Higgs/Amplitude Mode Dynamics From Holography

Aristomenis Donos1 and Christiana Pantelidou2

1 Centre for Particle Theory and Department of Mathematical Sciences,
Durham University, Durham, DH1 3LE, U.K.
2 School of Mathematics and Statistics,
University College Dublin, Belfield, Dublin 4, Ireland

Abstract

> Second order phase transitions are universally driven by an order parameter
> which becomes trivial at the critical point. At the same time, collective
> excitations which involve the amplitude of the order parameter develop a gap
> which smoothly closes to zero at criticality. We develop analytical
> techniques to study this “Higgs” mode in holographic systems which undergo a
> continuous phase transition at finite temperature and chemical potential.
> This allows us to study the linear response of the system at energy scales
> of the order of the gap. We express the Green’s functions of scalar
> operators in terms of thermodynamic quantities and a single transport
> coefficient which we fix in terms of black hole horizon data.

##  1 Introduction

Holography provides a powerful framework to study strongly coupled quantum
field theories [1] at finite temperature and chemical potential [2]. In
particular, in the large $N$ limit, it allows us to perform exact computations
which are impossible to carry out with standard quantum field theory
techniques.

Second order phase transitions are driven by an order parameter which becomes
non trivial in the broken phase. Depending on the nature of the system, the
phase transition can be accompanied by the breaking of a global symmetry. In
such a case, new gapless modes appear in the spectrum of the system in the
form of Goldstone modes. The universal feature is that the phase transition is
always accompanied by the emergence of a gapped mode which is exactly gapless
at the critical point, the amplitude/Higgs mode (from now on we will call it
the Higgs mode). At infinite wavelengths, the corresponding mode111Strictly
speaking, the mode we are referring to is an amplitude mode for the order
parameter of the system. is homogeneous and it is related to fluctuating
degrees of freedom which are intimately connected to the amplitude of the
order parameter and which decouple from the entropy and charge density.

The above highlights that the Higgs mode dominates critical phenomena as its
decay rate is parametrically small close to the critical point [3, 4]. Despite
its fundamental importance, the Higgs mode was observed in condensed matter
systems only very recently. For the case of superfluids see e.g. [5, 6]. In
view of these results, it is very important to obtain theoretical predictions
on potentially universal properties of the critical dynamics. In this paper,
we will study aspects of the dynamics close to a second order phase transition
by exploiting the holographic principle.

In holography, continuous phase transitions can be realised through black hole
instabilities. With fixed deformation parameters and chemical potential, a
unique black hole solution dominates the phase diagram at high temperatures.
Lowering the Hawking temperature, certain bulk perturbations can become
unstable below a critical temperature $T_{c}$. Following the instability leads
to new branches of black hole solutions corresponding to the broken phase. In
the context of applied holography, some of the most well studied examples are
the holgraphic superfluids [7, 8, 9] and phases which spontaneously break
translations [10, 11, 12].

It is natural to ask how much we can learn about the Higgs mode by considering
holographic systems near their critical points. The Higgs mode has been
identified in several holographic studies in the past by using numerical
techniques [13, 14] or analytically when full solutions in the bulk can be
obtained [15]. Given the power of holography in obtaining exact results, the
ultimate goal would be to construct an enlarged theory of hydrodynamics which
incorporates this universal nearly gapless mode in its description.

In this paper we will make a significant step in this direction by examining
the dynamics of holographic systems close to their phase transitions. More
specifically, we will obtain analytic expressions for the low frequency
retarded Green’s functions of scalar operators. As we show in section 5, these
are entirely determined by the thermodynamic properties of the system and a
single transport coefficient which is fixed by the black hole horizon data.
The linear response of the system is dominated by a single pole which
corresponds to the Higgs mode which decays slowly near the critical point. Our
main tool will be the techniques which have been recently exploited in [16] to
obtain a hydrodynamic description of holographic superfluids at zero charge
density. The main contribution was to write the transport coefficients of [17]
in terms of black hole horizon data.

In that construction, the basic ingredient is the symplectic current density
of the bulk theory [18]. This can be considered for any pair of perturbative
solutions in the bulk. Being a generalisation of Liouville’s theorem for
classical mechanics, the symplectic current is divergence free. This feature
will be very important to us since it allows us to extract useful information
for an unknown perturbative solution when it is parametrically close to a
known one.

In our context, the role of the known solution will be played by the static
solutions we can construct by varying the thermodynamic parameters of the
thermal state close to the critical point. At exactly the critical point, this
variation reduces to the static bulk perturbative solution that emerges at the
critical point and which leads to the broken phase black holes. This, is also
the limit of the time dependent Higgs mode we are after when the system is
exactly at the transition point. This suggests that the bulk dual of the Higgs
mode is parametrically close to the solution generated by varying the
thermodynamic parameters and will therefore be the unknown solution in the
symplectic current.

The paper is organised in five sections as follows. In Section 2 we introduce
the holographic framework in which we plan to study phase transitions driven
by a scalar order parameter. In this case, the phase transition is not
accompanied by the breaking of a continuous symmetry. In Section 3 we analyse
the time dependent perturbation which captures the Higgs mode and we use our
technique involving the symplectic current to extract its gap. In Section 4 we
generalise our holographic model to describe the breaking of a global symmetry
on the boundary at finite chemical potential. This is particularly interest
since, as we will see, the lack of particle-hole symmetry mixes the Goldstone
mode. In Section 5 we compute the Green’s functions of the scalar operators in
our holographic theory highlighting their domination by the pole relevant to
the Higgs mode. Finally, in Section 6 we perform numerical checks of our
analytic results for the gap and the Green’s functions we compute in Section
5. We conclude the paper with a summary of our main results and outlook in
Section 7.

##  2 Setup

We will consider a bulk theory which can model the phase transition of a
conformal field theory at finite chemical potential and which has been
deformed by a scalar operator $\mathcal{O}_{\phi}$ of dimension
$\Delta_{\phi}$ with bulk dual $\phi$. In addition to that, our bulk theory
will contain the metric $g_{\mu\nu}$, gauge field $A_{\mu}$ and scalar field
$\rho$ which will spontaneously give a VEV to the field theory dual operator
$\mathcal{O}_{\rho}$.

Without loss of generality, we will consider the bulk action,

$\displaystyle S_{b}=\int
d^{4}x\,\sqrt{-g}\,\left(R-V-\frac{1}{2}\left(\partial\phi\right)^{2}-\frac{1}{2}\left(\partial\rho\right)^{2}-\frac{Z}{4}F^{2}\right)\,,$
(2.1)

where $F=dA$ is the field strength of $A_{\mu}$. In general, the gauge
coupling $Z$ and potential $V$ can depend on the scalar fields $\phi$ and
$\rho$. In order for the bulk equations of motion to admit solutions which
asymptote to unit radius $AdS_{4}$, we will impose that for small values of
the scalar fields,

$\displaystyle V$ $\displaystyle\approx
6-\frac{1}{2}m_{\phi}^{2}\,\phi^{2}-\frac{1}{2}m_{\rho}^{2}\,\rho^{2}+\cdots\,,$
$\displaystyle Z$ $\displaystyle\approx
1+\zeta_{\phi}\,\phi^{2}+\zeta_{\rho}\,\rho^{2}+\cdots\,.$ (2.2)

Moreover, without any loss of generality, for small values of $\rho$ we can
assume that,

$\displaystyle V$ $\displaystyle\approx
v(\phi)-\frac{1}{2}m^{2}(\phi)\,\rho^{2}-\frac{1}{4!}\lambda(\phi)\,\rho^{4}+\cdots\,,$
$\displaystyle Z$
$\displaystyle\approx\zeta_{0}(\phi)+\zeta_{1}(\phi)\,\rho^{2}+\cdots\,.$
(2.3)

The background black holes that can capture the different phases of our theory
are described by the backround geometries with,

$\displaystyle ds^{2}$
$\displaystyle=-U(r)\,dt^{2}+\frac{dr^{2}}{U(r)}+e^{2g(r)}\,\left(dx^{2}+dy^{2}\right)\,,$
$\displaystyle\phi$ $\displaystyle=\phi(r),\qquad\rho=\rho(r),\qquad
A=a(r)\,dt\,.$ (2.4)

The scalar $\rho$ is identically zero in the normal phase, above the critical
temperature $T_{c}$.

We will choose coordinates so that the event horizon with Hawking temperature
$T$ is at $r=0$. Near the horizon, regularity dictates the Taylor expansions,

$\displaystyle U(r)\approx\,4\pi T\,r+\mathcal{O}(r^{2}),\qquad g(r)\approx
g^{(0)}+\mathcal{O}(r)\,,$
$\displaystyle\phi(r)\approx\phi^{(0)}+\mathcal{O}(r),\qquad\rho(r)\approx\rho^{(0)}+\mathcal{O}(r)\,,$
$\displaystyle a(r)\approx a^{(0)}\,r+\cdots\,.$ (2.5)

At the other end of the geometry, where $r\to\infty$, we want to impose
$AdS_{4}$ asymptotics with unit radius,

$\displaystyle U(r)\approx(r+R)^{2}+\mathcal{O}(r^{0}),\qquad
g(r)\approx\ln(r+R)+\mathcal{O}(r^{-1})\,,$
$\displaystyle\phi(r)\approx\phi_{(s)}\,(r+R)^{\Delta_{\phi}-3}+\cdots+\phi_{(v)}\,(r+R)^{-\Delta_{\phi}}+\cdots\,,$
$\displaystyle\rho(r)\approx\rho_{(s)}\,(r+R)^{\Delta_{\rho}-3}+\cdots\rho_{(v)}\,(r+R)^{-\Delta_{\rho}}+\cdots\,,$
$\displaystyle a(r)\approx\mu-\varrho\,(r+R)^{-1}+\cdots\,.$ (2.6)

The conformal dimensions $\Delta_{\phi}$ and $\Delta_{\rho}$ of the dual
operators $\mathcal{O}_{\phi}$ and $\mathcal{O}_{\rho}$ are determined in
terms of the masses according to $\Delta_{\phi}(\Delta_{\phi}-3)=m_{\phi}^{2}$
and $\Delta_{\rho}(\Delta_{\rho}-3)=m_{\rho}^{2}$. The constants of
integration $\phi_{(s)}$ and $\rho_{(s)}$ correspond to sources for the
boundary operator $\mathcal{O}_{\phi}$ and the “modulus” of the operator
$\mathcal{O}_{\rho}$. We will be mostly setting the latter to zero except for
section 5 where will turn it on perturbatively in order to extract a few of
the retarded Green’s functions of the system.

At this point, it is useful to note that the horizon charge density
$\varrho^{(0)}$ and entropy density $s$ satisfy,

$\displaystyle\varrho^{(0)}=Z^{(0)}\,e^{2\,g^{(0)}}\,a^{(0)},\qquad
s=4\pi\,e^{2\,g^{(0)}}\,,$ (2.7)

where $Z^{(0)}$ denotes the value of the coupling function $Z$ evaluated on
the horizon values of the background scalars $\phi$ and $\rho$. For normal
fluids, the horizon charge density $\rho^{(0)}$ coincides with the field
theory charge density $\varrho=\varrho_{(v)}$ which can be extracted from the
asymptotics (2). For the superfluid phase that we will consider in section 4,
this is no longer true. In contrast, the entropy density can always be
computed as a horizon quantity from (2.7).

##  3 The Higgs Mode from Holography

In this section we will develop our tools that will lead to the main results
of our paper. The time dependent perturbation capturing the Higgs mode we are
after will be constructed in an expansion of the thermodynamic and deformation
parameters around the transition point. In section 3.1 we will discuss the
static perturbations we can obtain by varying the thermodynamic and
deformation parameters of the black hole solutions which are dual to the
thermal states of our system. These will prove particularly useful in
constructing the next to leading order terms in the expansion of the time
dependent perturbation in section 3.2. Finally, in section 3.3 we will make
use of the symplectic current to fix the frequency of the quasinormal mode as
a function of our thermodynamics and black hole horizon data.

###  3.1 Expansions Around The Critical Point

The gapped mode that we will study in the following sections corresponds to a
time dependent perturbation around the background geometry (2). We will be
interested in studying this perturbation very close to the critical point
$(T_{c}(\mu,\phi_{(s)}),\mu,\phi_{(s)})$. For this reason, we will introduce a
parameter $\varepsilon$ which parametrises a curve in the $(T,\mu,\phi_{(s)})$
plane that originates from the point $(T_{c}(\mu,\phi_{(s)}),\mu,\phi_{(s)})$.
As we vary the parameter $\varepsilon$ we move in the space of thermodynamic
parameters according to $(T_{c}(\mu,\phi_{(s)})+\delta
T(\varepsilon),\mu+\delta\mu(\varepsilon),\phi_{(s)}+\delta\phi_{(s)}(\varepsilon))$.
To make our notation clear we consider the curve for small values of the
expanision parameter with,

$\displaystyle\delta T(\varepsilon)\approx\delta
T\,\frac{\varepsilon^{2}}{2},\qquad\delta\mu(\varepsilon)\approx\delta\mu\,\frac{\varepsilon^{2}}{2},\qquad\delta\phi_{(s)}(\varepsilon)\approx\delta\phi_{(s)}\frac{\varepsilon^{2}}{2}\,.$
(3.1)

For a general function $\Phi$ of the thermodynamic parameters, along the curve
we can write the variation
$\delta\Phi=\frac{\varepsilon^{2}}{2}\,\delta\Phi_{(2)}+\cdots$ with,

$\displaystyle\delta\Phi_{(2)}=\delta
T\,\partial_{T}\Phi+\delta\mu\,\partial_{\mu}\Phi+\delta\phi_{(s)}\,\partial_{\phi_{(s)}}\Phi\,.$
(3.2)

Right at the critical point, there exists a static perturbative solution
$\delta\rho_{\ast(0)}$ to the equation of motion for the scalar $\rho$ with
source free boundary conditions, just like in (2) for the non-linear problem.
Our goal is to track this perturbative mode right below the critical
temperature where it acquires a gap $\delta\omega_{g}$. As we will see later
more explicitly the gap is itself an expandable function of $\varepsilon$ and
we will have $\delta\omega_{g}\propto\varepsilon^{2}$.

In studying our gapped mode, we will find useful to consider the expansion of
the background geometry itself in $\varepsilon$. Constructing this
perturbative expansion is a rather non-trivial task. However, here we will
only need the fact that it exists and that it is analytic in $\varepsilon$,

$\displaystyle\rho$
$\displaystyle=\varepsilon_{\ast}\,\delta\rho_{\ast(0)}+\frac{\varepsilon_{\ast}^{3}}{3!}\,\delta\rho_{\ast(2)}+\cdots\,,$
$\displaystyle U$
$\displaystyle=U_{c}+\frac{\varepsilon_{\ast}^{2}}{2}\,\delta
U_{\ast(2)}+\frac{\varepsilon_{\ast}^{4}}{4!}\,\delta U_{\ast(4)}+\cdots,\quad
g=g_{c}+\frac{\varepsilon_{\ast}^{2}}{2}\,\delta
g_{\ast(2)}+\frac{\varepsilon_{\ast}^{4}}{4!}\,\delta g_{\ast(4)}+\cdots\,,$
$\displaystyle\phi$
$\displaystyle=\phi_{c}+\frac{\varepsilon_{\ast}^{2}}{2}\,\delta\phi_{\ast(2)}+\frac{\varepsilon_{\ast}^{4}}{4!}\,\delta\phi_{\ast(4)}+\cdots\,,\quad
a=a_{c}+\frac{\varepsilon_{\ast}^{2}}{2}\,\delta
a_{\ast(2)}+\frac{\varepsilon_{\ast}^{4}}{4!}\,\delta a_{\ast(4)}+\cdots\,.$
(3.3)

The leading terms in the above expansion represent the black hole solution (2)
at the critical point. For the curve of variation in the broken phase phase we
define the variation parameter $\varepsilon_{\ast}$ according to,

$\displaystyle\delta T(\varepsilon_{\ast})\approx\delta
T_{\ast(2)}\,\frac{\varepsilon_{\ast}^{2}}{2}+\cdots,\quad\delta\mu(\varepsilon_{\ast})\approx\delta\mu_{\ast(2)}\,\frac{\varepsilon_{\ast}^{2}}{2}+\cdots,\quad\delta\phi_{(s)}(\varepsilon_{\ast})\approx\delta\phi_{(s)\ast(2)}\frac{\varepsilon_{\ast}^{2}}{2}\,.$
(3.4)

Apart from the broken phase black holes, it is useful to consider the
expansion of the normal phase black hole solutions below $T_{c}$,

$\displaystyle\rho$ $\displaystyle=0\,,$ $\displaystyle U$
$\displaystyle=U_{c}+\frac{\varepsilon_{\\#}^{2}}{2}\,\delta
U_{\\#(2)}+\frac{\varepsilon_{\\#}^{4}}{4!}\,\delta U_{\\#(4)}+\cdots,\quad
g=g_{c}+\frac{\varepsilon_{\\#}^{2}}{2}\,\delta
g_{\\#(2)}+\frac{\varepsilon_{\\#}^{4}}{4!}\,\delta g_{\\#(4)}+\cdots\,,$
$\displaystyle\phi$
$\displaystyle=\phi_{c}+\frac{\varepsilon_{\\#}^{2}}{2}\,\delta\phi_{\\#(2)}+\frac{\varepsilon_{\\#}^{4}}{4!}\,\delta\phi_{\\#(4)}+\cdots\,,\quad
a=a_{c}+\frac{\varepsilon_{\\#}^{2}}{2}\,\delta
a_{\\#(2)}+\frac{\varepsilon_{\\#}^{4}}{4!}\,\delta a_{\\#(4)}+\cdots\,.$
(3.5)

Similarly to the case of the broken phase, we will define the expansion
parameter $\varepsilon_{\\#}$ according to,

$\displaystyle\delta T(\varepsilon_{\\#})\approx\delta
T_{\\#(2)}\,\frac{\varepsilon_{\\#}^{2}}{2}+\cdots,\quad\delta\mu(\varepsilon_{\\#})\approx\delta\mu_{\\#(2)}\,\frac{\varepsilon_{\\#}^{2}}{2}+\cdots,\quad\delta\phi_{(s)}(\varepsilon_{\\#})\approx\delta\phi_{(s)\\#(2)}\frac{\varepsilon_{\\#}^{2}}{2}\,.$
(3.6)

It is also useful to write thermodynamic quantities in terms of the
perturbative expansions of the horizon data. In particular, we can simply
expand the charge and the entropy densities of equation (2.7) in $\varepsilon$
to obtain,

$\displaystyle\delta T_{\ast}$
$\displaystyle=\frac{\varepsilon_{\ast}^{2}}{2}\,\delta
T_{\ast(2)}+\cdots,\quad\delta\varrho_{\ast}=\frac{\varepsilon_{\ast}^{2}}{2}e^{2g_{c}^{(0)}}\,\left(\delta
a^{(0)}_{\ast(2)}+2\,\delta g_{\ast(2)}^{(0)}\,a^{(0)}\right)+\cdots\,,$
$\displaystyle\delta s_{\ast}$
$\displaystyle=\varepsilon_{\ast}^{2}4\pi\,e^{2g_{c}^{(0)}}\,\delta
g_{\ast(2)}^{(0)}+\cdots\,,$ $\displaystyle\delta T_{\\#}$
$\displaystyle=\frac{\varepsilon_{\\#}^{2}}{2}\,\delta
T_{\\#(2)}+\cdots,\quad\delta\varrho_{\\#}=\frac{\varepsilon_{\\#}^{2}}{2}e^{2g_{c}^{(0)}}\,\left(\delta
a^{(0)}_{\\#(2)}+2\,\delta g_{\\#(2)}^{(0)}\,a^{(0)}\right)+\cdots\,,$
$\displaystyle\delta s_{\\#}=$
$\displaystyle\varepsilon_{\\#}^{2}4\pi\,e^{2g_{c}^{(0)}}\,\delta
g_{\\#(2)}^{(0)}+\cdots\,.$ (3.7)

The above horizon constants come from e.g. the near horizon expansions

$\displaystyle\delta U_{\ast(n)}(r)$ $\displaystyle=4\pi\,\delta
T_{\ast(n)}\,r+\cdots,\quad\delta g_{\ast(n)}(r)=\delta
g_{\ast(n)}^{(0)}+\cdots\,,$ $\displaystyle\delta a_{\ast(n)}(r)$
$\displaystyle=\delta
a_{\ast(n)}^{(0)}\,r+\cdots,\quad\delta\phi_{\ast(n)}=\delta\phi_{\ast(n)}^{(0)}+\cdots\,,$
(3.8)

and similarly for the normal phase $\varepsilon$ expansion. Close to the
conformal boundary we will have the asymptotic expansions,

$\displaystyle\delta U_{\ast(n)}(r)\approx\mathcal{O}(r^{0}),\qquad\delta
g_{\ast(n)}(r)\approx\mathcal{O}(r^{-1})\,,$
$\displaystyle\delta\phi_{\ast(n)}(r)\approx\delta\phi_{(s)\ast(n)}\,(r+R)^{\Delta_{\phi}-3}+\cdots+\delta\phi_{(v)\ast(n)}\,(r+R)^{-\Delta_{\phi}}+\cdots\,,$
$\displaystyle\delta\rho_{\ast(n)}(r)\approx\delta\rho_{(s)\ast(n)}\,(r+R)^{\Delta_{\rho}-3}+\cdots\delta\rho_{(v)\ast(n)}\,(r+R)^{-\Delta_{\rho}}+\cdots\,,$
$\displaystyle\delta
a_{\ast(n)}(r)\approx\delta\mu_{\ast(n)}-\delta\varrho_{\ast(n)}\,(r+R)^{-1}+\cdots\,.$
(3.9)

We will mostly keep $\delta\rho_{(s)\ast(n)}=0$, except in Section 5 where we
will study the retarded Green’s functions of our scalar operators. Note that
similar expansions hold for the functions appearing in the expansion around
the normal phase black holes.

For the VEVs of the scalar operators we can write the expansions,

$\displaystyle\delta\langle\mathcal{O}_{\rho}\rangle_{\ast}=\varepsilon_{\ast}\,\delta\langle\mathcal{O}_{\rho}\rangle_{\ast(0)}+\frac{\varepsilon_{\ast}^{3}}{3!}\,\delta\langle\mathcal{O}_{\rho}\rangle_{\ast(2)}+\cdots,\quad\delta\langle\mathcal{O}_{\phi}\rangle_{\ast}=\frac{\varepsilon_{\ast}^{2}}{2}\,\delta\langle\mathcal{O}_{\phi}\rangle_{\ast(2)}+\cdots\,,$
$\displaystyle\delta\langle\mathcal{O}_{\phi}\rangle_{\\#}=\frac{\varepsilon_{\\#}^{2}}{2}\,\delta\langle\mathcal{O}_{\phi}\rangle_{\\#(2)}+\cdots\,.$
(3.10)

###  3.2 Time Dependent Perturbation

In this section we will consider perturbative solutions of the equations of
motion around the backgrounds (2). Since our background geometries possess
spacetime translational invariance, it makes sense to consider the Fourier
decomposition of our perturbations. Since we are only interested in extracting
the gap, we will only consider Fourier modes with zero wavenumber according
to,

$\displaystyle\delta\mathcal{F}(t,\mathbf{x};r)=e^{-i\omega(t+S(r))}\,\delta
f(r)\,,$ (3.11)

where $\mathcal{F}$ repsesents perturbations of the scalars as well as the
metric and gauge field components. The function $S(r)$ is chosen so that it
drops faster that $\mathcal{O}(1/r^{3})$ close to the conformal boundary and
it therefore doesn’t interfere with holographic renormalisation. However,
close to the horizon, it is chosen so that it approaches $S(r)\to\frac{1}{4\pi
T}\ln r$ and the combination $t+S(r)$ is regular and ingoing.

For the system we are interested in, it is consistent to set the component
perturbations $\delta g_{ti}$, $\delta g_{ri}$, $\delta g_{xy}$ and $\delta
a_{i}$ to zero. Moreover, due to the unbroken rotational symmetry in the
$x$-$y$ plane we can set $\delta g_{xx}=\delta g_{yy}$. Imposing regular
ingoing boundary conditions close to the horizon, leads to the near horizon
expansions,

$\displaystyle\delta g_{tt}(r)$ $\displaystyle=4\pi T\,r\,\delta
g_{tt}^{(0)}+\cdots\,,\quad\delta g_{rr}(r)=\frac{\delta g_{rr}^{(0)}}{4\pi
T\,r}+\cdots\,\,,$ $\displaystyle\delta g_{ii}(r)$ $\displaystyle=\delta
g^{(0)}+r\,\delta g^{(1)}+\cdots\,,\quad\quad\delta g_{tr}(r)=\delta
g_{tr}^{(0)}+\cdots\,,$ $\displaystyle\delta a_{t}(r)$ $\displaystyle=\delta
a_{t}^{(0)}+\delta a_{t}^{(1)}\,r+\cdots\,,\quad\delta a_{r}(r)=\frac{1}{4\pi
T\,r}\delta a_{r}^{(0)}+\delta a_{r}^{(1)}+\cdots\,,$
$\displaystyle\delta\rho(r)$
$\displaystyle=\delta\rho^{(0)}+\cdots,\quad\delta\phi(r)=\delta\phi^{(0)}+\cdots\,,$
(3.12)

which are compatible with the equations of motion. In order to achieve
regularity, the above conditions need to be supplemented by,

$\displaystyle\delta g_{tt}^{(0)}+\delta g_{rr}^{(0)}=2\,\delta
g_{rt}^{(0)}\,,\quad\delta a_{r}^{(0)}=\delta a_{t}^{(0)}\,.$ (3.13)

The time dependent perturbation which is capturing the Higgs mode can be
expanded according to222Notice that we don’t discuss $\delta\tilde{g}_{tr}$ in
the expansion (3.2) as this can be changed by a coordinate transformation of
the form $t\rightarrow t+\delta f(r)$ for a perturbatively small function
$\delta f(r)$ and can be freely chosen as long as it satisfies the boundary
conditions (3.2) and (3.13).,

$\displaystyle\omega$
$\displaystyle=\varepsilon_{\ast}\,\omega_{[1]}+\varepsilon_{\ast}^{2}\,\omega_{[2]}+\cdots\,,$
$\displaystyle\delta\tilde{g}_{tt}$
$\displaystyle=-\varepsilon_{\ast}\,\delta\tilde{U}_{(2)}+\cdots,\quad\delta\tilde{g}_{rr}=-\varepsilon_{\ast}\,\frac{\delta\tilde{U}_{(2)}}{U_{c}^{2}}+\cdots\,,$
$\displaystyle\delta\tilde{g}_{ii}$
$\displaystyle=2\,\,e^{2g_{c}}\,\left(\varepsilon_{\ast}\,\delta\tilde{g}_{{2}}+\cdots\right),\quad\delta\tilde{a}_{t}=\varepsilon_{\ast}\,\delta\tilde{a}_{(2)}+\cdots\,,$
$\displaystyle\delta\tilde{\phi}$
$\displaystyle=\varepsilon_{\ast}\,\delta\tilde{\phi}_{(2)}+\cdots,\quad\delta\tilde{\rho}=\delta\rho_{\ast(0)}+\frac{\varepsilon_{\ast}^{2}}{2}\delta\tilde{\rho}_{(2)}+\cdots\,.$
(3.14)

The above expansion makes clear that our perturbation has to go to the zero
mode $\delta\rho_{\ast(0)}$ at the critical point with $\varepsilon_{\ast}=0$.
Notice that the choice of $\delta\tilde{g}_{rr}$ in the ansatz (3.2) fixes the
choice of the radial coordinate at order $\mathcal{O}(\varepsilon_{\ast})$.
The functions in equation (3.2) satisfy the boundary conditions (3.2) and
(3.13).

As we will see later, the leading term in the frequency expansion
$\omega_{[1]}$ is equal to zero. An important point in understanding the first
non-trivial $\varepsilon_{\ast}$ correction in our perturbation (3.2) is that
the equations that the functions $\delta\tilde{U}_{(2)}$,
$\delta\tilde{g}_{(2)}$, $\delta\tilde{a}_{(2)}$, $\delta\tilde{\phi}_{(2)}$
and $\delta\tilde{\rho}_{(2)}$ satisfy are a superset of the equations
saitisfied by the functions $\delta U_{\ast(2)}$, $\delta g_{\ast(2)}$,
$\delta a_{\ast(2)}$, $\delta\phi_{\ast(2)}$ and $\delta\rho_{\ast(2)}$. The
two extra equations are constraints and impose charge and energy conservation
from the field theory point of view. As such these constraints can be imposed
on any constant $r$ surface. However, as we will see in section 4, this is no
longer true for the equation of motion of the 1-form field due to the fact
that it becomes massive in the bulk. This is directly related to the
spontaneous symmetry breaking and the involvement of the field theory
Goldstone mode.

Returning to the case at hand, apart from the two constraints, this is an
inhomogeneous set of ordinary differential equations which are sourced by
quadratic terms in $\delta\rho_{\ast(0)}$. An important observation to make
here is that the homogeneous part of these equations is also solved by the
perturbations $\delta U_{\\#(2)}$, $\delta g_{\\#(2)}$, $\delta a_{\\#(2)}$,
$\delta\phi_{\\#(2)}$ defined by normal phase expansions of (3.1) defined by
any choice of expansion parameter $\varepsilon_{\\#}$. For such a solution of
the homogeneous part we will have a trivial contribution to
$\delta\tilde{\rho}_{(2)}$.

The final goal is the construction of a linear superposition of perturbations
coming from the first corrections of the broken phase branch (3.1) and normal
branch (3.1) such that it doesn’t carry any net charge or entropy density.
More precisely, after setting the value of the perturbative parametres equal
$\varepsilon_{\ast}=\varepsilon_{\\#}$, we have,

$\displaystyle\delta\tilde{U}_{(2)}$ $\displaystyle=\delta U_{\ast(2)}-\delta
U_{\\#(2)},\quad\delta\tilde{g}_{(2)}=\delta g_{\ast(2)}-\delta g_{\\#(2)}\,$
$\displaystyle\delta\tilde{a}_{(2)}$ $\displaystyle=\delta a_{\ast(2)}-\delta
a_{\\#(2)}-\delta\mu_{\ast(2)}+\delta\mu_{\\#(2)}\,,$
$\displaystyle\delta\tilde{\phi}_{(2)}$
$\displaystyle=\delta\phi_{\ast(2)}-\delta\phi_{\\#(2)}\,,$ (3.15)

with the parameters $\delta T_{\\#(2)}$ and $\delta\mu_{\\#(2)}$ such that the
total perturbation does not carry any net charge or entropy density. Moreover,
from the form of the solution (3.2), we can see that the source for scalar
field $\phi$ in the perturbation is,

$\displaystyle\delta\tilde{\phi}_{(s)(2)}=\delta\phi_{(s)\ast(2)}-\delta\phi_{(s)\\#(2)}\,.$
(3.16)

Therefore, when we look for quasi-normal modes when trying to compute the gap,
we will need to set $\delta\tilde{\phi}_{(s)(2)}=0$ in the perturbation.

Notice that in order to obtain a source free perturbation for the gauge field,
we also performed a gauge transformation with parameter
$\Lambda=(-\delta\mu_{\ast(2)}+\delta\mu_{\\#(2)})\,\left(t+S(r)\right)$ with
$S(r)$ as in (3.11). This step will not be possible in section 4 where we will
be dealing with superfluids and the bulk 1-form field will become massive due
to the spontaneous symmetry breaking. As we will see, the non-trivial
asymptotics of the bulk 1-form field will be rather related to the field
theory Goldstone which we expect to be involved in the case of a superfluid at
finite chemical potential, with no particle-hole symmetry.

In general we can write variations of the entropy, the charge densities and
the scalar VEV in terms of varations of the temperature, the chemical
potential and the scalar source according to,

$\displaystyle\begin{pmatrix}\delta s_{\ast}\\\ \delta\varrho_{\ast}\\\
\delta\langle\mathcal{O}_{\phi}\rangle_{\ast}\end{pmatrix}$
$\displaystyle=\frac{\varepsilon_{\ast}^{2}}{2}\begin{pmatrix}T_{c}^{-1}\,c_{\mu}^{\ast}&\xi^{\ast}&\nu_{T}^{\ast}\\\
\xi^{\ast}&\chi^{\ast}&\nu_{\mu}^{\ast}\\\
\nu_{T}^{\ast}&\nu_{\mu}^{\ast}&\nu_{\phi}^{\ast}\end{pmatrix}\begin{pmatrix}\delta
T_{\ast(2)}\\\ \delta\mu_{\ast(2)}\\\
\delta\phi_{(s)\ast(2)}\end{pmatrix}=\frac{\varepsilon_{\ast}^{2}}{2}\,\,\begin{pmatrix}\mathbf{\Xi}_{\ast}&\mathbf{\nu}_{\ast}\\\
\mathbf{\nu}_{\ast}^{T}&\nu^{\ast}_{\phi}\end{pmatrix}\begin{pmatrix}\delta
T_{\ast(2)}\\\ \delta\mu_{\ast(2)}\\\ \delta\phi_{(s)\ast(2)}\end{pmatrix}\,,$
$\displaystyle\begin{pmatrix}\delta s_{\\#}\\\ \delta\varrho_{\\#}\\\
\delta\langle\mathcal{O}_{\phi}\rangle_{\\#}\end{pmatrix}$
$\displaystyle=\frac{\varepsilon_{\ast}^{2}}{2}\begin{pmatrix}T_{c}^{-1}\,c_{\mu}^{\\#}&\xi^{\\#}&\nu_{T}^{\\#}\\\
\xi^{\\#}&\chi^{\\#}&\nu_{\mu}^{\\#}\\\
\nu_{T}^{\\#}&\nu_{\mu}^{\\#}&\nu_{\phi}^{\\#}\end{pmatrix}\begin{pmatrix}\delta
T_{\\#(2)}\\\ \delta\mu_{\\#(2)}\\\
\delta\phi_{(s)\\#(2)}\end{pmatrix}=\frac{\varepsilon_{\ast}^{2}}{2}\,\,\begin{pmatrix}\mathbf{\Xi}_{\\#}&\mathbf{\nu}_{\\#}\\\
\mathbf{\nu}_{\\#}^{T}&\nu^{\\#}_{\phi}\end{pmatrix}\begin{pmatrix}\delta
T_{\\#(2)}\\\ \delta\mu_{\\#(2)}\\\ \delta\phi_{(s)\\#(2)}\end{pmatrix}\,,$
(3.17)

where $\mathbf{\Xi}^{\ast}$ and $\mathbf{\Xi}^{\\#}$ are the thermodynamic
susceptibility matrices along the broken and normal phases correspondingly
evaluated at the critical point. Using the above we can express temperature
and chemical potential variations in terms of scalar deformations and entropy
and charge densities according to,

$\displaystyle\begin{pmatrix}\delta T_{\ast(2)}\\\
\delta\mu_{\ast(2)}\end{pmatrix}$
$\displaystyle=\mathbf{\Xi}_{\ast}^{-1}\,\left(\delta\mathbf{\Phi}_{\ast(2)}-\mathbf{\nu}_{\ast}\,\delta\phi_{\ast(s)(2)}\right)\,,$
$\displaystyle\begin{pmatrix}\delta T_{\\#(2)}\\\
\delta\mu_{\\#(2)}\end{pmatrix}$
$\displaystyle=\mathbf{\Xi}_{\\#}^{-1}\,\left(\delta\mathbf{\Phi}_{\\#(2)}-\mathbf{\nu}_{\\#}\,\delta\phi_{\\#(s)(2)}\right)\,.$
(3.18)

In the above we have defined the vectors,

$\displaystyle\delta\mathbf{\Phi}_{\ast(2)}=\begin{pmatrix}\delta
s_{\ast(2)}\\\
\delta\varrho_{\ast(2)}\end{pmatrix},\qquad\delta\mathbf{\Phi}_{\\#(2)}=\begin{pmatrix}\delta
s_{\\#(2)}\\\ \delta\varrho_{\\#(2)}\end{pmatrix}\,,$ (3.19)

which according to our earlier discussion, have to be equal to each other when
constructing the time dependent perturbation. Finally, using equations (3.2)
and (3.2), the variations of the scalar VEVs can be written as,

$\displaystyle\delta\langle\mathcal{O}_{\phi}\rangle_{\ast(2)}$
$\displaystyle=\mathbf{\nu}^{\ast\,T}\mathbf{\Xi}_{\ast}^{-1}\,\delta\mathbf{\Phi}_{\ast(2)}+\left(\nu^{\ast}_{\phi}-\mathbf{\nu}_{\ast}^{T}\,\mathbf{\Xi}_{\ast}^{-1}\,\mathbf{\nu}_{\ast}\right)\,\delta\phi_{\ast(s)(2)}\,,$
$\displaystyle\delta\langle\mathcal{O}_{\phi}\rangle_{\\#(2)}$
$\displaystyle=\mathbf{\nu}^{\\#\,T}\mathbf{\Xi}_{\\#}^{-1}\,\delta\mathbf{\Phi}_{\\#(2)}+\left(\nu_{\phi}^{\\#}-\mathbf{\nu}_{\\#}^{T}\,\mathbf{\Xi}_{\\#}^{-1}\,\mathbf{\nu}_{\\#}\right)\,\delta\phi_{\\#(s)(2)}\,.$
(3.20)

The above allows us to relate the thermodynamic susceptibilities between the
two different ensembles through,

$\displaystyle\begin{pmatrix}\left.\partial_{s}\langle\mathcal{O}_{\phi}\rangle\right|_{\varrho,\phi_{(s)}}\\\
\left.\partial_{\varrho}\langle\mathcal{O}_{\phi}\rangle\right|_{s,\phi_{(s)}}\end{pmatrix}=\mathbf{\nu}^{T}\,\mathbf{\Xi}^{-1},\qquad\left.\partial_{\phi_{(s)}}\langle\mathcal{O}_{\phi}\rangle\right|_{s,\varrho}=\nu_{\phi}-\mathbf{\nu}^{T}\,\mathbf{\Xi}^{-1}\,\mathbf{\nu}\,.$
(3.21)

From the horizon point of view, the zero net charge and entropy density
translates to the boundary conditions333The reader might wonder about the fact
that there is terms which involve different numbers of $\delta$ factors. We
note that $\delta$ itself is of order zero in $\varepsilon_{\ast}$ and that
depending on the origin of each factor, it can come with different powers of
$\varepsilon_{\ast}$.,

$\displaystyle\delta\tilde{g}_{(2)}^{(0)}=0,\quad
e^{2g_{c}^{(0)}}\left(Z_{c}^{(0)}\delta\tilde{a}_{(2)}^{\prime\,(0)}+\,a_{c}^{\prime\,(0)}\left(2\,Z_{c}^{(0)}\,\delta\tilde{g}_{(2)}^{(0)}+\partial_{\phi}Z_{c}^{(0)}\,\delta\tilde{\phi}_{(2)}^{(0)}+\partial_{\rho^{2}}Z_{c}^{(0)}\,\delta\rho_{\ast(0)}^{(0)\,2}\right)\right)=0$
(3.22)

This will be useful in the next section where we employ the symplectic current
to fix the term $\omega_{[2]}$ in the expansion (3.2) and also show that
$\omega_{[1]}=0$.

At this point, we understand the gapped mode in the bulk up to order
$\varepsilon_{\ast}$ in the perturbative expansion. The ingredients we need
are the static solutions of the broken and the normal phase bulk geometries.
The final piece of information is the characteristic frequency $\omega_{[2]}$
which will be the task of the next subsection.

###  3.3 The symplectic current

In this section, we will employ the Crnkovic-Witten sympectic current to
exract the final piece of information we are after, the frequency of the
gapped mode close to the phase transition. To define it, we consider a generic
classical Lagrangian field theory of a collection $\varphi^{I}$ of fields and
two perturbative solutions $\delta_{1}\phi^{I}$ and $\delta_{2}\phi^{I}$
around a background $\phi^{I}_{b}$. If the Lagrangian density
$\mathcal{L}(\varphi^{I},\partial_{\mu}\varphi^{I})$ can be written in terms
of the fields and their first derivatives then the vector density,

$\displaystyle
P^{\mu}=\delta_{1}\phi^{I}\,\delta_{2}\left(\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\phi^{I}}\right)-\delta_{2}\phi^{I}\,\delta_{1}\left(\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\phi^{I}}\right)\,,$
(3.23)

is divergence free,

$\displaystyle\partial_{\mu}P^{\mu}=0\,.$ (3.24)

The symplectic current (3.23) is antisymmetric in field space and as such,
when the second perturbative solution is infinitesimally close to the first
one, the symplectic form is expandable around zero.

As we will see, this observation is going to be particularly useful for us. In
particular, the role of the background will be played by the black hole
solution in the broken phase. We will take the first of the two perturbations
in (3.23) to be the static perturbation which is simply a derivative of the
expansion (3.1) with respect to $\varepsilon_{\ast}$ giving,

$\displaystyle\delta g^{\ast}_{tt}$ $\displaystyle=-\varepsilon_{\ast}\,\delta
U_{\ast(2)}-\frac{\varepsilon_{\ast}^{3}}{3!}\,\delta U_{\ast(4)}+\cdots,$
$\displaystyle\delta g^{\ast}_{rr}$
$\displaystyle=-\varepsilon_{\ast}\frac{\delta
U_{\ast(2)}}{U_{c}^{2}}-\frac{\varepsilon_{\ast}^{3}}{3!}\left(\frac{\delta
U_{\ast(4)}}{U_{c}^{2}}-\frac{6\,\delta
U_{\ast(2)}^{2}}{U_{c}^{3}}\right)+\cdots\,,$ $\displaystyle\delta
g^{\ast}_{ii}$
$\displaystyle=2\,\,e^{2g_{c}}\,\left(\varepsilon_{\ast}\,\delta
g_{\ast(2)}+\frac{\varepsilon_{\ast}^{3}}{3!}\left(6\,\delta
g_{\ast(2)}^{2}+\delta g_{\ast(4)}\right)+\cdots\right),\,$
$\displaystyle\delta a^{\ast}_{t}$ $\displaystyle=\varepsilon_{\ast}\,\delta
a_{\ast(2)}+\frac{\varepsilon_{\ast}^{3}}{3!}\,\delta
a_{\ast(4)}+\cdots,\quad\delta\rho^{\ast}=\delta\rho_{\ast(0)}+\frac{\varepsilon_{\ast}^{2}}{2}\,\delta\rho_{\ast(2)}+\cdots\,,$
$\displaystyle\delta\phi^{\ast}$
$\displaystyle=\varepsilon_{\ast}\,\delta\phi_{\ast(2)}+\frac{\varepsilon_{\ast}^{3}}{3!}\,\delta\phi_{\ast(4)}+\cdots\,.$
(3.25)

The second perturbative solution will be the time dependent perturbation (3.2)
and the expansion parameter will naturally be given by $\varepsilon_{\ast}$.

In order to obtain the symplectic current for the theory of equation (2.1), we
first need to write it in a form where all the fields will appear with their
first derivative at most. This will yield an equivalent action $\tilde{S}_{b}$
such that,

$\displaystyle\tilde{S}_{b}=S_{b}+S_{GH}=\int
d^{d}x\,\mathcal{L}(\varphi^{I},\partial_{\mu}\varphi^{I})\,,$ (3.26)

where we have introduced the Gibbons-Hawking term,

$\displaystyle S_{GH}=2\,\int_{\partial M}d^{3}x\,\sqrt{-h}\,K\,,$ (3.27)

and $\varphi^{I}$ include the metric along with the rest of the matter fields
of our bulk theory. As usual, $K=\nabla_{\mu}n^{\mu}$ is the trace of the
extrinsic curvature of the conformal boundary $\partial M$ and
$n=dr/\sqrt{\mathcal{N}}$ is its normal one-form of unit norm. The integration
measure is with respect to the induced metric
$h_{\mu\nu}=g_{\mu\nu}-n_{\mu}n_{\nu}$.

For our bulk gravitational theory originating from (2.1), the sympectic
current will read,

$\displaystyle P^{\mu}=$
$\displaystyle\delta_{1}g_{\alpha\beta}\,\delta_{2}\left(\frac{\partial\mathcal{L}}{\partial\partial_{\mu}g_{\alpha\beta}}\right)-\delta_{2}g_{\alpha\beta}\,\delta_{1}\left(\frac{\partial\mathcal{L}}{\partial\partial_{\mu}g_{\alpha\beta}}\right)+\delta_{1}A_{\alpha}\,\delta_{2}\left(\frac{\partial\mathcal{L}}{\partial\partial_{\mu}A_{\alpha}}\right)-\delta_{2}A_{\alpha}\,\delta_{1}\left(\frac{\partial\mathcal{L}}{\partial\partial_{\mu}A_{\alpha}}\right)$
$\displaystyle+\delta_{1}\phi\,\delta_{2}\left(\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\phi}\right)-\delta_{2}\phi\,\delta_{1}\left(\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\phi}\right)+\delta_{1}\rho\,\delta_{2}\left(\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\rho}\right)-\delta_{2}\rho\,\delta_{1}\left(\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\rho}\right)\,.$
(3.28)

In Appendix A we evaluate the derivatives of the bulk action density with
respect to the partial derivatives of our fields.

The next step is to evaluate the symplectic current (3.3) for the pair of
perturbations (3.3) and (3.2) around the expanded broken phase background
(3.1). Before we do that, it is good to understand how we can benefit from
considering the condition (3.24) in our setup. Since we are Fourier expanding
our time dependent modes, we will write the non-trivial components as,

$\displaystyle\mathcal{P}^{t}=e^{-i\omega(t+S(r))}\,P^{t}(r),\quad\mathcal{P}^{r}=e^{-i\omega(t+S(r))}\,P^{r}(r)\,.$
(3.29)

The continuity equation (3.24) then becomes,

$\displaystyle-i\omega\,(P^{t}+S^{\prime}\,P^{r})+P^{r\prime}=0\,.$ (3.30)

Considering now the expansion in $\varepsilon_{\ast}$, we can write,

$\displaystyle P^{t}$
$\displaystyle=\varepsilon_{\ast}\,P^{t}_{(1)}+\varepsilon_{\ast}^{2}\,P^{t}_{(2)}+\cdots\,,$
$\displaystyle P^{r}$
$\displaystyle=\varepsilon_{\ast}\,P^{r}_{(1)}+\varepsilon_{\ast}^{2}\,P^{r}_{(2)}+\cdots\,,$
(3.31)

from where we we see that at leading order (3.30) implies,

$\displaystyle P^{r\prime}_{(1)}=0\,.$ (3.32)

We therefore see that the term involving the time derivative in (3.24), is
subleading in the $\varepsilon_{\ast}$ expansion. Keeping up to order
$\varepsilon_{\ast}$ in the radial component of the symplectic current we
obtain,

$\displaystyle
P^{r}_{(1)}=-i\omega_{[1]}\,e^{2g_{c}}\,U_{c}\,S^{\prime}\,\delta\rho_{\ast(0)}^{2}\,.$
(3.33)

After integrating (3.32) from the horizon up to infinity, we can show that
$\omega_{[1]}=0$, as promised. This implies that after moving on to higher
order in the perturbative expansion we will have that,

$\displaystyle P^{r\prime}_{(2)}=0\,,$ (3.34)

which will allow us to determine $\omega_{[2]}$, as we will see shortly.
Expanding the sympectic current at next to leading order we obtain,

$\displaystyle P^{r}_{(2)}$
$\displaystyle=e^{2g_{c}}\,\left(-i\omega_{[2]}\,U_{c}\,S^{\prime}\,\delta\rho_{\ast(0)}^{2}+Z_{c}\,\left(\delta\tilde{a}_{(2)}\,\delta
a_{\ast(2)}^{\prime}-\delta\tilde{a}^{\prime}_{(2)}\,\delta
a_{\ast(2)}\right)\right)$ $\displaystyle+$ $\displaystyle
2e^{2g_{c}}\,\left(\delta
g_{\ast(2)}\,\left(Z_{c}\,\delta\tilde{a}_{(2)}\,a_{c}^{\prime}-\delta\tilde{U}_{(2)}^{\prime}\right)-\delta\tilde{g}_{(2)}\,\left(Z_{c}\,\delta
a_{\ast(2)}\,a_{c}^{\prime}-\delta U_{\ast(2)}^{\prime}\right)\right)$
$\displaystyle+$ $\displaystyle
2\,U_{c}\,e^{2g_{c}}\,\left(\delta\tilde{g}_{(2)}\left(2\,\delta
g_{\ast(2)}^{\prime}+\delta\rho_{\ast(0)}\,\delta\rho_{\ast(0)}^{\prime}\right)-\delta
g_{\ast(2)}\left(2\,\delta\tilde{g}_{(2)}^{\prime}+\delta\rho_{\ast(0)}\,\delta\rho_{\ast(0)}^{\prime}\right)\right)$
$\displaystyle-$
$\displaystyle\frac{1}{2}e^{2g_{c}}\,U_{c}\,\left(\delta\rho_{\ast(0)}^{\prime}\,\left(\delta\tilde{\rho}_{(2)}-\delta\rho_{\ast(2)}\right)-\delta\rho_{\ast(0)}\,\left(\delta\tilde{\rho}^{\prime}_{(2)}-\delta\rho^{\prime}_{\ast(2)}\right)\right)$
$\displaystyle+$ $\displaystyle e^{2g_{c}}\left(-\delta
U_{\ast(2)}\,\delta\tilde{\phi}_{(2)}+\delta\tilde{U}_{(2)}\,\delta\phi_{\ast(2)}+2\,U_{c}\,\left(-\delta
g_{\ast(2)}\,\delta\tilde{\phi}_{(2)}+\delta\tilde{g}_{(2)}\,\delta\phi_{\ast(2)}\right)\right)\,\phi^{\prime}_{c}$
$\displaystyle+$ $\displaystyle
e^{2g_{c}}\,U_{c}\,\left(\delta\phi_{\ast(2)}\,\delta\tilde{\phi}_{(2)}^{\prime}-\delta\tilde{\phi}_{(2)}\,\delta\phi_{\ast(2)}^{\prime}\right)$
$\displaystyle-$ $\displaystyle e^{2g_{c}}\delta
U_{\ast(2)}\,\left(2\,\delta\tilde{g}_{(2)}^{\prime}+\delta\rho_{\ast(0)}\,\delta\rho_{\ast(0)}^{\prime}\right)+e^{2g_{c}}\,\delta\tilde{U}_{(2)}\,\left(2\,\delta
g_{\ast(2)}^{\prime}+\delta\rho_{\ast(0)}\,\delta\rho_{\ast(0)}^{\prime}\right)$
$\displaystyle-$ $\displaystyle\partial_{\phi}Z_{c}\,e^{2g_{c}}\,\left(\delta
a_{\ast(2)}\,\delta\tilde{\phi}_{(2)}-\delta\tilde{a}_{(2)}\,\delta\phi_{\ast(2)}\right)\,a_{c}^{\prime}-\partial_{\rho^{2}}Z_{c}\,e^{2g_{c}}\,\left(\delta
a_{\ast(2)}-\delta\tilde{a}_{(2)}\right)\,\delta\rho_{\ast(0)}^{2}\,a_{c}^{\prime}\,.$
(3.35)

Integrating equation (3.34) from the horizon to infinity yields a term on the
horizon and a term evaluated at the conformal boundary which have to be equal
to each other. The expression we obtain reads,

$\displaystyle-i\,\omega_{[2]}\,e^{2g_{c}^{(0)}}\,\delta\rho_{\ast(0)}^{(0)\,2}-2\,e^{2g_{c}^{(0)}}\,\delta
g_{\ast(2)}\,\delta\tilde{U}_{(2)}^{\prime\,(0)}$
$\displaystyle+\delta\tilde{a}_{(2)}^{(0)}\,e^{2g_{c}^{(0)}}\left(Z_{c}^{(0)}\delta
a_{\ast(2)}^{\prime\,(0)}+\,a_{c}^{\prime\,(0)}\left(2\,Z_{c}^{(0)}\,\delta
g_{\ast(2)}^{(0)}+\partial_{\phi}Z_{c}^{(0)}\,\delta\phi_{\ast(2)}^{(0)}+\partial_{\rho^{2}}Z_{c}^{(0)}\,\delta\rho_{\ast(0)}^{(0)\,2}\right)\right)=$
$\displaystyle-\left(2\Delta_{\phi}-3\right)\delta\phi_{(s)\ast(2)}\,\left(\delta\phi_{(v)\ast(2)}-\delta\phi_{(v)\\#(2)}\right)\,,$
(3.36)

where we imposed the absence of scalar sources on the boundary by setting
$\delta\phi_{(s)\ast(2)}=\delta\phi_{(s)\\#(2)}$ in equation (3.16). After
recognising the variations of the horizon charge and entropy densities (2.7),
we can write,

$\displaystyle\qquad-i\,\omega_{[2]}\,\frac{s_{c}}{4\pi}\,\delta\rho_{\ast(0)}^{(0)\,2}=$
$\displaystyle\delta s_{\ast(2)}\,(\delta T_{\ast(2)}-\delta
T_{\\#(2)})+\delta\varrho_{\ast(2)}\,(\delta\mu_{\ast(2)}-\delta\mu_{\\#(2)})-\delta\phi_{(s)\ast(2)}\,\left(\delta\langle\mathcal{O}_{\phi}\rangle_{\ast(2)}-\delta\langle\mathcal{O}_{\phi}\rangle_{\\#(2)}\right)\,,$
(3.37)

with the variations of the scalar VEVs defined according to (3.1). After
introducing appropriate factors of $\varepsilon_{\ast}$, setting,

$\displaystyle\delta\mathbf{\Phi}=\frac{\varepsilon_{\ast}^{2}}{2}\,\delta\mathbf{\Phi}_{\ast(2)}\,,\qquad\delta\phi_{(s)}=\frac{\varepsilon_{\ast}^{2}}{2}\,\delta\phi_{(s)\ast(2)}\,.$
(3.38)

and using equations (3.2) and (3.2), the frequency of our Higgs takes the
remarkably simple form,

$\displaystyle\omega=i\frac{8\,\Delta E}{\varpi}\,.$ (3.39)

In the above, we have defined the horizon quantity,

$\displaystyle\varpi=\frac{s_{c}\,\rho^{(0)\,2}}{4\,\pi}\,,$ (3.40)

as well as

$\displaystyle\Delta E=$
$\displaystyle\frac{1}{2}\delta\mathbf{\Phi}^{T}\,\left(\mathbf{\Xi}_{\ast}^{-1}-\mathbf{\Xi}_{\\#}^{-1}\right)\delta\mathbf{\Phi}-\delta\mathbf{\Phi}^{T}\left(\mathbf{\Xi}_{\ast}^{-1}\mathbf{\nu}_{\ast}-\mathbf{\Xi}_{\\#}^{-1}\mathbf{\nu}_{\\#}\right)\delta\phi_{(s)}$
$\displaystyle-\frac{1}{2}\left(\nu^{\ast}_{\phi}-\mathbf{\nu}_{\ast}^{T}\mathbf{\Xi}_{\ast}^{-1}\mathbf{\nu}_{\ast}-\nu^{\\#}_{\phi}+\mathbf{\nu}_{\\#}^{T}\mathbf{\Xi}_{\\#}^{-1}\mathbf{\nu}_{\\#}\right)\,\delta\phi_{(s)}^{2}\,.$
(3.41)

As we show in Appendix B, $\Delta E$ is the energy density difference between
the normal and broken phase thermal states at entropy density $s+\delta
s_{\ast}$, charge density $\varrho+\delta\varrho_{\ast}$ and scalar
deformation parameter $\phi_{(s)}+\delta\phi_{(s)\ast}$. As we would expect,
the frequency of the Higgs mode (3.3) is in the lower half of the complex
plane as long as the energy of the broken phase black holes is lower than then
normal phase in the microcanonical ensemble.

As we will see later, apart from the energy, there will be other quantities
which will have to compare between the two phases. For this reason, if a
thermodynamic quantity $\mathcal{O}$ can be expressed in terms of the entropy
density, the charge density and the scalar deformation, $\Delta\mathcal{O}$
will denote the difference,

$\displaystyle\Delta\mathcal{O}=\mathcal{O}_{\ast}(s_{c}+\delta
s_{\ast},\varrho+\delta\varrho_{\ast},\phi_{(s)}+\delta\phi_{(s)\ast})-\mathcal{O}_{\\#}(s_{c}+\delta
s_{\ast},\varrho+\delta\varrho_{\ast},\phi_{(s)}+\delta\phi_{(s)\ast})\,,$
(3.42)

with $\mathcal{O}_{\ast}$ and $\mathcal{O}_{\\#}$ the values of $\mathcal{O}$
in the broken and normal phase respectively. Moreover, $s_{c}$ is the entropy
at the critical point and is set by $\rho$ and $\phi_{(s)}$. In our work we
will only need the leading order approximation in $\varepsilon_{\ast}$ where
as we know $\delta
s_{\ast}\sim\delta\varrho_{\ast}\sim\delta\phi_{(s)\ast}\sim
O(\varepsilon_{\ast}^{2})$.

##  4 Charged Superfluids

In this section, we will examine the case in which the broken phase describes
a superfluid. To achieve this, the bulk theory will need to contain a complex
scalar field $\psi$ with a $U(1)$ gauged symmetry which will drive the phase
transition. For concreteness, we will consider the bulk action,

$\displaystyle S_{b}=\int
d^{4}x\,\sqrt{-g}\,\left(R-V-\frac{1}{2}\left(\partial\phi\right)^{2}-\frac{1}{2}\left(D\psi\right)^{2}-\frac{Z}{4}F^{2}\right)\,,$
(4.1)

with $D_{\mu}\psi=\left(\nabla_{\mu}+i\,q\,A_{\mu}\right)\,\psi$.

In the superfluid phase, the operator $\mathcal{O}_{\psi}$ which is dual to
the bulk field $\psi$ takes a non-trivial VEV yielding a corresponding non-
trivial bulk field $\psi$. In order to proceed, we will perform a field
transformation to write the complex scalar field in a polar decomposition
according to,

$\displaystyle\psi=\rho\,e^{i\,q\,\theta}\,,$ (4.2)

with $\rho>0$ and $0\leq\theta<2\pi/q$. The above transformation brings the
bulk action to the form,

$\displaystyle S_{b}=\int
d^{4}x\,\sqrt{-g}\,\left(R-V-\frac{1}{2}\left(\partial\phi\right)^{2}-\frac{1}{2}\left(\partial\rho\right)^{2}-\frac{Z}{4}F^{2}-\frac{q^{2}}{2}\rho^{2}\,B^{2}\right)\,,$
(4.3)

where we have set $B=A+\partial\theta$ and therefore $F=dB$. This form is very
similar to the bulk action (2.1) that we considered in the previous sections
for neutral scalar fields which are dual to real order parameters. The crucial
difference is that in the broken phase, the 1-form field does not enjoy gauge
invariance due to the Stueckelberg mechanism.

We would now like to consider the asymptotic expansion of 1-form field
$B_{\alpha}$ fluctuations. For the components along the conformal boundary
directions we obtain,

$\displaystyle\delta
B_{\alpha}=\frac{\partial_{\alpha}\delta\theta_{(s)}}{(r+R)^{3-2\,\Delta_{\rho}}}+\cdots+\delta
A_{\alpha}+\partial_{\alpha}\delta\theta_{(v)}+\cdots+\frac{\delta
j_{\alpha}}{r+R}+\cdots\,,$ (4.4)

with the radial component $B_{r}$ completely determined by this. The
asymptotic expansion of the 1-form field $B$ is therefore fixed by a scalar
function $\theta_{(s)}$ and the two 1-forms
$A_{\alpha}+\partial_{\alpha}\,\theta_{(v)}$ and $j_{\alpha}$.

The function $\theta_{(v)}$ and the parameter $\rho_{(v)}$ that appears in the
expansion (2) of the modulus $\rho$ of the complex field $\psi$ combine
together to parametrise the VEV,

$\displaystyle\langle\mathcal{O}_{\psi}\rangle=\frac{1}{2}(2\Delta_{\rho}-3)\,\rho_{(v)}e^{i\,q\,\theta_{(v)}}\,,$
(4.5)

from which we obtain the perturbative epxression

$\displaystyle\delta\langle\mathcal{O}_{\psi}\rangle=\langle\mathcal{O}_{\psi}\rangle_{b}\,i\,q\,\delta\theta_{(v)}+e^{i\,\arg(\langle\mathcal{O}_{\psi}\rangle_{b})}\,\delta|\langle\mathcal{O}_{\psi}\rangle|\,,$
(4.6)

in terms of the VEV of the thermal state
$\langle\mathcal{O}_{\psi}\rangle_{b}$. The combination $\delta
A_{\alpha}+\delta\partial_{\alpha}\theta_{(v)}$ is then a gauge invariant
combination of the perturbative external source $\delta A_{\alpha}$ and the
phase of the VEV $\delta\theta_{(v)}$. In the absence of external sources, the
constants of integration $j_{\alpha}$ are directly related to VEV of the
boundary charge density $\langle J\rangle_{\alpha}$ satisfying the continuity
equation $\partial_{\alpha}j^{\alpha}=0$.

The treatment of the gapped mode is identical to that of the neutral scalar
field with the gauge field $A$ replaced by the massive 1-form $B$. The whole
argument would go through apart from one technical point of particular
physical significance. This is related to the gauge transformation that we
dicuss below equation (3.2) and which removes the sources from the asymptotic
of the gauge field $A$. Since we cannot perform gauge transformations to $B$,
the same treatment is not possible in this case. However, as we can see from
the expansion (4.4), a non zero value for $B$ on the conformal boundary can be
treated as the Goldstone mode $\delta\theta_{(v)}$ as long as it is an exact
form i.e. it can be expressed as the derivative of the function
$\delta\theta_{(v)}$. This is certainly possible for the perturbations we are
studying in this paper since only the time and the radial components of the
1-form $B$ will be involved. The constant on the boundary coming from the time
component can then always be expressed as a partial derivative of a phase
$\delta\theta_{(v)}$ with respect to time.

To proceed, we introduce the Fourier modes of equation (3.11) according to,

$\displaystyle\delta B_{\mu}=e^{-i\omega(t+S(r))}\,\delta b_{\mu}(r)\,.$ (4.7)

Another important ingredient in order to construct the perturbation of order
$\varepsilon_{\ast}$ for the 1-form field is the discussion below equation
(3.2). We can indeed form zero total charge linear combinations of the static
perturbations $\delta a_{\ast(2)}$ and $\delta a_{\\#(2)}$ as we did in the
case of the neutral order parameter in order to obtain solutions of the time
component of the equations of motion. The important difference now is that the
radial component is no longer a constraint which can be solved everywhere in
the bulk by imposing charge continuity on the boundary.

This is a signal that, in addition to our previous considerations, the phase
$\delta\theta$ in the bulk is going to get involved in the perturbation at
order $\varepsilon$. More concretly, we can expand the bulk phase in
$\varepsilon_{\ast}$ according to,

$\displaystyle\delta\tilde{\theta}=\varepsilon_{\ast}^{-1}\left(\delta\tilde{\theta}_{(0)}+\varepsilon_{\ast}^{2}\,\delta\tilde{\theta}_{(2)}+\cdots\right)\,,$
(4.8)

where $\delta\tilde{\theta}_{(0)}$ is constant everywhere in the bulk and
$\delta\tilde{\theta}_{(2)}$ an analytic function of the radial coordinate.
The above implies that at order $\varepsilon_{\ast}$ the bulk perturbation for
the one form field takes the form,

$\displaystyle\delta\tilde{b}_{t(2)}$ $\displaystyle=\delta a_{\ast(2)}-\delta
a_{\\#(2)}-i\omega_{[2]}\,\delta\tilde{\theta}_{(0)}\,,$
$\displaystyle\delta\tilde{b}_{r(2)}$
$\displaystyle=-i\,S^{\prime}\,\omega_{[2]}\,\delta\tilde{\theta}_{(0)}+\delta\tilde{\theta}_{(2)}^{\prime}\,.$
(4.9)

A crucial point to note here is that the terms related to
$\delta\tilde{\theta}_{(0)}$ and $\delta\tilde{\theta}_{(2)}$ enter the time
component of the equation of motion of $\delta B$ at order
$\varepsilon_{\ast}^{3}$. This shows that the above solves the time dependent
equation at order $\varepsilon_{\ast}$ even after the addition of the terms
which are an exact form.

On the other hand, as one can easily see, both $\delta\tilde{\theta}_{(0)}$
and $\delta\tilde{\theta}_{(2)}$ will enter the radial component of the
equation of motion of $\delta B$ at order $\varepsilon_{\ast}^{3}$. Expanding
this equation close to the horizon with the boundary conditions given in (2),
(3.2) and (3.13) we obtain,

$\displaystyle\delta\tilde{\theta}_{(0)}$
$\displaystyle=\frac{1}{q^{2}\,\delta\rho_{\ast(0)}^{(0)\,2}}\,\left(Z_{c}^{(0)}\delta\tilde{a}_{(2)}^{\prime\,(0)}+\,a_{c}^{\prime\,(0)}\left(2\,Z_{c}^{(0)}\,\delta\tilde{g}_{(2)}^{(0)}+\partial_{\phi}Z_{c}^{(0)}\,\delta\tilde{\phi}_{(2)}^{(0)}+\partial_{\rho^{2}}Z_{c}^{(0)}\,\delta\tilde{\rho}_{(0)}^{(0)\,2}\right)\right)$
(4.10)
$\displaystyle=\frac{4\pi}{s_{c}\,q^{2}\,\delta\rho_{\ast(0)}^{(0)\,2}}\,\left(\delta\varrho^{(0)}_{\ast(2)}-\delta\varrho_{\\#(2)}\right)=\frac{4\pi}{s_{c}\,q^{2}\,\delta\rho_{\ast(0)}^{(0)\,2}}\,\left(\delta\varrho^{(0)}_{\ast(2)}-\delta\varrho_{\ast(2)}\right)\,.$

To obtain the first equality in the second line of (4.10), we used equation
(2.7) which relates the charge density perturbation of the normal phase
$\delta\varrho_{\\#(2)}$ to the flux density of the gauge field at the
horizon. At the same time, we have defined $\delta\varrho^{(0)}_{\ast(2)}$,
the perturbation of a horizon charge density related to the 1-form field $B$
in the broken phase. In the case of superfluids, an important point is that
this is not equal to the charge density of the dual field theory which needs
to be read off from the bulk fields after expanding close to the conformal
boundary. Finally, to obtain the last equality, we have used that the two
perturbations that we obtain from the static backgrounds have to be chosen so
that $\delta\varrho_{\\#(2)}=\delta\varrho_{\ast(2)}$. We see that the bulk
constant $\delta\tilde{\theta}_{(0)}$ is directly proportional to the
difference between the horizon and the conformal boundary charge densities.

We now turn our attention to the symplectic current built from the background
solution and the time dependent perturbation of our gapped mode. We follow the
same logic as in section 3.3 for the neutral case. In fact, the bulk actions
(2.1) and (4.3) have identical kinetic terms and they will therefore yield the
same form for the symplectic current. Keeping terms up to order
$\varepsilon^{2}_{\ast}$ yields the same form (3.3) for the radial component.

Following the argument of section 3.3, we equate the value of the radial
component $P_{(2)}^{r}$ on the horizon to its value at the conformal boundary.
In this case however, there is a non-zero contribution from the conformal
boundary due to the non-trivial asymptotics of the vector field component
$\delta\tilde{b}_{t(2)}$. This allows us to write,

$\displaystyle-i\,\omega_{[2]}\,e^{2g_{c}^{(0)}}\,\delta\rho_{\ast(0)}^{(0)\,2}-i\omega_{[2]}\,\delta\tilde{\theta}^{(0)}\,\left(\delta\varrho^{(0)}_{\ast(2)}-\delta\varrho_{\ast(2)}\right)$
$\displaystyle=\delta s_{\ast(2)}\,(\delta T_{\ast(2)}-\delta
T_{\\#(2)})+\delta\varrho_{\ast(2)}\,(\delta\mu_{\ast(2)}-\delta\mu_{\\#(2)})-\delta\phi_{(s)\ast(2)}\,\left(\delta\langle\mathcal{O}_{\phi}\rangle_{\ast(2)}-\delta\langle\mathcal{O}_{\phi}\rangle_{\\#(2)}\right)\,.$
(4.11)

After using equation (4.10) for the zero order perturbation of the phase
$\delta\tilde{\theta}_{(0)}$, we obtain the simple result,

$\displaystyle\omega=i\,\frac{8\,\Delta E}{\varpi}\,,$ (4.12)

where we have defined the transport coefficient,

$\displaystyle\varpi=\frac{s_{c}\,\rho^{(0)\,2}}{4\,\pi}+\frac{16\pi}{s_{c}\,q^{2}\,\rho^{(0)\,2}}\left(\varrho^{(0)}-\varrho\right)^{2}\,,$
(4.13)

and $\Delta E$ remains the energy difference of equation (3.3).

##  5 Green’s Functions of Scalar Operators

In this section we wish to study the effect of the Higgs mode we constructed
in the previous section on the linear response of the system near the critical
point. For this reason, we will need to introduce boundary sources for the
scalar fields in the time dependent perturbation that we constructed in
section 3.2.

As we will see shortly, the scalar sources will have to be turned on at orders
$\delta\rho_{(s)}\sim O(\varepsilon^{2}_{\ast})$ and $\delta\phi_{(s)}\sim
O(\varepsilon_{\ast})$ in the $\varepsilon_{\ast}$ expansion of equation
(3.2). Including sources does not change the arguments around the construction
of sections 3.2 and 4. As we have seen there, at next to leading order in
$\varepsilon_{\ast}$, the perturbation can is simply a linear combination of a
variation of the broken and the unbroken phase black hole backgrounds. The
former variation is fixed by how we move on the phase diagram and does not
allow us to introduce new sources in the perturbation.

This shows that the only way to introduce perturbative source for the scalar
$\phi$, which are not present in the background thermal state, is through the
variation of the normal phase branch of the black holes $\delta\phi_{\\#(2)}$
in equation (3.2). However, for the scalar which is relevant to the amplitude
of the order parameter $\langle\mathcal{O}_{\rho}\rangle$, the source will
need to be introduced in the $O(\varepsilon^{3})$ term of equation (3.2). This
can be achieved through the asymptotics of the bulk function
$\delta\tilde{\rho}_{(2)}$.

We will start our treatment by considering a source $\delta\tilde{\rho}_{(s)}$
for the operator $\mathcal{O}_{\rho}$. As described above, this appears as a
constant of integration in the asymptotic expansion,

$\displaystyle\delta\tilde{\rho}_{(2)}=\delta\tilde{\rho}_{(s)(2)}\,(r+R)^{\Delta_{\rho}-3}+\cdots+\delta\tilde{\rho}_{(v)(2)}\,(r+R)^{-\Delta_{\rho}}+\cdots\,.$
(5.1)

for the function $\delta\tilde{\rho}_{(2)}$ defined through equation (3.2).
This proves our earlier statement that the source for $\mathcal{O}_{\rho}$
should be of order $\varepsilon_{\ast}^{2}$ with
$\delta\tilde{\rho}_{(s)}=\frac{\varepsilon_{\ast}^{2}}{2}\,\delta\tilde{\rho}_{(s)(2)}$.
By equating the value of the radial component of the symplectic current (3.3)
on the horizon and the conformal boundary we obtain,

$\displaystyle-i\,\omega_{[2]}\,e^{2g_{c}^{(0)}}\,\delta\rho_{\ast(0)}^{(0)\,2}-i\omega_{[2]}\,\delta\tilde{\theta}^{(0)}\,\left(\delta\varrho^{(0)}_{\ast(2)}-\delta\varrho_{\ast(2)}\right)$
$\displaystyle=\delta s_{\ast(2)}\,(\delta T_{\ast(2)}-\delta
T_{\\#(2)})+\delta\varrho_{\ast(2)}\,(\delta\mu_{\ast(2)}-\delta\mu_{\\#(2)})-\delta\phi_{(s)\ast(2)}\,\left(\delta\langle\mathcal{O}_{\phi}\rangle_{\ast(2)}-\delta\langle\mathcal{O}_{\phi}\rangle_{\\#(2)}\right)$
$\displaystyle\quad+\frac{1}{2}\,\delta\tilde{\rho}_{(s)(2)}\,\delta\langle\mathcal{O}_{\rho}\rangle_{\ast(0)}\,.$
(5.2)

with $\delta\tilde{\theta}^{(0)}$ given once again by equation (4.10). The
above allows us to relate the frequency
$\omega=\varepsilon_{\ast}^{2}\,\omega_{[2]}$ to the source
$\delta\tilde{\rho}_{(s)}$. On the other hand, the expansion (3.1) for the
VEVs of the operators $\mathcal{O}_{\rho}$ and $\mathcal{O}_{\phi}$ is enough
to fix the retarded Green’s function,

$\displaystyle G_{\mathcal{O}_{\rho}\mathcal{O}_{\rho}}(\omega)$
$\displaystyle=\frac{\delta\tilde{\langle\mathcal{O}_{\rho}\rangle}}{\delta\tilde{\rho}_{(s)}}=\frac{\delta\langle\mathcal{O}_{\rho}\rangle_{\ast}^{2}}{\varpi\,\left(-i\,\omega+\omega_{gap}\right)}$
$\displaystyle=$
$\displaystyle\frac{(\Delta\langle\mathcal{O}_{\rho}\rangle)^{2}}{\varpi\,\left(-i\,\omega+\omega_{gap}\right)}\,,$
(5.3)

where the symbol $\Delta$ is as defined in equation (3.42). We have defined
the transport coefficient $\varpi$ and gap frequency $\omega_{gap}$ according
to,

$\displaystyle\varpi=\frac{s_{c}\,\rho^{(0)\,2}}{4\,\pi}+\frac{16\pi}{s_{c}\,q^{2}\,\rho^{(0)\,2}}\left(\varrho^{(0)}-\varrho\right)^{2},\qquad\omega_{gap}=-\frac{8\,\Delta
E}{\varpi}\,.$ (5.4)

This is the first Green’s function we determine and as we anticipated it is
dominated by a single simple pole at $\omega=-i\,\omega_{gap}$ agreeing with
the gap of equation (4.12). Notice, that equation (5) allows us to write the
expressions,

$\displaystyle\omega_{gap}$
$\displaystyle=\frac{(\Delta\langle\mathcal{O}_{\rho}\rangle)^{2}}{\chi_{\mathcal{O}_{\rho}\mathcal{O}_{\rho}}}\,\varpi^{-1}\,,$
$\displaystyle\chi_{\mathcal{O}_{\rho}\mathcal{O}_{\rho}}$
$\displaystyle=-\frac{(\Delta\langle\mathcal{O}_{\rho}\rangle)^{2}}{8\,\Delta
E}\,.$ (5.5)

where $\chi_{\mathcal{O}_{\rho}\mathcal{O}_{\rho}}$ is the susceptibility of
the operator $\mathcal{O}_{\rho}$. The above allows us to write the Kubo
formula,

$\displaystyle\frac{\varpi}{(\Delta\langle\mathcal{O}_{\rho}\rangle)^{2}}=\frac{1}{\chi_{\mathcal{O}_{\rho}\mathcal{O}_{\rho}}^{2}}\lim_{\omega\to
0}\frac{\mathrm{Im}G_{\mathcal{O}_{\rho}\mathcal{O}_{\rho}}(\omega)}{\omega}\,,$
(5.6)

noting that the left hand side remains finite at the critical point.

The next Green’s function we can compute after having turned on the source for
$\mathcal{O}_{\rho}$ is,

$\displaystyle
G_{\mathcal{O}_{\phi}\mathcal{O}_{\rho}}(\omega)=\frac{\delta\tilde{\langle\mathcal{O}_{\phi}\rangle}}{\delta\tilde{\rho}_{(s)}}=2\,\frac{\delta\langle\mathcal{O}_{\rho}\rangle_{\ast}\,\left(\delta\langle\mathcal{O}_{\phi}\rangle_{\ast}-\delta\langle\mathcal{O}_{\phi}\rangle_{\\#}\right)}{\varpi\,\left(-i\,\omega+\omega_{gap}\right)}\,,$
(5.7)

where once again we used the relation (5) for the source
$\delta\tilde{\rho}_{(s)}$. In the current situation, we are not turning on
the source $\delta\tilde{\phi}_{(s)}$ for the scalar operator. Therefore, in
this case we simply have,

$\displaystyle\Delta\langle\mathcal{O}_{\phi}\rangle=\delta\langle\mathcal{O}_{\phi}\rangle_{\ast}-\delta\langle\mathcal{O}_{\phi}\rangle_{\\#}\,,$
(5.8)

allowing us to write,

$\displaystyle
G_{\mathcal{O}_{\phi}\mathcal{O}_{\rho}}(\omega)=2\,\frac{\Delta\langle\mathcal{O}_{\rho}\rangle\,\Delta\langle\mathcal{O}_{\phi}\rangle}{\varpi\,\left(-i\,\omega+\omega_{gap}\right)}\,.$
(5.9)

Once again, we see that the retarded Green’s functions are determined by the
transport coefficient $\varpi$ and thermodynamic properties of the broken and
normal phases.

We now turn our attention to the source of the scalar operator
$\mathcal{O}_{\phi}$. Introducing a source $\delta\tilde{\phi}_{(s)(2)}$ as in
equation (3.16) allows us to express the scalar source variation for the
normal phase as,

$\displaystyle\delta\phi_{(s)\\#}=\delta\phi_{(s)\ast}-\delta\tilde{\phi}_{(s)}\,.$
(5.10)

The final ingredient left in order to compute the retarded Green’s function is
a relation that we can obtain from the symplectic current (3.3). By evaluating
it on the horizon and the conformal boundary we obtain the relation,

$\displaystyle-i\,\omega_{[2]}\,e^{2g_{c}^{(0)}}\,\delta\rho_{\ast(0)}^{(0)\,2}-i\omega_{[2]}\,\delta\tilde{\theta}^{(0)}\,\left(\delta\varrho^{(0)}_{\ast(2)}-\delta\varrho_{\ast(2)}\right)$
$\displaystyle=\delta s_{\ast(2)}\,(\delta T_{\ast(2)}-\delta
T_{\\#(2)})+\delta\varrho_{\ast(2)}\,(\delta\mu_{\ast(2)}-\delta\mu_{\\#(2)})-\delta\phi_{(s)\ast(2)}\,\left(\delta\langle\mathcal{O}_{\phi}\rangle_{\ast(2)}-\delta\langle\mathcal{O}_{\phi}\rangle_{\\#(2)}\right)$
$\displaystyle\quad+\delta\tilde{\phi}_{(s)(2)}\,\delta\langle\mathcal{O}_{\phi}\rangle_{\ast(2)}\,.$
(5.11)

After using equations (3.2) and (3.2) as well as the definitions (3.42) and
(5.4), the above takes the form,

$\displaystyle
2\,\varepsilon_{\ast}\,\delta\tilde{\phi}_{(s)}\,\Delta\langle\mathcal{O}_{\phi}\rangle$
$\displaystyle=-\varpi\,\left(i\,\omega-\omega_{gap}\right)\,.$ (5.12)

The combination of this and relation (3.1) allows us to compute the retarded
Green’s functions,

$\displaystyle G_{\mathcal{O}_{\phi}\mathcal{O}_{\phi}}(\omega)$
$\displaystyle=\frac{\delta\tilde{\langle\mathcal{O}_{\phi}\rangle}}{\delta\tilde{\phi}_{(s)}}=2\,\frac{\delta\langle\mathcal{O}_{\phi}\rangle_{\ast}-\delta\langle\mathcal{O}_{\phi}\rangle_{\\#}}{\varepsilon_{\ast}\,\delta\tilde{\phi}_{(s)}}$
$\displaystyle=4\,\frac{\left(\Delta\langle\mathcal{O}_{\phi}\rangle\right)^{2}}{\varpi\,\left(-i\,\omega+\omega_{gap}\right)}+\left.\partial_{\phi_{(s)}}\langle\mathcal{O}_{\phi}\rangle_{\\#}\right|_{s,\varrho}\,,$
(5.13)

where the last term is computed in the normal phase, at the critical point. To
see this relation, we have combined equations (3.2) and (3.21) along with
(5.10). Finally, we can use the result of equation (5.12) to compute,

$\displaystyle G_{\mathcal{O}_{\rho}\mathcal{O}_{\phi}}(\omega)$
$\displaystyle=\frac{\delta\tilde{\langle\mathcal{O}_{\rho}\rangle}}{\delta\tilde{\phi}_{(s)}}=\frac{\delta\langle\mathcal{O}_{\rho}\rangle_{\ast}}{\varepsilon_{\ast}\,\delta\tilde{\phi}_{(s)}}$
$\displaystyle=2\,\frac{\Delta\langle\mathcal{O}_{\phi}\rangle\,\Delta\langle\mathcal{O}_{\rho}\rangle}{\varpi\,\left(-i\,\omega+\omega_{gap}\right)}\,.$
(5.14)

From the form of the Green’s functions in equations (5.9) and (5), it is
trivial to check the validity of the Onsager relation
$G_{\mathcal{O}_{\rho}\mathcal{O}_{\phi}}(\omega)=G^{\ast}_{\mathcal{O}_{\phi}\mathcal{O}_{\rho}}(-\omega)$.

It is interesting to examine the behaviour of the boundary theory Goldstone
mode which is represented by the angle $\delta\theta_{(v)}$ of equation (4.6).
This can be read off from the asymptotics of the time component of the 1-form
field $\delta\tilde{b}_{t}$ and interpreting the constant term as the time
derivative $\partial_{t}\delta\theta_{(v)}$. This gives,

$\displaystyle\delta\tilde{\theta}_{(v)(0)}=\delta\tilde{\theta}_{(0)}+\frac{2\,i}{\omega}\,\left(\delta\mu_{\ast}-\delta\mu_{\\#}\right)\,,$
(5.15)

with the variation of the normal phase chemical potential $\delta\mu_{\\#(2)}$
being fixed by equation (3.2).

The VEV of the operator $\mathcal{O}_{\rho}$ is related to the amplitude of
the order parameter, according to the decomposition of fluctuations (4.6).
More specifically, in the broken phase, where the polar coordinate
decomposition (4.2) makes sense, we have,

$\displaystyle\delta\langle\mathcal{O}_{\rho}\rangle=2\,\delta|\langle\mathcal{O}_{\psi}\rangle|=\frac{1}{|\langle\mathcal{O}_{\psi}\rangle_{b}|}\,\left(\langle\mathcal{O}_{\bar{\psi}}\rangle_{b}\,\delta\langle\mathcal{O}_{\psi}\rangle+\langle\mathcal{O}_{\psi}\rangle_{b}\,\delta\langle\mathcal{O}_{\bar{\psi}}\rangle\right)\,.$
(5.16)

It is interesting to also consider the operator $\mathcal{O}_{Y}$ which is
related to the phase of the order parameter and therefore the Goldstone mode
of the theory. For the fluctuations of this operator we can write,

$\displaystyle\delta\langle\mathcal{O}_{Y}\rangle=q\,\langle\mathcal{O}_{\rho}\rangle_{b}\,\delta\theta_{(v)}=\frac{1}{i\,|\langle\mathcal{O}_{\psi}\rangle_{b}|}\,\left(\langle\mathcal{O}_{\bar{\psi}}\rangle_{b}\,\delta\langle\mathcal{O}_{\psi}\rangle-\langle\mathcal{O}_{\psi}\rangle_{b}\,\delta\langle\mathcal{O}_{\bar{\psi}}\rangle\right)\,.$
(5.17)

Given the expression (5.15) for the phase, we can easily compute the retarded
Green’s fucntions measuring the response of the operator $\mathcal{O}_{Y}$
against sources for the operators $\langle\mathcal{O}_{\rho}\rangle$ and
$\langle\mathcal{O}_{\phi}\rangle$,

$\displaystyle G_{\mathcal{O}_{Y}\mathcal{O}_{\phi}}(\omega)$
$\displaystyle=\frac{\delta\tilde{\langle\mathcal{O}_{Y}\rangle}}{\delta\tilde{\phi}_{(s)}}=\frac{2\,q\,\Delta\langle\mathcal{O}_{\rho}\rangle\,\Delta\langle\mathcal{O}_{\phi}\rangle}{\omega}\,\frac{\omega\,\vartheta+2\,i\,\Delta\mu}{\varpi\,\left(-i\,\omega+\omega_{gap}\right)}+\frac{2\,q\,\Delta\langle\mathcal{O}_{\rho}\rangle}{i\,\omega}\,\left.\frac{\partial\mu_{\\#}}{\partial\phi_{(s)}}\right|_{s,\varrho}\,,$
$\displaystyle G_{\mathcal{O}_{Y}\mathcal{O}_{\rho}}(\omega)$
$\displaystyle=\frac{\delta\tilde{\langle\mathcal{O}_{Y}\rangle}}{\delta\tilde{\rho}_{(s)}}=\frac{q\,\Delta\langle\mathcal{O}_{\rho}\rangle^{2}}{\omega}\,\frac{\omega\,\vartheta+2\,i\,\Delta\mu}{\varpi\,\left(-i\,\omega+\omega_{gap}\right)}\,.$
(5.18)

In the above we have set,

$\displaystyle\vartheta=\frac{8\,\pi}{s_{c}\,q^{2}\,\rho^{(0)2}}\,\left(\varrho^{(0)}-\varrho\right)\,.$
(5.19)

An obvious remaining question regards turning on a source
$\delta\tilde{s}_{Y}$ for the operator $\mathcal{O}_{Y}$. As explained in
[16], in the spontaneous case this is achieved by considering a non-trivial
source term $\delta\theta_{(s)}$ in the UV expansion (4.4) for the bulk vector
field. However, as shown in [16], turning a source $\lambda$ for the complex
operator $\mathcal{O}_{\psi}$ would imply the non-conservation of electric
charge through the Ward identity,

$\displaystyle\nabla_{\alpha}\langle
J^{\alpha}\rangle=iq\,\left(\langle\mathcal{O}_{\psi}\rangle\lambda^{\ast}-\langle\mathcal{O}_{\psi}^{\ast}\rangle\lambda\right)\,.$
(5.20)

Applying this for the case of our perturbative setup gives,

$\displaystyle\nabla_{\alpha}\langle
J^{\alpha}\rangle=q\,\langle\mathcal{O}_{\rho}\rangle_{b}\,\delta\tilde{s}_{Y}\,.$
(5.21)

Having it mind that the frequency is of order
$\mathcal{O}(\varepsilon_{\ast}^{2})$ and the leading correction to the charge
density is of order $\mathcal{O}(\varepsilon_{\ast})$, we see that the source
$\delta\tilde{s}_{Y}$ should be of order $\mathcal{O}(\varepsilon_{\ast}^{2})$
and we will write
$\delta\tilde{s}_{Y}=\frac{\varepsilon_{\ast}^{2}}{2}\,\delta\tilde{s}_{Y(2)}+\cdots$.
Combining the above ingredients leads us to the charge imbalance,

$\displaystyle-i\omega_{[2]}\,\left(\delta\varrho_{\ast(2)}-\delta\varrho_{\\#(2)}\right)=\frac{q}{2}\,\delta\langle\mathcal{O}_{\rho}\rangle_{\ast(0)}\,\delta\tilde{s}_{Y(2)}\,,$
(5.22)

when linearly superposing the static perturbations obtained from the broken
and the normal phase black holes.

We see that this source implies a slight modification of our previous
derivations to account for the non-zero difference between the charge
densities. One of the crucial steps concerns equation (4.10) which now gets
modified to,

$\displaystyle\delta\tilde{\theta}_{(0)}=\frac{4\pi}{s_{c}\,q^{2}\,\delta\rho_{\ast(0)}^{(0)\,2}}\,\left(\delta\varrho^{(0)}_{\ast(2)}-\delta\varrho_{\ast(2)}+\frac{i\,q\,\delta\langle\mathcal{O}_{\rho}\rangle_{\ast(0)}}{2\,\omega_{[2]}}\,\delta\tilde{s}_{Y(2)}\right)\,.$
(5.23)

Given the conditions (5.21), (5.22) and (5.23), evaluated the radial component
of the symplectic current (3.3) yields,

$\displaystyle\delta\tilde{s}_{Y}=-\frac{\varpi\,\left(-i\omega+\omega_{gap}\right)}{\varepsilon_{\ast}\,\Delta\langle\mathcal{O}_{\rho}\rangle}\,\frac{\omega}{\omega\,\vartheta+2\,i\,\Delta\mu}\,.$
(5.24)

We are now in the position to evaluate the retarded Green’s functions,

$\displaystyle G_{\mathcal{O}_{Y}\mathcal{O}_{Y}}(\omega)$
$\displaystyle=\frac{q\,\langle\mathcal{O}_{\rho}\rangle\,\delta\tilde{\theta}_{(v)}}{\delta\tilde{s}_{Y}}$
$\displaystyle=-\frac{q^{2}\,\Delta\langle\mathcal{O}_{\rho}\rangle^{2}\,\left(\omega\,\vartheta+2\,i\,\Delta\mu\right)^{2}}{\omega^{2}\,\varpi\,\left(-i\,\omega+\omega_{gap}\right)}+\frac{i\,q^{2}\,\gamma\,\Delta\langle\mathcal{O}_{\rho}\rangle^{2}}{\omega}-\frac{2\,q^{2}\,\Delta\langle\mathcal{O}_{\rho}\rangle^{2}}{\omega^{2}}\,\left.\frac{\partial\mu_{\\#}}{\partial\varrho}\right|_{s,\phi_{(s)}}\,,$
$\displaystyle G_{\mathcal{O}_{\phi}\mathcal{O}_{Y}}(\omega)$
$\displaystyle=\frac{\delta\tilde{\langle\mathcal{O}_{\phi}\rangle}}{\delta\tilde{s}_{Y}}=\frac{2\,\Delta\langle\mathcal{O}_{\phi}\rangle}{\varepsilon_{\ast}\,\delta\tilde{s}_{Y}}+\frac{2\,i\,q\,\Delta\langle\mathcal{O}_{\rho}\rangle}{\omega}\,\left.\frac{\partial\langle\mathcal{O}_{\phi}\rangle_{\\#}}{\partial\rho}\right|_{s,\phi_{(s)}}$
$\displaystyle=-\frac{2\,q\,\Delta\langle\mathcal{O}_{\rho}\rangle\,\Delta\langle\mathcal{O}_{\phi}\rangle}{\omega}\,\frac{\omega\,\vartheta+2\,i\,\Delta\mu}{\varpi\,\left(-i\,\omega+\omega_{gap}\right)}+\frac{2\,i\,q\,\Delta\langle\mathcal{O}_{\rho}\rangle}{\omega}\,\left.\frac{\partial\langle\mathcal{O}_{\phi}\rangle_{\\#}}{\partial\rho}\right|_{s,\phi_{(s)}}\,,$
$\displaystyle G_{\mathcal{O}_{\rho}\mathcal{O}_{Y}}(\omega)$
$\displaystyle=\frac{\delta\tilde{\langle\mathcal{O}_{\rho}\rangle}}{\delta\tilde{s}_{Y}}=-\frac{q\,\Delta\langle\mathcal{O}_{\rho}\rangle^{2}}{\omega}\,\frac{\omega\,\vartheta+2\,i\,\Delta\mu}{\varpi\,\left(-i\,\omega+\omega_{gap}\right)}\,,$
(5.25)

after fixing,

$\displaystyle\gamma=\frac{4\,\pi}{s_{c}\,q^{2}\,\rho^{(0)2}}=\frac{1}{\varpi}\left(\vartheta^{2}+\frac{1}{q^{2}}\right)\,.$
(5.26)

The last equality can be used to show that our expression for
$G_{\mathcal{O}_{Y}\mathcal{O}_{Y}}$ leads to positive spectral weight.
Moreover, the double pole in the same Green’s function is due to the presence
of the Goldstone mode, directly related to the phase of the condensate.

After noting that $\mathcal{O}_{Y}$ transforms as a pseudoscalar operator
under time reversal, our expressions for the retarded Green’s funcitons
satisfy the Onsager relations444In order to check this, it is useful to note
the Maxwell type of relation
$\left.\frac{\partial\langle\mathcal{O}_{\phi}\rangle}{\partial\varrho}\right|_{s,\phi_{(s)}}=-\left.\frac{\partial\mu}{\partial\phi_{(s)}}\right|_{s,\varrho}$
coming from the enlarged first law (B.2). Finally, our Green’s function
$G_{\mathcal{O}_{Y}\mathcal{O}_{Y}}$ agrees555In order to see the agreement,
one has to write the transport coefficient $\Xi$ of [16] as
$\Xi\approx\gamma\,\chi_{QQ}^{2}/2$ in the near critical point limit.
Moreover, at zero chemical potential we have that
$\frac{\partial\mu_{\ast}}{\partial\varrho}=\chi_{QQ}^{-1}$. Note also that
there is a mismatch of a factor of $2$ due to the different normalisation of
the kinetic term of the complex scalar in our bulk action (4.1). with our
previous result in [16] at zero chemical potential and when we take the near
critical point and zero pinning limit of the result presented in [16].

##  6 Numerical checks

The aim of this section is to provide numerical evidence in support of the
analytic expression for the gapped mode given in equation (3.39) and (4.12).
We consider the model (2.1) and (4.3) respectively, together with,

$\displaystyle
V=-6+\frac{m_{\rho}^{2}}{2}\rho^{2}+\frac{\lambda_{\rho}}{2}\,\rho^{4}+\frac{m_{\phi}^{2}}{2}\phi^{2}+\frac{\lambda_{\phi}}{2}\phi^{4}+\lambda\,\rho^{2}\phi^{2}\,$
$\displaystyle\tau=1+\zeta_{\rho}\rho^{2}+\zeta_{\phi}\phi^{2}$ (6.1)

and $m_{\rho}^{2}=-2\,,m^{2}_{\phi}=-2$. As described in section 2, we choose
to deform our boundary theory by a chemical potential $\mu=1$ and a relevant
operator, $O_{\phi}$, with scaling dimension $\Delta_{\phi}=2$; in particular
we pick the source for $\phi$ to be given by $\phi_{s}=1$. The corresponding
backreacted solution will then be given by black holes supported by a non-
trivial profile for the scalar field $\phi$, while the scalar field $\rho$
remains trivial. These solutions correspond to the normal phase of the system.

###  6.1 Ungauged model

For the ungauged model with
$\zeta_{\rho}=1,\zeta_{\phi}=0,\lambda_{\phi}=1,\lambda_{\rho}=1,\lambda=-1$,
the normal phase black holes exhibit a second order phase transition at
temperature $T_{c}=0.115$ to a configuration supported by a non-trivial scalar
field $\rho$. The latter describe the broken phase that we are interested in.

With these broken phase black holes at hand, we now turn our attention to
studying perturbations around them. In order to check numerically the validity
of the analytic expressions for the gap, we need to construct the quasinormal
modes. In particular, we consider the following perturbations,

$\displaystyle\delta g_{\mu\nu}=e^{-i\omega\,v(t,r)}\left(-U\delta
g_{tt}(r)+e^{2g(r)}\delta g_{xx}(r)(dx^{2}+dy^{2})\right)$
$\displaystyle\delta A=e^{-i\omega\,v(t,r)}\delta a_{t}(r)\,dt\,,$
$\displaystyle\delta\phi=e^{-i\omega\,v(t,r)}\delta\phi(r)\,,$
$\displaystyle\delta\rho=e^{-i\omega\,v(t,r)}\delta\rho(r)\,,$ (6.2)

where $v$ is the infalling Eddington-Finkelstein coordinate defined as,

$v(t,r)=t+\int_{\infty}^{r}\frac{dy}{U(y)}\,.$ (6.3)

These perturbations satisfy three first order and two second order
differential equations, which we solve using a shooting method subject to
appropriate boundary conditions. In particular, in the horizon (located at
r=0) we impose,

$\displaystyle\delta g_{tt}=c_{1}\,r+\dots\,,$ $\displaystyle\delta
g_{xx}=c_{2}\,r+\dots\,,$ $\displaystyle\delta a_{t}=c_{3}\,r+\dots\,,$
$\displaystyle\delta\rho=\delta\rho_{h}+\dots\,,$
$\displaystyle\delta\phi=\delta\phi_{h}+\dots\,,$ (6.4)

where $c_{1},c_{2},c_{3}$ are fixed, while close to the boundary we impose,

$\displaystyle\delta g_{tt}=\delta g_{tt}^{(s)}+\dots\,,$ $\displaystyle\delta
g_{xx}=\delta g_{xx}^{(s)}+\dots\,,$ $\displaystyle\delta a_{t}=\delta
a^{(s)}+\dots\,,$
$\displaystyle\delta\rho=\frac{\delta\rho_{s}}{r}+\frac{\delta\rho_{v}}{r^{2}}+\dots\,,$
$\displaystyle\delta\phi=\frac{\delta\phi_{s}}{r}+\frac{\delta\phi_{v}}{r^{2}}+\dots\,,$
(6.5)

For the computation of quasinormal modes, we need to ensure that we remove all
the sources from the UV expansion up to a combination of coordinate
reparametrisations and gauge transformations,

$\displaystyle[\delta g_{\mu\nu}+\mathcal{L}_{\zeta}g_{\mu\nu}]\to 0\,,$
$\displaystyle[\delta A+\mathcal{L}_{\zeta}A+d\Lambda]\to 0\,,$
$\displaystyle[\delta\phi+\mathcal{L}_{\zeta}\phi]\to 0\,,$
$\displaystyle[\delta\rho+\mathcal{L}_{\zeta}\rho]\to 0\,.$ (6.6)

where the gauge transformations are of the form,

$\displaystyle x^{\mu}\to x^{\mu}+\zeta^{\mu}\,\quad\zeta$
$\displaystyle=e^{-i\omega t}\,\zeta^{\mu}\,\partial_{\mu}\,,$ $\displaystyle
A_{\mu}\to A_{\mu}+\partial_{\mu}\Lambda\,\quad\Lambda$
$\displaystyle=e^{-i\omega t}\,\lambda\,,$ (6.7)

for $\zeta$, $\lambda$ constants. This requirement boils down to the sources
appearing in (6.1) taking the form,

$\displaystyle\delta g_{tt}^{(s)}=2i\omega\,\zeta_{1}-2\zeta_{2}\,,$
$\displaystyle\delta g_{xx}^{(s)}=-2\,\zeta_{2}\,,$ $\displaystyle\delta
a^{(s)}=i\omega(\mu\,\zeta_{1}+\lambda)\,,$ $\displaystyle\delta\rho_{s}=0\,,$
$\displaystyle\delta\phi_{s}=\phi_{s}\,\zeta_{2}\,.$ (6.8)

Overall, in addition to the frequency $\omega$, we have two constants in the
IR ($\delta\rho_{h},\delta\phi_{h}$) and five constants in the UV
($\zeta_{1},\zeta_{2},\lambda,\delta\rho^{(v)},\delta\phi^{(v)}$). This gives
a total of eight free constants, out of which one can be set to unity due to
the linearity of the equations. The remaining seven parameters match exactly
the integration constants of the problem, giving rise to a single (discrete
set) solution. In figure 1, we compare the results for gap coming from the
direct calculation of the quasinormal modes and from the analytic expression
(3.39) for $\zeta_{\rho}=1,\zeta_{\phi}=0,\lambda_{\phi}=1,\lambda_{\rho}=1$,
and $\lambda=-1$. In particular, we plot the ratio
$\omega_{analytic}/\omega_{numerics}$ as a function of $T/T_{c}$. We see good
quantitative agreement as $T\to T_{c}$.

Figure 1: Plot of the ratio of the analytic result for the gap coming from
3.39 and the gap computed numerically through the calculation of quasinormal
modes as a function of $T/T_{c}$. We see good quantitative agreement close to
the critical temperature. Here
$\phi_{s}=1,\mu=1,\zeta_{\rho}=1,\zeta_{\phi}=0,\lambda_{\rho}=1,\lambda_{\phi}=1$
and $\lambda=-1$.

###  6.2 Superfluid model

For superfluids with
$\zeta_{\rho}=1,\zeta_{\phi}=0,\lambda_{\phi}=1,\lambda_{\rho}=1,\lambda=-1$,
the critical temperature is $T_{c}=0.128$ and it also corresponds to a second
order phase transition. The broken phase is supported by a non-trivial charged
condensate, $\rho$. Given these backreacted black holes, we now turn our
attention to studying perturbations around them.

Similarly to above, to check the validity of the analytic expressions for the
gap, we construct the quasinormal modes. In particular, we consider the
following perturbations,

$\displaystyle\delta g_{\mu\nu}=e^{-i\omega\,v(t,r)}\left(-U\delta
g_{tt}(r)+e^{2g(r)}\delta g_{xx}(r)(dx^{2}+dy^{2})\right)$
$\displaystyle\delta B=e^{-i\omega\,v(t,r)}(\delta b_{t}(r)\,dt+\delta
b_{r}(r)dr)\,,$ $\displaystyle\delta\phi=e^{-i\omega\,v(t,r)}\delta\phi(r)\,,$
$\displaystyle\delta\rho=e^{-i\omega\,v(t,r)}\delta\rho(r)\,,$ (6.9)

where $v$ is the infalling Eddington-Finkelstein coordinate. These
perturbations satisfy three first order and three second order differential
equations, which we solve using a shooting method subject to the following
boundary conditions. Close to the horizon (located at r=0) we impose,

$\displaystyle\delta g_{tt}=c_{1}\,r+\dots\,,$ $\displaystyle\delta
g_{xx}=c_{2}\,r+\dots\,,$ $\displaystyle\delta b_{t}=\delta
b_{t}^{h}+\dots\,,$ $\displaystyle\delta b_{r}=\frac{\delta
b_{t}^{h}}{4\pi\,T\,r}+\dots\,,$
$\displaystyle\delta\rho=\delta\rho^{h}+\dots\,,$
$\displaystyle\delta\phi=\delta\phi^{h}+\dots\,,$ (6.10)

where $c_{1},c_{2}$ are fixed, while close to the boundary we require that,

$\displaystyle\delta g_{tt}=\delta g_{tt}^{(s)}+\dots\,,$ $\displaystyle\delta
g_{xx}=\delta g_{xx}^{(s)}+\dots\,,$ $\displaystyle\delta b_{t}=\delta
b_{t}^{(s)}+\dots\,,$ $\displaystyle\delta b_{r}=0+\dots\,,$
$\displaystyle\delta\rho=\frac{\delta\rho_{s}}{r}+\frac{\delta\rho_{v}}{r^{2}}+\dots\,,$
$\displaystyle\delta\phi=\frac{\delta\phi_{s}}{r}+\frac{\delta\phi_{v}}{r^{2}}+\dots\,,$
(6.11)

Demanding the absence of sources, the sources appearing in (6.2) take the
form,

$\displaystyle\delta g_{tt}^{(s)}=2i\omega\,\zeta_{1}-2\zeta_{2}\,,$
$\displaystyle\delta g_{xx}^{(s)}=-2\,\zeta_{2}\,,$ $\displaystyle\delta
b_{t}^{(s)}=i\omega(\mu\,\zeta_{1}+\lambda-\delta\theta_{(v)})\,,$
$\displaystyle\delta\rho_{s}=0\,,$
$\displaystyle\delta\phi_{s}=\phi_{s}\,\zeta_{2}\,,$ (6.12)

Overall, in addition to the frequency $\omega$, we have three constants in the
IR ($\delta\rho^{h},\delta\phi^{h},\delta b_{t}^{h}$) and six constants in the
UV
($\zeta_{1},\zeta_{2},\lambda,\delta\rho^{(v)},\delta\phi^{(v)},\delta\theta_{(v)}$).
Due to the linearity of the equations, one out of these ten free constants can
be set to unit. One is then left with nine parameters which match exactly the
integration constants of the problem, giving rise to a single (discrete set)
solution. Just like above, in figure 2, we plot the ratio
$\omega_{analytic}/\omega_{numerics}$ coming from the analytic expression
(4.12) and from the direct calculation of the quasinormal modes for
$\zeta_{\rho}=1,\zeta_{\phi}=0,\lambda_{\phi}=1,\lambda_{\rho}=1$, and
$\lambda=-1$. We see good quantitative agreement as $T\to T_{c}$.

Figure 2: Comparison of the analytic and numerical result for the gap for
superfluids. Here
$\phi_{s}=1,q=1,\mu=1,\zeta_{\rho}=1,\zeta_{\phi}=0,\lambda_{\rho}=1,\lambda_{\phi}=1$
and $\lambda=-1$.

For this model with
$\zeta_{\rho}=0,\zeta_{\phi}=0,\lambda_{\phi}=1,\lambda_{\rho}=1/4,\lambda=-3/4,$
and $T_{c}=0.0484121$, we also compute the Green’s functions for the scalar
fields for $T/T_{c}=0.999957$, corresponding to $\omega_{gap}=4\cdot 10^{-5}$.
In this case, the IR and UV expansions stay unchanged. For the metric and
gauge field sources we impose absence of sources just like above. For the
scalar field sources, we impose,

$\displaystyle\delta\rho_{s}=s_{1}\,,$
$\displaystyle\delta\phi_{s}=s_{2}+\phi_{s}\,\zeta_{2}\,,$ (6.13)

and we set either $(s_{1},s_{2})=(1,0)$ or $(s_{1},s_{2})=(0,1)$ depending on
which correlator we want to compute. Note that under the reparametrisation
(6.1), the VEVs for the scalars transform as

$\displaystyle\delta\rho_{v}\to\delta\rho_{v}-2\rho_{v}\,\zeta_{2}\,,$
$\displaystyle\delta\phi_{v}\to\delta\phi_{v}-2\phi_{v}\,\zeta_{2}+i\omega\,\phi_{s}\,\zeta_{2}\,,$
(6.14)

Overall, for fixed $s_{1},s_{2}$, we have nine constants in the expansion
$\delta\rho^{h},\delta\phi^{h},\delta b_{t}^{h},\zeta_{1}$,
$\zeta_{2},\lambda,\delta\rho^{(v)},\delta\phi^{(v)},\delta\theta_{(v)}$ in
addition to the frequency $\omega$. Given that the problem is fixed in terms
of nine integration constants, we conclude that we expect to find a one
parameter family of solutions labeled by $\omega$. In figure 3 and 4, we plot
the real and imaginary parts of the four Green’s functions in terms of the
frequency: the red dots correspond to the numerical results, while the solid
blue lines indicate the analytic expressions of the previous section. We see
excellent quantitative agreement. Note that the agreement is expected to
improve as one approaches the critical temperature. The location of the peak
in the imaginary part of the Green’s functions corresponds to the location of
the gap.

Figure 3: Real and Imaginary part of the Green’s functions. Analytic results
are displayed in dashed lines, while numerical results are show with black
dots. Here
$\phi_{s}=1,q=1,\mu=1,\zeta_{\rho}=0,\zeta_{\phi}=0,\lambda_{\phi}=1,\lambda_{\rho}=1/4,\lambda=-3/4$
and $T/T_{c}=0.999957$.

Figure 4: Real and Imaginary part of the Green’s functions. Analytic results
are displayed in dashed lines, while numerical results are show with black
dots. Here
$\phi_{s}=1,q=1,\mu=1,\zeta_{\rho}=0,\zeta_{\phi}=0,\lambda_{\phi}=1,\lambda_{\rho}=1/4,\lambda=-3/4$
and $T/T_{c}=0.999957$.

##  7 Discussion

We considered the perturbative dynamics of holographic systems which are
parametrically close to a thermal phase transition. In particular, we focused
on gravitational fluctuations capturing the Higgs mode which universally
emerges in the broken phase of all second order phase transitions.

One of the crucial ingredients in our construction is the utilisation of the
Crnkovic-Witten symplectic current. In the case of the source free quasinormal
mode, dual to the Higgs mode, this allowed us to obtain the gap of equations
(3.39) and (4.12). In the case of Section 5, where the frequency was fixed by
an external source, the Crnkovic-Witten symplectic current allowed us to
relate the external source to the VEVs of our scalar operators, fixing the
form of the Green’s functions. The second important ingredient was the
construction of the terms which are next to leading order in the $\varepsilon$
expansion of equation (3.2). As we explained in section 3.2, these are simply
linear combinations of the static perturbations that can be obtained by
varying the thermodynamic parameters of the broken and normal phase black
holes.

The results of Section 5 confirm that the Higgs/amplitude mode dominates the
linear response of the scalar operators in the infinite wavelength limit. At
finite wavelengths, the Higgs mode will mix with the charge and current
densities of theory, affecting the hydrodynamics. It is natural to consider
holographic techniques in order to derive an enlarged hydrodynamic description
which includes the dynamics of the Higgs mode. This, would increase the range
of validity of hydrodynamics close to the critical point.

In this paper, we have considered phase transitions which preserve the
spacetime symmetries of the boundary theory. However, over the past few years
a plethora of gravitational instabilities which spontaneously break
translations have been discovered in the context of holography. These have
been shown to lead to second order phase transitions in general. It is
interesting to examine the dynamics of the Higgs/amplitude mode for the order
parameter of density waves.

From a slightly different angle, in [16] we used the Crnkovic-Witten current
to obtain the dissipative terms in the constitutive relations for the electric
current in the superfluid phase at zero charge density. Even though that setup
is simple due to the non-mixing of the normal and the superfluid components,
it can be seen as a proof of principle in extracting the transport
coefficients of holographic theories.We expect that the techniques we have
developed here and in [16] will be central in the directions outlined above
and we wish to report more on these in the near future.

Finally, it is necessary to make the connection between our results and the
field theory approach on critical phenomena e.g. [4, 19, 20]. We plan to
report more on this connection in the near future [21].

## Acknowledgements

We would like to thank P. Kailidis and V. Ziogas for discussions and
collaboration in related topics. AD is supported by STFC grant ST/T000708/1.
C.P. acknowledges support from a Royal Society - Science Foundation Ireland
University Research Fellowship via grant URF/R1/211027.

##  Appendix A Symplectic Current Contributions

Here we list all the relevant terms that enter in the construction of the
symplectic current (3.3). After writing the bulk action (4.3) in terms of
first derivatives of the metric, we obtain,

$\displaystyle\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\phi}$
$\displaystyle=-\sqrt{-g}\,\partial^{\mu}\phi,\quad\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\rho}=-\sqrt{-g}\,\partial^{\mu}\rho,\quad\frac{\partial\mathcal{L}}{\partial\partial_{\mu}A_{\alpha}}=-\sqrt{-g}\,Z\,F^{\mu\alpha}\,,$
$\displaystyle\frac{\partial\mathcal{L}}{\partial\partial_{\mu}g_{\alpha\beta}}$
$\displaystyle=\sqrt{-g}\,\Gamma^{\mu}_{\gamma\delta}\left(\,g^{\gamma\alpha}\,g^{\delta\beta}-\frac{1}{2}\,g^{\gamma\delta}g^{\alpha\beta}\right)-\sqrt{-g}\,\Gamma^{\kappa}_{\kappa\lambda}\,\left(g^{\mu\left(\alpha\right.}g^{\left.\beta\right)\lambda}-\frac{1}{2}\,g^{\mu\lambda}g^{\alpha\beta}\right)\,,$
(A.1)

where $\Gamma^{\alpha}_{\beta\gamma}$ are the Christoffel symbols compatible
with our perturbed metric.

##  Appendix B The Energy Difference $\Delta E$

In this appendix we will show that equation (3.3) gives the energy difference
of the broken and the normal phase at fixed scalar deformation, entropy and
charge density. The energy density is,

$\displaystyle E=F+T\,s+\mu\,\varrho,$ (B.1)

satisfying the First Law of thermodynamics,

$\displaystyle\delta E=T\,\delta
s+\mu\,\delta\varrho-\langle\mathcal{O}_{\phi}\rangle\,\delta\phi_{(s)}\,,$
(B.2)

given that the gravitation free energy $F$ is the appropriate potential for
the grand canonical ensemble,

$\displaystyle\delta F=-s\,\delta
T-\varrho\,\delta\mu-\langle\mathcal{O}_{\phi}\rangle\,\delta\phi_{(s)}\,.$
(B.3)

As we would expect the above suggests that energy should be considered as a
function of $s$, $\varrho$ and $\phi_{(s)}$.

We could think of expanding the energy at the critical point along the broken
and the normal phases. Instead of doing that separately, we will do this
collectively for both the broken and the normal phase and specialise in the
end, when we take the difference. Doing so we obtain,

$\displaystyle\delta E=T\,\delta
s+\mu\,\delta\varrho-\langle\mathcal{O}_{\phi}\rangle\,\delta\phi_{(s)}+\frac{1}{2}\,\delta\Gamma^{T}\begin{pmatrix}\frac{\partial^{2}E}{\partial
s^{2}}&\frac{\partial^{2}E}{\partial
s\,\partial\varrho}&\frac{\partial^{2}E}{\partial s\,\partial\phi_{(s)}}\\\
\frac{\partial^{2}E}{\partial
s\,\partial\varrho}&\frac{\partial^{2}E}{\partial\varrho^{2}}&\frac{\partial^{2}E}{\partial\varrho\,\partial\phi_{(s)}}\\\
\frac{\partial^{2}E}{\partial
s\,\partial\phi_{(s)}}&\frac{\partial^{2}E}{\partial\varrho\,\partial\phi_{(s)}}&\frac{\partial^{2}E}{\partial\phi_{(s)}^{2}}\end{pmatrix}\delta\Gamma+\cdots\,,$
(B.4)

where we have set,

$\displaystyle\delta\Gamma=\begin{pmatrix}\delta s\\\ \delta\varrho\\\
\delta\phi_{(s)}\end{pmatrix}=\begin{pmatrix}\delta\mathbf{\Phi}\\\
\delta\phi_{(s)}\end{pmatrix}\,.$ (B.5)

Equation (B.4) can be written as,

$\displaystyle\delta E=$ $\displaystyle T\,\delta
s+\mu\,\delta\varrho-\langle\mathcal{O}_{\phi}\rangle\,\delta\phi_{(s)}$
$\displaystyle+\frac{1}{2}\,\delta\mathbf{\Phi}^{T}\,\begin{pmatrix}\frac{\partial^{2}E}{\partial
s^{2}}&\frac{\partial^{2}E}{\partial s\,\partial\varrho}\\\
\frac{\partial^{2}E}{\partial
s\,\partial\varrho}&\frac{\partial^{2}E}{\partial\varrho^{2}}\end{pmatrix}\,\delta\mathbf{\Phi}+\delta\mathbf{\Phi}^{T}\,\begin{pmatrix}\frac{\partial^{2}E}{\partial
s\,\partial\phi_{(s)}}\\\
\frac{\partial^{2}E}{\partial\varrho\,\partial\phi_{(s)}}\end{pmatrix}\,\delta\phi_{(s)}+\frac{1}{2}\,\frac{\partial^{2}E}{\partial\phi_{(s)}^{2}}\,\delta\phi_{(s)}^{2}+\cdots$
$\displaystyle=$ $\displaystyle T\,\delta
s+\mu\,\delta\varrho-\langle\mathcal{O}_{\phi}\rangle\,\delta\phi_{(s)}$
$\displaystyle+\frac{1}{2}\,\delta\mathbf{\Phi}^{T}\,\begin{pmatrix}\frac{\partial
T}{\partial s}&\frac{\partial T}{\partial\varrho}\\\
\frac{\partial\mu}{\partial
s}&\frac{\partial\mu}{\partial\varrho}\end{pmatrix}\,\delta\mathbf{\Phi}-\delta\mathbf{\Phi}^{T}\,\begin{pmatrix}\frac{\partial}{\partial
s}\langle\mathcal{O}_{\phi}\rangle\\\
\frac{\partial}{\partial\varrho}\langle\mathcal{O}_{\phi}\rangle\end{pmatrix}\,\delta\phi_{(s)}-\frac{1}{2}\,\delta\phi_{(s)}^{2}\,\frac{\partial}{\partial\phi_{(s)}}\langle\mathcal{O}_{\phi}\rangle+\cdots$
$\displaystyle=$ $\displaystyle T\,\delta
s+\mu\,\delta\varrho-\langle\mathcal{O}_{\phi}\rangle\,\delta\phi_{(s)}$
$\displaystyle+\frac{1}{2}\,\delta\mathbf{\Phi}^{T}\,\mathbf{\Xi}^{-1}\,\delta\mathbf{\Phi}-\delta\mathbf{\Phi}^{T}\,\mathbf{\Xi}^{-1}\,\mathbf{\nu}\,\delta\phi_{(s)}-\frac{1}{2}\,\delta\phi_{(s)}^{2}\,\frac{\partial}{\partial\phi_{(s)}}\langle\mathcal{O}_{\phi}\rangle+\cdots\,,$
(B.6)

where in the last line we recognised the matrix of susceptibilities as the
Jacobian of the transformation when passing from $(s,\varrho)$ to $(T,\mu)$.
For the last term we will consider the chain rule,

$\displaystyle\left.\frac{\partial}{\partial\phi_{(s)}}\langle\mathcal{O}_{\phi}\rangle\right|_{s,\varrho}$
$\displaystyle=\left.\frac{\partial}{\partial\phi_{(s)}}\langle\mathcal{O}_{\phi}\rangle\right|_{T,\mu}+\left.\frac{\partial
T}{\partial\phi_{(s)}}\right|_{s,\varrho}\,\left.\frac{\partial}{\partial
T}\langle\mathcal{O}_{\phi}\rangle\right|_{\mu,\phi_{(s)}}+\left.\frac{\partial\mu}{\partial\phi_{(s)}}\right|_{s\varrho}\,\left.\frac{\partial}{\partial\mu}\langle\mathcal{O}_{\phi}\rangle\right|_{T,\phi_{(s)}}$
$\displaystyle=\nu_{\phi}-\mathbf{\nu}^{T}\,\mathbf{\Xi}^{-1}\,\mathbf{\nu}\,.$
(B.7)

where in the last line we used equation (3.2) to compute the partial
derivatives of $T$ and $\mu$ at fixed $s$ and $\varrho$. Putting everything
together we have that,

$\displaystyle\delta E=$ $\displaystyle T\,\delta
s+\mu\,\delta\varrho-\langle\mathcal{O}_{\phi}\rangle\,\delta\phi_{(s)}$
$\displaystyle+\frac{1}{2}\,\delta\mathbf{\Phi}^{T}\,\mathbf{\Xi}^{-1}\,\delta\mathbf{\Phi}-\delta\mathbf{\Phi}^{T}\,\mathbf{\Xi}^{-1}\,\mathbf{\nu}\,\delta\phi_{(s)}-\frac{1}{2}\,\left(\nu_{\phi}-\mathbf{\nu}^{T}\,\mathbf{\Xi}^{-1}\,\mathbf{\nu}\right)\,\delta\phi_{(s)}^{2}+\cdots\,,$
(B.8)

The above shows that after expanding along the broken and the normal phases
and taking the difference, the terms linear in the variations will cancel out
and the quadratic ones will give the result of equation (3.3).

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