
under which:

$\delta=\frac{\Delta V}{V}=-2\,\Delta\alpha,$ (38)

where we have fixed $\alpha=1$. This minus sign is consistent with previous
arguments baggioli2015electron ; Baggioli:2019elg ; Armas:2020bmo  and with
the physical picture that $\Delta\alpha>0$ implies a reduction of the solid
length-scale $L_{\alpha}$.

The second scenario, that of a deviatoric strain that causes no changes in the
volume of any subregion of the medium, can be realized by setting $\alpha=1$
and by noticing that the shear strain is parameterized by:

$\Xi_{xy}\equiv\frac{1}{2}\,\epsilon=-\sinh{\left(\frac{\Omega}{2}\right)}\,.$
(39)

As anticipated, at nonlinear level, we define $\epsilon$ through the off-
diagonal component of the deformation gradient tensor, which no longer exactly
coincides with the off-diagonal component of the nonlinear strain tensor
$E_{xy}$.999For completeness, we report that:
$E_{xy}=-\frac{\sinh{\Omega}}{2},$ (40) which is slightly different from
$\Xi_{xy}$ in Eq.(39) in the nonlinear regime. In the linear regime, the two
coincide:  $\Xi^{lin}_{xy}=E^{lin}_{xy}=-\frac{\Omega}{2}\,.$ (41)

Finally, the minus sign appearing in (39) can be dropped if we extend the
$SO(2)$ symmetry to the full orthogonal group $O(2)$. It can indeed easily be
seen that, given the rank 2 representation of the reflection with respect to
the $x$ axis, which we denote by $\sigma_{x}$:

$\sigma_{x}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\,\Rightarrow\,\begin{pmatrix}\cosh{\left(\Omega/2\right)}&-\sinh{\left(\Omega/2\right)}\\\
-\sinh{\left(\Omega/2\right)}&\cosh{\left(\Omega/2\right)}\end{pmatrix}=\sigma_{x}\begin{pmatrix}\cosh{\left(\Omega/2\right)}&\sinh{\left(\Omega/2\right)}\\\
\sinh{\left(\Omega/2\right)}&\cosh{\left(\Omega/2\right)}\end{pmatrix}\sigma_{x},$
(42)

so that the minus sign appearing in (39) can be reabsorbed by the action of
the reflection $\sigma_{x}$. Because of this reason, the minus sign will be
dropped in the rest of the manuscript.

In summary, to compare the effective description presented in Section 2 with
the results from holography, we just need to use the simple identifications:

$\delta=\left(\frac{1}{\alpha^{2}}-1\right)\,,\qquad\epsilon=2\sinh\left(\frac{\Omega}{2}\right)\,,$
(43)

where now the parameters $\delta,\,\epsilon$ are taken at full nonlinear
level. The linear regime can be re-obtained by taking the limits
$\alpha,\,\Omega\ll 1$.
Let us emphasize that since the reference configuration defined in Eq.(32)
does not minimize the free energy, all configurations $\phi^{I}=\alpha
x^{I}/L_{0}$ could be taken as legitimate reference choices. Given an initial
state with $\alpha=\alpha_{i}$ and a final state with $\alpha=\alpha_{f}$, the
parameter $\delta$ encoding a purely volumetric deformation can be more
generally defined as:

$\delta=\left(\frac{1}{\alpha_{f}^{2}}-\frac{1}{\alpha_{i}^{2}}\right),$ (44)

which boils down to the previous definition (43) when $\alpha_{i}=1$ and
$\alpha_{f}=\alpha$, as in our choice of reference configuration.

##  4 Results

In this section, we outline the results obtained via the holographic model
(18) and we compare them with the effective theory of Section 2.

###  4.1 The phase diagram of the unsheared solid

As a first step of our study, we build the phase diagram of the theory
described by (18) for unsheared ($\epsilon=0$) configurations. More precisely,
we compute the dimensionless critical temperature $T_{c}L_{\alpha}$, at which
the global $U(1)$ symmetry is spontaneously broken, in terms of the
dimensionless charge density $\rho L_{\alpha}^{2}$. Unlike the global $U(1)$,
translational invariance is always broken, indicating that our system in the
normal phase ($T>T_{c}$) is a solid phase with finite shear rigidity. From a
physical perspective, we can therefore identify the broken phase with the
supersolid phase in which both translations and the $U(1)$ symmetry are
spontaneously broken.

Figure 2: Left: The critical temperature as a function of the charge density
for unsheared configurations ($\epsilon=0$). Colored circles indicate the
following values: green ($\rho L_{\alpha}^{2}=1$), red ($\rho
L_{\alpha}^{2}=2$) and cyan ($\rho L_{\alpha}^{2}=4$). The grey area is the
set of normal states that would be superfluid in absence of elasticity (cf.
Hartnoll_2008 ). Right: The normalized condensate as a function of the
temperature, normalised by its critical value, for the highlighted values of
$\rho L_{\alpha}^{2}$. Inset: A zoom close to the transition, where the
condensate scales like $|\Psi|\sim\sqrt{t}$, with $t\equiv|T-T_{c}|/T_{c}$
being the reduced temperature.

After these few premises, we present, in the left panel of Fig.2, the phase
diagram for the unsheared configurations of our system. The blue shaded area
below the solid curve indicates the supersolid phase. The dashed grey line,
instead, is the critical temperature in absence of any solid component,
corresponding to the original model of Hartnoll_2008 , with the grey area
indicating all those states that would break the $U(1)$-symmetry if the system
were not a solid. The grey shaded curve follows a square-root law as a
consequence of the fact that, in the model of Hartnoll_2008 , the
dimensionless critical temperature cannot be tuned,
$T_{c}\,/\sqrt{\rho}=const$. This does not happen in presence of translational
symmetry breaking, due to which a second dimensional scale – the lattice
spacing $L_{\alpha}$ – appears.

We observe that higher values for the adimensional charge density correspond,
in general, to a higher critical temperature $T_{c}$. As expected, we also
find that the effects of a solid component, $L_{\alpha}\neq 0$, become more
important in the small charge density limit. On the contrary, for large $\rho
L_{\alpha}^{2}$, the bulk axions fields can be neglected and the system
effectively behaves as in the original superfluid model of Hartnoll_2008 .

In the right panel of Fig.2 we show the superfluid condensate
$\langle\Psi\rangle$ as a function of the temperature for three different
values of parameters (highlighted in the left panel); the larger the value of
the adimensional charge density, the larger the (adimensional) superfluid
condensate. We also find, in accordance with Hartnoll_2008 , that the
supersolid transition is a mean field one, as emphasized in the inset where
the typical square-root in the reduced temperature $t$ behavior of the
condensate is observed:

$\langle\Psi\rangle\sim\Psi_{0}\,t\,,\qquad t\equiv\frac{T-T_{c}}{T_{c}},$
(45)

with the slope $\Psi_{0}$ too increasing with the charge density.

###  4.2 Response to shear deformations across the supersolid phase transition

Among the reasons behind holography’s success is the relative ease with which
it allows to compute Green functions. In this section, we take advantage of
this feature to characterise how the holographic supersolid system reacts
mechanically under an infinitesimal (within linear response theory) shear
deformation. We will use the holographic prescription (see for example
Baggioli:2019rrs ) to calculate numerically the retarded Green function of the
off-diagonal component of the stress tensor $T^{xy}$, which at zero momentum
and small frequency is101010Here, we have already subtracted the standard
contribution $\delta T^{xy}\sim p\,h_{xy}$, with $p$ being the thermodynamic
pressure, which appears in the holographic computation as a counterterm from
the holographic renormalization. See for example Burikham:2016roo .:

$\mathcal{G}^{R}_{T^{xy}T^{xy}}(k=0,\omega)\simeq\mathcal{G}-i\omega\eta+O(\omega^{2})\,,$
(46)

where $\mathcal{G}$ is the shear modulus, and $\eta$ the corresponding
viscosity. We will take advantage of this expansion to write the corresponding
Kubo formulas:

$\mathcal{G}=\lim_{\omega\to
0}\mathrm{Re}\,\mathcal{G}^{R}_{T^{xy}T^{xy}}(0,\omega),\quad\eta=\lim_{\omega\to
0}-\frac{\mathrm{Im}\,\mathcal{G}^{R}_{T^{xy}T^{xy}}(0,\omega)}{\omega}\,.$
(47)

From a holographic point of view, the Green function (46) is computed by
introducing a spatial, transverse metric perturbation of the metric, $\delta
g_{xy}(u,t)=\zeta(u)e^{-i\omega t}$. One then needs to solve numerically (with
appropriate boundary conditions, outlined e.g. in Alberte:2016xja ) its
finite-frequency equation of motion:

$\zeta^{\prime\prime}+\left(\frac{f^{\prime}}{f}-\frac{\chi^{\prime}}{2}-\frac{2}{u}\right)\zeta^{\prime}+\left(\frac{\omega^{2}e^{\chi}}{f^{2}}-\frac{L^{2}m^{2}\partial_{u}V(X)}{uf}\right)\zeta=0\,.$
(48)

Near the UV boundary:

$\zeta(u)\simeq\zeta_{0}\,(1+...)+\zeta_{3}\,u^{3}\,(1+...)\,,$ (49)

so that we can read off:

$\mathcal{G}^{R}_{T^{xy}T^{xy}}(\omega)=-\frac{3}{2}\frac{\zeta_{3}}{\zeta_{0}}\,.$
(50)

Albeit the theory outlined in Section 2 provides predictions only for the real
part of the Green function (46) (the shear modulus), we expand the scope of
this section to include results for the shear viscosity $\eta$ as well. We do
this for two reasons: on one hand, holography provides such results with no
additional effort, making our analysis more complete; on the other hand, and
most importantly, because we are interested in studying the Kovtun-Son-
Starinets (KSS) bound violation in the system at hand. Such violation is
expected on the general ground that the metric perturbation is massive – as
outlined in Hartnoll_bumpyBH . KSS bound violations in a holographic axion
models are widely reported in the literature Hartnoll_bumpyBH ;
Burikham:2016roo ; Alberte:2016xja ; Ling:2016ien  but definitely not totally
understood from a physical perspective Baggioli:2020ljz . The addition of a
superfluid transition on top of these affects has not been the subject of a
study yet and it is therefore interesting to be pursued.

For completeness, we show the response of our holographic model to shear
deformations at finite charge density in Appendix C. From there, one can see
that the presence of charge density diminishes the rigidity of the system (the
shear modulus) and increases the value of the $\eta/s$ ratio towards its
maximal value corresponding to the KSS bound. We also find that the effects of
the charge density are more relevant at small temperature. Finally, we briefly
show how the charge density affects the nonlinear elastic response. Also in
that case we observe a decrease in the stress which is more dramatic at small
shear strains.

####  4.2.1 Shear modulus

From a macroscopic point of view, the most important physical difference
between a fluid and a solid is the presence, in the latter, of a finite static
shear modulus, which defines the shear rigidity of the system; such a quantity
vanishes in fluid phases of matter.111111See Baggioli:2019jcm ;
Baggioli:2021ntj  for more details about the subtle distinction between solids
and liquids. In a wide class of holographic axion models, to which the one
considered in this work belongs, the shear modulus has already been computed
(in Alberte:2015isw ; Alberte:2016xja ; Baggioli:2019elg ; Andrade:2019zey ),
and it has been directly used to match the speed of the transverse phonon
excitations extracted numerically from the QNMs of the system with the
hydrodynamic expectations Alberte:2017oqx . In this work, we investigate how
the additional $U(1)$ symmetry breaking affects the shear rigidity properties
of a system that condenses in a supersolid phase below a given temperature.

Figure 3: Left: Dimensionless shear modulus as a function of $TL_{\alpha}$.
Data series are identified by their colour, see Fig. 2. Dashed lines report
the would-be value for the shear modulus in absence of a superfluid
condensate. Inset: The correction to the shear modulus near the transition, as
a function of $|\Psi|^{2}$, showing the linear behaviour predicted by (16).
Right: The coefficient $a_{2}^{(\mathcal{G})}$ appearing in (16) as a function
of $\rho L_{\alpha}^{2}$.

In Fig.3, we plot the dimensionless shear modulus in function of the
adimensional temperature, spanning both the normal and the supersolid phases.
We observe a strong enhancement of the shear modulus $\mathcal{G}$ taking
place in the supersolid phase: coloured solid lines indicate $\mathcal{G}$ in
the full system, while dashed ones report its would-be value in a system that
does not condense. Upon the transition the shear modulus is continuous, but
its first derivative is not. Near the phase transition, at small values of the
superfluid condensate $\langle\Psi\rangle$, we observe a linear-
in-$|\Psi|^{2}$ increase of the shear modulus:

$\delta\mathcal{G}\sim|\Psi|^{2},$ (51)

where $\delta\mathcal{G}$ is defined as the difference between the shear
modulus in the supersolid phase and the corresponding value for the solid
phase with no superfluid condensate. The inset of the left panel of Fig.3
shows more clearly this feature.

Our numerical findings are in agreement with the EFT expectations of Eq. (16),
thus allowing us to derive the value of the EFT parameter denoted therein as
$a_{2}^{(\mathcal{G})}$ for a set of values of the adimensional charge
density. In the right panel of Fig.3, we show that such a parameter displays a
distinct non-monotonic behaviour in function of $\rho L_{\alpha}^{2}$. More
precisely, it increases for small values of $\rho L_{\alpha}^{2}$ up to a
maximum $(\rho L_{\alpha}^{2})_{max}\approx 1$, before decreasing
monotonically towards zero. We find it significant that the maximum value,
which signals the maximum interplay between superfluidity and elasticity,
occurs exactly when the parameter $\rho L_{\alpha}^{2}$ is $\mathcal{O}(1)$.
Physically that is the point at which the effects of charge density and shear
ridigity are comparable and none of the them can be neglected. On the
contrary, the regimes where $\rho L_{\alpha}^{2}\ll 1$ or $\rho
L_{\alpha}^{2}\gg 1$, corresponds to situations where one of the two
dimensional scales prevails on the other rendering the other irrelevant.
Finally, we find that for, large charge density $\rho L_{\alpha}^{2}\gg 1$,
the effects of a small superfluid condensate on the solid behaviour
(parameterized by the leading order coefficient $a_{2}^{(\mathcal{G})}$) are
negligible. Away from the transition (for large values of the condensate), on
the contrary, the effect seems to be stronger for those states. At the same
time, we do expect that for small $\rho L_{\alpha}^{2}$ the effects of the
superfluid condensate are vanishing as well since the latter tends to zero
(see left panel of Fig.2).

Interestingly, a sudden increase of the shear modulus at the supersolid
transition has been reported experimentally doi:10.1063/1.2883895 ; Day2007 ;
Syshchenko_2009 .

####  4.2.2 Shear viscosity

Figure 4: Left: Dimensionless shear viscosity as a function of $TL_{\alpha}$.
Data series are identified by their colour, see Fig. 2. Dashed lines report
the would-be value for the shear viscosity in absence of a superfluid
condensate. The discontinuous points correspond to the critical temperature
and the onset of supersolidity. Right: KSS bound violation for the same data
series; the dashed lines indicate would-be values in absence of condensation,
while the dot-dashed one is the KSS bound: $\frac{\eta}{s}=\frac{1}{4\pi}$.

We now turn to the information that we can extract from the imaginary part of
the Green function (46) – the shear viscosity $\eta$. Results coming from the
numerical integration of the equation of motion for the transverse metric
perturbation are reported in Fig.4. The qualitative features of the
adimensionalized $\eta$ at the transition temperature – left panel – are
similar to the ones reported for the shear modulus $\mathcal{G}$. The shear
viscosity appears to be continuous upon the transition, unlike its first
derivative. Importantly, the correction is negative implying that the
viscosity in the supersolid phase is always lower than in the solid one (with
the same choice of parameters). This is intuitively in accordance with the
physical idea that in the supersolid phase part of the system, i.e. the
superfluid component (cf. two-fluid model), flows without friction, resulting
in an overall lower viscosity. In the right panel of Fig.4 we report the
behavior of the viscosity-to-entropy ratio $\eta/s$. There, $s$ is the entropy
density of the dual field theory extracted from the Bekenstein-Hawking entropy
of the black hole in the bulk. With our notations, we have:

$s=\frac{2\pi}{u_{h}^{2}}\,,$ (52)

with $u_{h}$ being the radial coordinate’s value on the horizon. As
anticipated before, the KSS bound is violated for all values of the
adimensional charge density considered, at all temperatures and both in the
solid and supersolid phases. This is expected because of the massive nature of
gravity in the model at hand. However, we find that the condensation makes
such violation even more pronounced. Interestingly, a pure holographic
$s$-wave superfluid is reported to not violate the KSS bound
Natsuume_shearvisc . From this fact, we draw the conclusion that the mechanism
that enhances the KSS bound violation is to be found in the gravity-mediated
interaction between the condensate and the axions – that, in our model, play
the role of the phonons. From the dual field theory perspective, this
interaction is rooted in the low-energy description of the supersolid state
and responsible for most of the effects described in this work.

###  4.3 Critical temperature under mechanical deformations

Now we turn to investigate the supersolid critical temperature $T_{c}$ of the
system, with the specific aim to understand how it is affected by the
parameters describing the deformations of the solid.

We start by considering a purely deviatoric deformation of the system,
described by the shear strain parameter $\epsilon$.

Figure 5: Left: The critical temperature as function of the shear parameter
$\epsilon$ for the notable values of $\rho L_{\alpha}^{2}$ introduced in
Fig.2. The starting ($\epsilon=0$) point of each curve should be identified
with the corresponding highlighted point in Fig.2. The inset shows how small
deformations induce a quadratic decrease of the critical temperature as a
function of the shear parameter $\epsilon$, again in agreement with the GL
theory (11). Right: The normalized coefficient $a_{0}$ appearing in (11) as a
function of the adimensional charge density.

In the left panel of Fig.5, we show the dimensionless supersolid critical
temperature in function of the shear strain $\epsilon$ for some values of the
dimensionless charge density. The value of $\rho L_{\alpha}^{2}$ does not seem
to be affecting the qualitative behaviour of the curves, which all start
horizontally with a negative concavity, then later have an inflection point
while continuing to decrease monotonically. No curve is observed to reach a
vanishing critical temperature for a finite value of the shear strain
parameter $\epsilon$; it is possible indeed to check that the zero temperature
state violates the near horizon BF bound, so that the supersolid phase is
expected to survive up to arbitrarily large strains and arbitrarly low
temperatures. Let us notice that the larger the charge density the larger the
shear strain needed to affect significantly the critical temperature.

By zooming in at small values of the strain, we observe a universal behavior:

$|\delta T_{c}|\,\sim\,\epsilon^{2}\,,$ (53)

where $\delta T_{c}$ is the difference between the critical temperature at
zero strain and that at finite strain. Such behavior is plotted, for some
values of $\rho L_{\alpha}^{2}$, in the inset of the left panel of Fig.5.

Once more, our numerical results are consistent with the prediction of the
effective field theory in Section 2, specifically with Eq.(11). Using the
values of the coefficient $a_{2}^{(\mathcal{G})}$ previously extracted, we are
now able to obtain the second parameter appearing in the EFT description:
$a_{0}$. We show the value of this parameter in function of the dimensionless
charge density in the right panel of Fig.5. In the regime considered, we
observe a monotonically decreasing behaviour. Unfortunately, our numerical
scheme does not allow us to have control over the computations at small $\rho
L_{\alpha}^{2}$ and we cannot therefore exclude a possible non-monotonic
behaviour for small values of the charge density. However, $a_{0}$ is a
parameter of the pure superfluid phase, unlike $a_{2}^{(\mathcal{G})}$, which
describes its interplay with the breaking of translations; therefore, a
qualitatively different behavior would not be surprising.

Figure 6: Left: Effects of a pure shear transformation ($\epsilon=0.5$) on a
square lattice, and definition of the pure shear angle $\Delta\Theta_{xy}$.
Right: An alternative visualization of the phase diagram at fixed charge
density ($\rho L_{\alpha}^{2}=4$) in terms of the shear angle
$\Delta\Theta_{xy}$. The blue dashed lines indicates the range of values in
which our numerical method loses precision and numerical extrapolation must be
used.

Notice that the curves of the left panel of Fig.5 provide the same information
of the phase diagram of the system at fixed charge density $\rho
L_{\alpha}^{2}$: indeed every $(T,\,\epsilon)$ point below the appropriate
critical curve lies within the supersolid phase, while the ones above in the
normal solid phase. Here, we present a more direct way to visualize the finite
shear strain by introducing, instead of the shear strain parameter $\epsilon$,
the shear angle $\Delta\Theta_{xy}$, whose meaning is illustrated in the left
panel of Fig.6. In terms of our original parameter $\epsilon$, which is
defined as the off-diagonal component of the deformation gradient tensor
$\Xi$, the finite angle can be expressed as:

$\Delta\Theta_{xy}=\arctan\left(\frac{\epsilon}{\sqrt{4+\epsilon^{2}}}\right)\,.$
(54)

Notice that the support of this function is the interval
$(-\frac{\pi}{4},\,\frac{\pi}{4})$. This is indicative of the fact that for a
pure shear , as opposed to a simple shear, the maximum shear angle, defined as
the one where the systems collapses on a line, is $\pm\frac{\pi}{4}$.

Exploiting this relation, we can provide an alternative visualization for the
fixed-density phase diagram, presented in the right panel of Fig.6 for one
$\rho L_{\alpha}^{2}$ value. Inside the lobe-shaped curve, the system is
supersolid, outside it is in its normal phase. Qualitatively similar curves
can be found for different values of the dimensionless charge density.

Then, we move to analyze the opposite scenario of a purely volumetric
deformation in which the shear strain $\epsilon$ is taken to be zero. A finite
volume change $\delta=\Delta V/V$ is parameterized by the bulk parameter
$\alpha$ as shown in Eq.(44).

As a consequence of keeping constant the combination $\rho L_{\alpha}^{2}$, we
find that:

$T_{c}\,L_{\alpha_{i}}=T_{c}^{\prime}\,L_{\alpha_{f}}\quad\longrightarrow\quad\frac{T_{c}^{\prime}}{T_{c}}\,=\,\frac{\alpha_{i}}{\alpha_{f}}$
(55)

where $T_{c}^{\prime}$ is the critical temperature after the deformation, and
$T_{c}$ the one before.

Figure 7: Left:  The universal function regulating the behavior of the
critical temperature under isotropic compressions ($\delta<0$) and dilatations
($\delta>0$). Right:  The corrections to the critical temperature from a
purely volumetric deformation $\delta$, for the notable values of adimensional
charge density defined in Fig.2.

For simplicity, and without loss of generality, let us then consider (55) with
the reference state $\alpha_{i}=1$. One would obtain:

$T_{c}^{\prime}=T_{c}\,\alpha,$ (56)

after the deformation, where we have identified $\alpha_{f}=\alpha$. To
compare with the results of Section 2, we need to use the volumetric
deformation parameter $\delta$ as defined in (43), obtaining thus:

$T_{c}^{\prime}=\frac{T_{c}}{\sqrt{1+\delta}}\approx
T_{c}\left(1-\frac{1}{2}\delta+\frac{3}{8}\delta^{2}+...\right),$ (57)

where the first half of the equality is valid all orders in $\delta$, while
the second half follows from the Taylor expansion of the right hand side for
small deformations, $\delta\rightarrow 0$. The fully non-linear results can be
then expressed as:

$\frac{T_{c}(\delta)}{T_{c}(0)}=\frac{1}{\sqrt{1+\delta}}\,,$ (58)

plotted in the left panel of Fig.7. The behavior of the function suggests
that, whenever the system is compressed ($\delta<0$), the critical temperature
rises, viceversa, whenever it undergoes a dilatation ($\delta>0$) it
decreases. Some notable additional features are the singularity at
$\delta=-1$, which is the limit of infinite compression at which the system
shrinks to a point, and the fact that the function has no zero; this latter
fact provides a good consistency check, as in our model it is the value of
$\rho L_{\alpha}^{2}$ which decides whether the system must have a supersolid
phase or not, regardless of the value of $L_{\alpha}$; as such, the zero
temperature broken phase must resist for all values of the deformation
parameter. In the right panel of Fig.7, we show the effect of small
deformations around the reference configuration for various values of $\rho
L_{\alpha}^{2}$. The linear slope increases with the charge density $\rho
L_{\alpha}^{2}$.

Figure 8: The corrections to the critical temperature from a volumetric strain
appearing in Eq.(12) in function of $\rho L_{\alpha}^{2}$. Left:  The linear
coefficient $a_{1}$. Right:  The quadratic term $a_{2}^{(\mathcal{K})}$.

Using the small deformation expansion in Eq. (57), it is possible to extract
the coefficients $a_{1}$ and $a_{2}^{(\mathcal{K})}$ of the EFT of Section 2
by comparing it with (12). They are:

$a_{1}=\frac{1}{2}\,a_{0}\,T_{c},\qquad
a_{2}^{(\mathcal{K})}=-\frac{3}{8}\,a_{0}\,T_{c}\,.$ (59)

The results are reported in Fig.8. The behavior of these coefficients, which
is qualitatively the same as one is a multiple of the other, is similar to the
one of the $a_{2}^{(\mathcal{G})}$ coefficient, reported in the right panel of
Fig.3. The only notable differences come from the fact that the maximum seems
to be shifted towards higher values of $\rho L_{\alpha}^{2}$, but still around
order $\mathcal{O}(1)$, and that it does not appear evident whether or not the
coefficient is going to vanish for large values of the adimensional charge
density.

###  4.4 Interplay between large shear deformations and condensation

In the previous sections we outlined extensive evidence of the fact that the
model at hand exhibits a rich interplay between elasticity and superfluidity.
We characterised how deformations (small and large) change the critical
properties of the system, and how the condensation – even far from the
transition – does have an impact on the response to linear mechanical
deformations. In this section, we investigate the fully nonlinear regime, to
estimate how large (shear) deformations affect the condensation, and how the
stress response to large deformations is changed by condensation.

####  4.4.1 Strain-induced transition

By looking at the curves of Fig.5, it seems quite clear that this model
predicts the possibility to have an isothermal, shear-induced transition back
to the normal state.

We report the results for such a study in Fig.9. We find that, starting from
an adimensional temperature below the critical one
($T^{*}L_{\alpha}<T_{c}L_{\alpha}$) and keeping it fixed throughout the
process, the adimensional condensate starts decreasing as the shear strain
parameter $\epsilon$ is increased, until a critical value of the latter occurs
where the supersolid phase disappears. Such critical value should be
identified with the solution of the equation:

$T_{c}L_{\alpha}\left(\rho L_{\alpha},\,\epsilon\right)=T^{*}L_{\alpha},$ (60)

and indeed this is what we find, up to the precision of our numerical methods.

Interestingly, we can also study the condensate near the transition, to see
what kind of transition the shear-strain induced one is. In the inset of
Fig.9, evidence that it is a mean field transition is reported. The condensate
indeed obeys a square-root law in the reduced shear strain parameter, which we
have defined in analogy with the reduced temperature as:

$\epsilon_{r}=\frac{|\epsilon-\epsilon_{c}|}{\epsilon_{c}},$ (61)

with $\epsilon_{c}$ being the critical shear strain parameter above which no
supersolid phase is present.

Figure 9: The superfluid condensate throughout an isothermal pure shear
deformation process. Solid lines indicate processes at $TL_{\alpha}=0.2$,
while the dashed line at $TL_{\alpha}=0.05$. The endpoints of the curve are in
accordance with the ones predicted in Fig.5. Inset:  The dimensionless
condensate close to the strain induced transition, as a function of the square
root of the reduced strain parameter
$\epsilon_{r}\equiv|\epsilon-\epsilon_{c}|/\epsilon_{c}$ with $\epsilon_{c}$
the point where superconductivity disappears. The linear behavior of the curve
points out the mean-field theory nature of the transition. Notice that in this
panel, $\rho$ has been used to adimensionalise the condensate, for the sake of
a better graphical representation.

####  4.4.2 Condensate-enhanced stress response

Another natural study is to extend the results of Section 4.2.1 to the non-
linear regime and to see how the condensation affects the stress response when
large deformations are involved. Instead of working with the shear modulus
$\mathcal{G}$, which is a linear quantity, we work with the full stress-strain
relation. In order to do so, we will generalise the results of
baggioli2020black , which laid the groundwork for the calculation of such
relations in a fully nonlinear, holographic setup.

Let us review briefly what we mean by this. In a linear theory of elasticity –
i.e. when deformations are small –, the effective free energy regulating the
processes described by the theory is (3), so that the stress response
$\sigma_{\alpha\beta}$ that follows is:

$\sigma_{\alpha\beta}=C_{\alpha\beta\gamma\delta}\,\epsilon_{\gamma\delta},$
(62)

with $\epsilon_{\gamma\delta}$ the usual linear strain tensor. In the case of
shear deformations, for a homogeneous and isotropic medium, and adopting our
parameterization, one finds:

$\sigma=\mathcal{G}\epsilon,$ (63)

where $\sigma=\sigma_{xy}$. When deformations become large enough, the
response is described by a more general relation:

$\sigma=\sigma(\epsilon),$ (64)

where now $\sigma(\epsilon)$ is some generic nonlinear function of the
parameter $\epsilon$, that we expect to be linear in the limit
$\epsilon\rightarrow 0$. Holographically, $\sigma$ can be shown to be
proportional to the subleading term of the UV expansion of the $h(u)$ function
appearing in the metric. Such expansion can be found to be, in our case of
interest:

$h(u)\simeq h_{3}\,u^{3}\,(1+...).$ (65)

Figure 10: The stress-strain curve for $TL_{\alpha}=0.2$ and $\rho
L_{\alpha}^{2}=4$. The dashed line represent the would-be mechanical response
in absence of condensation (i.e. in the solid phase without superfluid
component). The red star indicates where the two curves join – at the critical
strain parameter $\epsilon_{c}$, whose value is further stressed by the
vertical red, dashed line.

We refer to the original work baggioli2020black  for further details. Here, we
limit ourselves to report the useful final formula:

$\sigma=\frac{3}{2}\,h_{3},$ (66)

that can be used to compute the shear stress numerically. Fig.10 reports one
comparison between the stress-strain relation in the supersolid and solid
states. The results are consistent with what found before: the presence of the
condensate enhances the stress response. Clearly, this enhancement vanishes as
the system transitions back into the normal phase due to the increased strain.

##  5 Outlook

In this work we have initiated the study of supersolid phases of matter using
the holographic framework by considering solutions which break spontaneously
translations together with a global $U(1)$ symmetry.

The first interesting observed feature is an increase of the shear modulus in
the supersolid phase which can be rationalized as a contribution from the
superfluid component. As expected from the effective theory description
PhysRevLett.96.055301  reviewed in Section 2, close to the supersolid critical
temperature the increase is linearly proportional to the square of the
superfluid condensate $|\Psi|^{2}$. This result is compatible with the
experimental results appeared so far in the literature Day2007 ;
doi:10.1063/1.2883895 ; Syshchenko_2009 . The same behavior – with opposite
sign though – has been found by studying the shear viscosity. The latter
further decreases in the supersolid phase and presents a discontinuous
derivative at the supersolid critical point. Qualitatively, this behavior is
in accordance with the notion that, upon a superfluid transition, the normal
component of the system, and therefore the associate friction, decrease.
Additionally, we find the Kovtun-Son-Starinets bound violation is enhanced by
the presence of a condensate and we indicate as a potential responsible for
this fact the interactions between the solid and superfluid degrees of
freedom. Interestingly, in absence of the solid component and these new
interactions, no effects on the $\eta/s$ ratio was found in the literature (at
least for the $s$-wave case) Natsuume_shearvisc .

Moreover, we have characterized the nature of the supersolid phase transition
as a function of the mechanical deformation parameters. In the linear
elasticity regime – for small deformations – the qualitative behavior of the
critical temperature is once more the one predicted by the effective field
theory (Section 2). We find that a shear strain renders the critical
temperature smaller and therefore it tends to destroy the supersolid phase. On
the contrary, we observe that the behavior in terms of a volumetric
deformation depends on its sign. A volumetric compression increases the
supersolid critical temperature, while an isotropic expansion disfavours the
broken phase.

Finally, we have studied the regime of large mechanical deformations and how
they do affect the supersolid critical temperature. We observe that the
superfluid condensate goes to zero at a critical value of the strain following
a mean-field behaviour. On the other hand, the condensed system appears to be
stiffer even when deformation become large.

An obvious extension of our work would be to build a relativistic
hydrodynamics framework for supersolids and compare it with our holographic
models. For example, it is well known that a supersolid supports two different
longitudinal sound modes with different speeds of propagation
PhysRevLett.108.175301 . All this information can be obtained by extracting
numerically the quasinormal modes of our system. It would be also interesting
to think about apply an external rotation to our supersolid system in order to
check in more detail how close is the holographic model to what expected for
real supersolid systems.

Let us conclude with two comments about the phenomenology of our holographic
toy model. Pair-density waves (PDWs) have been constructed holographically in
Cai:2017qdz ; Cremonini:2017usb ; Cremonini:2016rbd . To the best of our
knowledge, without appealing to microscopic details, the symmetry properties
of PDWs and supersolids are very similar – they both break spatial
translations together with a global $U(1)$ symmetry. Therefore, at least at
low energy (late time and large scales) we do not expect major differences
between our findings and the physics of PDWs. It would be fruitful to make
this point clearer and discuss the commonalities/differences between our model
and results and the holographic setups for PDWs mentioned above. Another
object of prospective further studies is the question regarding strain-induced
instabilities. In real-world materials, stress-strain curves are known to have
an endpoint – usually corresponding to the point where the sample breaks –. A
first study on how to determine these elasticity bounds in zero temperature
effective field theories is included in the analysis of baggioli2020black  and
used afterwards in Pan:2021cux . Mechanisms proposed there as possible
signatures of strain-induced instabilities are imaginary speeds of sound in
the bulk or the appearance of ghosts. Such proposals, though, have been
brought forward in the context of the decoupling limit; the only reliable way
to obtain such bounds – at least numerically – would be to check whether the
dispersion relation of perturbations around the large-strained state develop
any pathology. This requires a considerable effort and is left for future
investigations. From a holographic perspective, the analysis should be
analogous to that of holographic superfluids with background superfluid
velocity Amado:2013aea . Importantly, this approach could put important bounds
on supersolidity and/or shed some lights on their instability properties in
the nonlinear regime PhysRevLett.104.075302 .

Finally, despite it is well-known (but too often ignored) that the HHH model
Hartnoll:2008vx  does not represent the gravity dual of a superconducting
state of matter (unless one imposes mixed boundary conditions for the bulk
gauge field Domenech:2010nf ), we could still ask ourselves whether our
results might be relevant to discuss the impact of mechanical deformations in
the context of superconductors, and especially non BCS-like ones.
Unfortunately, it seems that no universal trend has emerged so far and all
possible behaviors have been experimentally observed buehler1965effect ;
zhai2000effect ; medvedev2009electronic ; han2010superconductivity ;
qing2012interface ; hicks2014strong ; steppke2017strong ; ahadi2019enhancing ;
ruf2021strain ; ghini2021strain ; locquet1998doubling ;
malinowski2020suppression . By abusing of the model and considering the
critical temperature as the superconducting (SC) one, our computations suggest
that a compression always increases the SC critical temperature while a
dilatation or a shear strain decrease it. It would be interesting in the
future to assess whether the holographic methods could provide any useful
information in this direction as well and compare to the experimental trends
reported above.

## Acknowledgments

We especially thank Alessio Zaccone and Oriol Pujolàs for their collaboration
at an early stage of this project and for uncountable suggestions and
discussions. We thank Wei-Jia Li and Li Li for several discussions on the
topics of this work and for providing useful comments on a first version of
this manuscript. We thank Blaise Goutéraux for a related collaboration which
helped a lot in clarifying several aspects of this work. We thank Sašo
Grozdanov and the Holography, Strings and Transport group at University of
Ljubljana for the logistic support in these hard times. M.B. acknowledges the
support of the Shanghai Municipal Science and Technology Major Project (Grant
No.2019SHZDZX01). G.F. is profoundly thankful to his parents for their loving
support, and to all the academics that provided him with their precious
mentorship throughout the time, especially to those in Milan.

##  Appendix A A brief review of finite elasticity theory

In this Appendix, we briefly review some basics of finite elasticity theory
following the arguments and notations of ZAMM:ZAMM19850650903 . For the sake
of the present work, we will restrict our attention to non-dissipative
processes and materials with no defects. In such framework, one can see any
deformation process as a smooth, invertible map from one initial (reference)
to a final (current) configuration.

Given a reference frame, one way to uniquely identify any geometrical
configuration is by specifying the position of every of its constituents – be
they atoms or more generic infinitesimal volume elements –. This can be
obtained by introducing a set of scalar fields $\varphi^{I}(x^{i})$, often
referred to as crystal fields Armas:2019sbe , in a number equal to the spatial
dimensions of the system121212Strictly speaking, the crystal fields could be
less than the number of dimensions, as it is the case for smectic crystals;
for isotropic solids, though, the equality must hold.. Symbolically:

$\text{Field profile }\varphi^{I}\Longleftrightarrow\text{Material
configuration}\,.$

Notice that we used a different convention for the index of the coordinates
$x^{i}$ (lowercase Latin) and of the crystal fields $\varphi^{I}$ (uppercase
Latin). The meaning of such distinction will be clarified in short.

With this in mind, we proceed to give a more precise mathematical formulation
of our problem. As we said, a deformation is a regular map going from one
configuration to the other. Because we are free to pick our reference frame, a
very natural choice is to choose one where the reference configuration
coincides with the coordinates, in the sense that:

$\varphi_{0}^{I}(x^{i})={\delta^{I}}_{i}\,x^{i}.$ (67)

where $x_{j}$ are the (spatial) coordinates adopted. Notice that in doing so
we did not state anything regarding the nature of this reference
configuration; rather, we limited ourselves to pick a convenient framework to
work with. In most applications, such a configuration is assumed to be the
equilibrium one (for which the total stress vanishes), but there is no
necessity to assume this, so that the formalism presented here is valid even
outside of such hypothesis.

Another important observation regarding (67) is that in such a state there
exist virtually no distinction between upper- and lowercase indices. This is a
feature of the choice of reference frame, and we learn that, until we stick
with this convention, lowercase indices are associated to the reference
configuration. In the literature, they are also called Lagrangian. In more
general terms, the Lagrangian framework of elasticity is the one where the
current configuration is expressed as a function of the reference one, while
the opposite standpoint – made possible by the invertibility of each
deformation map – is referred to as Eulerian.

In elasticity theory, the fundamental object is the displacement vector, a
vectorial field that specifies where each of the points is mapped when a
deformation occurs. In our language, this is equivalent to subtracting the
reference configuration from the current one $\varphi^{I}(x^{i})$. In flat
space, this gives the displacement vector:

$\zeta^{I}(x^{i})\equiv\varphi^{I}(x^{i})-{\delta^{I}}_{j}\,x^{j}\,,$ (68)

where we assumed a Lagrangian point of view. A representation of this
framework is provided in Fig. 11.

Figure 11: A representation of the reference configuration
$\varphi_{0}^{I}=x^{I}$ (left), the current configuration $\varphi^{I}$
(center) and the displacement field $\zeta_{I}$ (right).

In the theory of finite deformations, a prominent role is played by the mixed
deformation gradient tensor, defined as:

${\Xi^{I}}_{j}\equiv\frac{\partial\varphi^{I}}{\partial x^{j}}.$ (69)

This is nothing but the Jacobian of the transformation, which can be used as a
building block to construct various (nonlinear) versions of the strain tensor.
Two of those can be obtained by comparing the distance between two neighboring
points before and after the deformation. We have, indeed, that:

$d\varphi^{I}={\Xi^{I}}_{j}\,dx^{j},$ (70)

together with an inverse relation. Continuing on these lines, we can define
the Green-Lagrange strain tensor $E_{ij}$ via the relation:

$d\varphi^{2}-dx^{2}\,\equiv dx^{T}\,\cdot\,2\,E\,\cdot dx.$ (71)

Using Eq.(71) together with Eq.(69), we then obtain:

$E\,=\,\frac{1}{2}\left(\Xi^{T}\Xi-\mathbb{I}\right)\,,$ (72)

where $\mathbb{I}$ is the identity. In components, we have:

${E^{i}}_{j}=\frac{1}{2}\left({\Xi^{i}}_{K}{\Xi^{K}}_{j}-{\delta^{i}}_{j}\right),$
(73)

so that we are left with Lagrangian indices only. In terms of the
displacements – with an abuse of notation, all the indices have become
Lagrangian – the relation reads:

$E_{ij}=\frac{1}{2}\left(\partial_{i}\zeta_{j}+\partial_{j}\zeta_{i}+\partial_{i}\zeta_{k}\,\partial_{j}\zeta_{k}\right).$
(74)

An analogous procedure can be followed to obtain a Eulerian strain tensor. It
will suffice to use the inverse of (70) in the defining relation:

$d\varphi^{2}-dx^{2}\,\equiv d\varphi^{T}\,\cdot\,2\,E^{\prime}\,\cdot
d\varphi.$ (75)

One obtains:

$E^{\prime}\,=\,\frac{1}{2}\left(\mathbb{I}-(\Xi^{-1})^{T}\Xi^{-1}\right)\,,$
(76)

which has indeed two Eulerian indices. Its expression in terms of the
(Eulerian) displacement vector $\bar{\zeta}$ is:

$E_{IJ}=\frac{1}{2}\left(\partial_{I}\bar{\zeta}_{J}+\partial_{J}\bar{\zeta}_{I}-\partial_{I}\bar{\zeta}_{K}\,\partial_{J}\bar{\zeta}_{K}\right).$
(77)

It is worth noticing that at the linear level, when quadratic terms in the
displacements are negligible, both (72) and (76) reduce to the linear strain
tensor $\epsilon_{ij}$ typically used in linear elasticity theory
chaikin_lubensky_1995 . This can be seen as a consequence of the fact that, in
that regime, the difference between the two standpoint blurs, as the
transformations themselves are infinitesimal. In any case, the key lesson to
be learned here is that in the nonlinear regime multiple definitions for the
strain tensor are available, and which to use reduces to a matter of
mathematical convenience. In this work, we adopt the Green-Lagrange one (72).

To conclude this appendix, let us analyze in more detail the structure of the
deformation gradient tensor $\Xi$ in light of some consistency and symmetry-
imposed restrictions. For convenience, let us stick to two spatial dimensions,
where we would expect, on general grounds, such an object to carry $4$
independent degrees of freedom. Importantly, there is a redundancy in this
description, which reduces their number. Let us see why. Because $\Xi$ is a
non-singular matrix, the polar decomposition theorem holds, which means that
there exist two (uniquely defined) symmetric matrices $U$ and $W$, and a
rotation matrix $R$ such that:

$\Xi=U\cdot R=R\cdot W.$ (78)

Using the definition of the strain tensor in Eq.(72), and the property that
$RR^{T}=I$, it is straightforward to verify that:

$E(\Xi)\,=\,E(U)\,=\,E(W)\,.$ (79)

This implies that all the matrices $\Xi$, $U$ and $W$ give the same strain
tensor: from the point of view of mechanical deformations, they are completely
equivalent131313We remind that the energy functional depends exclusively on
the strain tensor and not on the deformation gradient tensor. Another way to
see this is by recurring to the notion of material objectivity, mentioned in
ZAMM:ZAMM19850650903 .. In operative terms, the equivalence relation defined
in Eq.(79) allows us to always choose a symmetric deformation gradient tensor
with only $3$ independent parameters.

In this work, we deal with isotropic systems. This notion allows us to push
this reduction further using the invariance of the systems under rotations by
defining an equivalence class under the transformation:

$\Xi^{\prime}=R^{T}\cdot\Xi\cdot R,$ (80)

which tells us that all deformation gradient tensors are equivalent up to a
rigid rotation $R$. Using this assumption, the resulting number of independent
parameters appearing in the gradient deformation tensor is reduced to $2$ (in
two dimensions).

In case of homogeneous deformations, then, the deformation gradient $\Xi$ is
constant. Convenient parameterizations for matrices with the same symmetry
properties that used in this manuscript are:

$\Xi=\alpha\begin{pmatrix}\cosh{(\Omega/2)}&\sinh{(\Omega/2)}\\\
\sinh{(\Omega/2)}&\cosh{(\Omega/2)}\end{pmatrix}=\alpha\begin{pmatrix}\sqrt{1+\epsilon^{2}/4}&\epsilon/2\\\
\epsilon/2&\sqrt{1+\epsilon^{2}/4}\end{pmatrix}\,,$ (81)

where $\alpha$ parameterize a purely volumetric (bulk) deformation in case
$\Omega$ vanishes, while $\Omega$ (with $\alpha=1$) a purely deviatoric term
which modifies the shape of the material but not its volume
($\mathrm{det}\left[\Xi\right]=1$).

From the last equality, it is possible to derive the parametrization of the
linear strain tensor shown in the main text in Eq.(9) by simply expanding (72)
near the reference configuration $\Xi=\mathbb{I}$. An efficient way to do so
is by operating the substitution:

$\Xi=\alpha\begin{pmatrix}\sqrt{1+\epsilon^{2}/4}&\epsilon/2\\\
\epsilon/2&\sqrt{1+\epsilon^{2}/4}\end{pmatrix}\longrightarrow(1+\tilde{\delta})\begin{pmatrix}\sqrt{1+\epsilon^{2}/4}&\epsilon/2\\\
\epsilon/2&\sqrt{1+\epsilon^{2}/4}\end{pmatrix},$ (82)

and consider both $\tilde{\delta}$ and $\epsilon$ as small parameters. Then,
inserting the new expression into (72), one gets:

$E=\frac{1}{2}\left((1+\tilde{\delta})^{2}\begin{pmatrix}1+\epsilon^{2}/2&\epsilon\sqrt{1+\epsilon^{2}/4}\\\
\epsilon\sqrt{1+\epsilon^{2}/4}&1+\epsilon^{2}/2\end{pmatrix}-\mathbb{I}\right)=\begin{pmatrix}\tilde{\delta}&\epsilon/2\\\
\epsilon/2&\tilde{\delta}\end{pmatrix}+O(\tilde{\delta}^{2},\epsilon^{2},\tilde{\delta}\epsilon).$
(83)

Keeping the linear order in the deviation parameters gives the linear strain
tensor. To arrive to (9) one just need to recall that:

$\delta\equiv\Tr(u)=2\tilde{\delta}.$ (84)

Using this relation one gets to the final form (9) reported in the main body
of this work.

Importantly, this also shows that the $\epsilon$ parameter appearing in the
linear strain tensor coincides with the off-diagonal component of the
nonlinear deformation gradient $\Xi_{xy}$ as stated in the main text as well.

##  Appendix B Details of the holographic model

This appendix is devoted to explain some details of the holographic model
discussed in the main text. The covariant equations of motion of the system
can be retrieved by applying the usual Euler-Lagrange procedure to the action
(18) with respect to the fields $\phi^{I}$, $\psi$, $A_{\mu}$, and
$g_{\mu\nu}$. One obtains:

$\displaystyle\nabla\left(\frac{\partial V}{\partial
X}\nabla\phi^{I}\right)=0,$ (85) $\displaystyle\nabla^{2}\psi-
iq\left[(\nabla\cdot A)+2(A\cdot\nabla)\right]\psi-(q^{2}A^{2}+M^{2})\psi=0,$
(86)
$\displaystyle\nabla_{\nu}F^{\nu\mu}-e_{0}^{2}\left[iq\left(\psi^{*}\partial^{\mu}\psi-\psi\partial^{\mu}\psi^{*}\right)+2q^{2}|\psi|^{2}A^{\mu}\right]=0,$
(87)
$\displaystyle\begin{split}&R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\left(R-2\Lambda\right)-2m^{2}\left(\frac{\partial
V}{\partial
g^{\mu\nu}}-\frac{1}{2}g_{\mu\nu}V(X)\right)-\frac{\kappa^{2}}{e_{0}^{2}}\left(F_{\mu\sigma}F_{\nu}^{\sigma}-\frac{1}{4}g_{\mu\nu}F^{2}\right)-\\\\[1.0pt]
&-\kappa^{2}\left[\left(\partial_{\mu}\psi-
iqA_{\mu}\psi\right)\left(\partial_{\nu}\psi^{*}+iqA_{\nu}\psi^{*}\right)+(\mu\leftrightarrow\nu)-g_{\mu\nu}|(\partial-
iqA)\psi|^{2}-g_{\mu\nu}M^{2}|\psi|^{2}\right]=0,\end{split}$ (88)

where $\nabla$ is defined to be the covariant derivative associated to the
metric connection. As argued in the main text, (85) is solved by (20), and to
restrict to the homogeneous, isotropic case we adopt the ansatz (22) for the
rest of the bulk fields. After writing $\psi(u)=|\psi(u)|e^{i\theta(u)}$, one
can readily check that the radial component of Maxwell’s equation reads:

$\theta^{\prime}(u)=0\,,$ (89)

so that $\theta(u)=const$ and we may make a phase choice for the condensate
field. This reflects the fact that only phase differences are physical, and
give rise to measurable phenomena (Josephson effect). Thus, we pick
$\theta(u)=0$ and work with a real scalar field. Following the notations
presented in the main text, then, the bulk equations of motion are given by:

$\displaystyle\psi^{\prime\prime}+\left(\frac{f^{\prime}}{f}-\frac{\chi^{\prime}}{2}-\frac{2}{u}\right)\psi^{\prime}+\left(\frac{q^{2}A_{t}^{2}e^{\chi}}{f^{2}}-\frac{L^{2}M^{2}}{u^{2}f}\right)\psi=0,$
$\displaystyle
A_{t}^{\prime\prime}+\frac{\chi^{\prime}}{2}\,A_{t}^{\prime}-\frac{2e_{0}^{2}L^{2}q^{2}\psi^{2}}{u^{2}f}A_{t}=0$
$\displaystyle
2uf^{\prime}h^{\prime}+f\left(2uh^{\prime\prime}-h^{\prime}\left(u\chi^{\prime}+4\right)\right)+\frac{4\alpha^{2}m^{2}uV_{X}(X)}{L_{0}^{2}}\sinh(\Omega-h)=0,$
$\displaystyle-\frac{2\kappa^{2}q^{2}uA_{t}^{2}e^{\chi}\psi^{2}}{f^{2}}-\frac{1}{2}uh^{\prime
2}-2\kappa^{2}u\psi^{\prime 2}+\chi^{\prime}=0,$
$\displaystyle-\frac{\kappa^{2}u^{4}e^{\chi}A_{t}^{\prime
2}}{e_{0}^{2}L^{2}}+2uf^{\prime}-uf\chi^{\prime}-6f-2L^{2}m^{2}V(X)-2\kappa^{2}L^{2}M^{2}\psi^{2}+6=0,$
(90)

where we used that:

$X=\frac{\alpha^{2}\,u^{2}}{L_{0}^{2}L^{2}}\cosh(\Omega-h).$ (91)

The arguments of the functions involved are not reported for readability’s
sake and primes indicate differentiation with respect to the radial coordinate
$u$, while the $X$ subscript with respect to $X$.

It is vital to notice that the bulk equations of motion (90) do possess
several symmetries, which turn out to be useful for both practical
computations and to shed light on some physical ambiguities as well.

  * •

The first symmetry is given by the transformation:

$e^{\chi}\rightarrow a^{2}e^{\chi},\hskip 12.0ptt\rightarrow at,\hskip
12.0ptA_{t}\rightarrow\frac{1}{a}A_{t}\,.$ (92)

This is a $\chi$-shift symmetry, which can be used to set
$\chi_{UV}\equiv\chi(u=0)=0$ and identify the bulk time coordinate with the
boundary one. As argued in the main text, this can be used to reduce the
number of physical parameters to be specified to select one particular
solution.

  * •

A second symmetry is given by:

$\begin{gathered}\\{t,x,y,u\\}\rightarrow a\\{t,x,y,u\\}\\\
\Rightarrow\\{L,\kappa\\}\rightarrow
a\\{L,\kappa\\},\,\\{m,M,A_{t},\psi\\}\rightarrow\frac{1}{a}\\{m,M,A_{t},\psi\\},\end{gathered}$
(93)

which we use to conveniently set the AdS length $L$ to $1$.

  * •

The last transformation is:

$\begin{gathered}\\{t,x,y,u\\}\rightarrow a\\{t,x,y,u\\}\\\ \Rightarrow
L_{0}\rightarrow aL_{0},\,A_{t}\rightarrow\frac{1}{a}A_{t},\end{gathered}$
(94)

which leaves all fields invariant, as well as $X$. From a practical point of
view, this can be understood as the possibility of consistently setting
$L_{0}=1$ throughout the calculations; importantly, the way $L_{0}$ scales in
this symmetry is what suggests us that $L_{0}$ is indeed a length. There is
also an additional redundancy in the sense that, at the level of the equations
(90), the following is a symmetry as well:

$L_{0}\rightarrow
aL_{0}\Longleftrightarrow\alpha\rightarrow\frac{1}{a}\alpha\,.$ (95)

Finally, let us comment on the numerical method used to solve the bulk
equations and construct the gravitational solution.

We use a shooting method that starts integrating the EOMs from the black
hole’s horizon $u_{h}$ to the UV boundary, where the holographic dictionary
allows us to read the physical quantities we are interested into. By examining
such equations in the vicinity of the horizon – and requiring the fields to be
regular on it –, we discover that they define a map:

$\left(\psi(u_{h}),\,A_{t}^{\prime}(u_{h}),\,\chi(u_{h}),\,h(u_{h}),\,u_{h};\,\alpha,\,\Omega,\,L_{0}\right)\longrightarrow\text{
Bulk fields},$ (96)

where the three parameters on the right of the semicolon are the ones that
define which axion field profile we are considering. Once all eight parameters
on the left hand side of the map are picked, a unique profile for the bulk
fields is selected. By using the above symmetries, it is possible to
considerably lower the amount of free parameters to be specified.

To begin, one has to restrict the space of physical solutions by requiring the
following conditions:

  1. 1.

$\psi^{(1)}=0$ along with $\psi^{(2)}\neq 0$ (in the broken phase). This
corresponds to the condensate assuming a nonzero vacuum expectation value in
absence of a source, i.e. to the spontaneous global-$U(1)$ breaking.

  2. 2.

$h(0)=0$, so that the boundary metric is Minkowski’s.

  3. 3.

$\chi(0)=0$, so that boundary temperature corresponds to the one computed in
the bulk’s deep interior.

The first condition can be imposed algorithmically by creating a feedback loop
in the numerical integration of the equations of motion – thus fixing one of
the initial values, in our case the electric field on the horizon
$A_{t}^{\prime}(u_{h})$ –; the second, instead, can be attained by defining an
auxiliary field: $\eta(u)=\Omega-h(u)$, so that $\Omega$ becomes a quantity to
be read on the UV boundary,making it trivial to impose $h(0)=0$; fixing
$\eta(u_{h})$, then, imposes condition 2 and avoids the need to specify
$\Omega$ before integrating the equations of motion.

Additionally, we learn that the value of $\chi(u_{h})$ is totally irrelevant,
as a suitable rescaling of the time coordinate $t$ will do the trick of
setting $\chi(0)=0$. In other terms: we are allowed to start with an arbitrary
value of $\chi(u_{h})$, then to rescale the time coordinate. This will shift
the value of $\chi(u_{h})$ to the one realizing $\chi(0)=0$.

At last, it is possible to notice that the two quantities $L_{0}$ and $\alpha$
appear, in the equations of motion, always in the combination $L_{0}/\alpha$,
making them virtually indistinguishable. This does not come as totally
unexpected: being $L_{0}$ a generic length scale, and $\alpha$ a parameter
describing shape-preserving deformations, it makes sense that they are tied
together. This allows us to define a single scale length:

$L_{\alpha}\equiv\frac{L_{0}}{\alpha},$ (97)

that substitutes the two distinct parameters $L_{0}$ and $\alpha$. After this
few considerations, then, the map (96) becomes:

$\left(\psi(u_{h}),\,\eta(u_{h}),\,u_{h};\,L_{\alpha}\right)\longrightarrow\text{
Bulk fields},$ (98)

We can do better, and give a boundary-oriented physical meaning to this map.
By making use of these relations, the map (98), as seen from the boundary
theory, is then:

$\left(T,\,\rho,\,\Omega;\,L_{\alpha}\right)\longrightarrow\text{ Bulk
fields}.$ (99)

This means that once we fix the temperature, charge density and deformation
state of the system, we are provided with a unique set of profiles for the
bulk fields. In this, we should remember that the boundary theory still
retains a property of scale invariance, so that the system is invariant under
rescalings:

$T\rightarrow aT,\qquad\rho\rightarrow a^{2}\rho,\qquad L_{\alpha}\rightarrow
a^{-1}L_{\alpha},$ (100)

as fixed by the dimensionality of these parameters.

All in all, a good choice to identify a state in the dual field theory is
given by:

$\\{TL_{\alpha},\,\rho L_{\alpha}^{2},\,\Omega\\}$ (101)

which corresponds to the choice made in the main text.

##  Appendix C The effects of the charge density on the shear response

For completeness, in this Appendix, we report some results which regard the
effect of the background charge density $\rho$ on the mechanical response of
the holographic system to shear deformations. In particular, we are interested
in the behavior of the shear modulus and the shear viscosity in the solid
phase in function of the charge density. Despite these same results have been
used in the previous literature Ammon:2020xyv ; Baggioli:2020edn , we find
useful to explicitly show them again here. Additionally, we also consider the
full nonlinear response at finite charge density which, to the best of our
knowledge, has not been considered explicitly in the literature.

Figure 12: Adimensional shear modulus (Left) and viscosity-over-entropy ratio
(Right), for a set of adimensional temperatures:
$TL_{\alpha}=0.05,\,0.1,\,0.2,\,0.4$ (from yellow to red).

In Fig.12, we show the (adimensional) shear modulus and the viscosity-over-
entropy ratio as a function of the charge density in the normal (solid) phase,
at fixed temperatures. From there, we notice that the presence of finite
charge density universally decreases the shear modulus and increases the value
of the $\eta/s$ ratio towards its KSS value. The effects of the charge density
are more pronounced for small temperatures and they become very mild at high
temperature.

Figure 13: Adimensional stress response as a function of the shear strain
parameter $\epsilon$, at fixed adimensional temperature ($TL_{\alpha}=0.2$)
for a set of increasing (adimensional) charge densities: $\rho
L_{\alpha}^{2}=0,\,2,\,4,\,8$ (from lighter to darker).

In Fig.13, we focus on how the charge density affects the isothermal stress
response of the system in the nonlinear regime. We find that the effects of
the charge density are important only for small values of the strain and are
consistent with the previous discussion about the shear modulus. In general,
we see that the presence of charge density diminishes the elastic response of
our system and that the effect is more important in the linear regime, i.e. at
small shear strain.

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