aainstitutetext: Asia Pacific Center for Theoretical Physics, Pohang, 37673,
Koreabbinstitutetext: Departamento de Física de Altas Energías, Instituto de
Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal
70-543, CDMX 04510, Méxicoccinstitutetext: Institute for Theoretical Physics,
Wroclaw University of Science and Technology, 50-370 Wroclaw,
Polandddinstitutetext: Department of Physics, Pohang University of Science and
Technology, Pohang, 37673, Korea

# Modulated instabilities and the AdS2 point in dense holographic matter

Jesús Cruz Rojas c Tuna Demircik a,d Matti Järvinen
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate fluctuations of hot and dense QCD plasma by using the
gauge/gravity duality. To this end, we carry out a comprehensive
classification and analysis of quasinormal modes of charged black holes in the
holographic V-QCD model. It turns out that the Chern-Simons term determined by
the flavor anomalies of QCD is strong enough to drive a modulated instability.
While such an instability is expected at high densities, we find that the
unstable phase extends to surprisingly low densities and high temperatures,
close to the region where data from lattice simulations is available. We also
analyze the limit of small temperature which is controlled by a quantum
critical AdS2 point. We study in detail the signatures of the critical point
in the quasinormal mode spectrum, focusing on the interplay between the
hydrodynamic modes and other modes.

††preprint: APCTP Pre2024 - 009

##  1 Introduction and summary

Recent observations of binary neutron star mergers by LIGO and Virgo
interferometers have provided new constraints for the behavior of QCD matter
at high densities. These results are complemented by ongoing heavy ion
collisions experiments at RHIC, i.e., the beam energy scan program, as well as
future heavy-ion experiments at facilities such as FAIR, NICA and J-PARC,
which will push the range of data towards higher densities. Thanks to this
incoming new data, there has also been enhanced interest in theoretical
studies of dense QCD recently. Obtaining sound theoretical predictions for
dense matter is however known to be particularly challenging. This is because
none of the usual first-principles methods (perturbation theory, lattice
simulations, or effective theory) work reliably at the relevant densities,
i.e. the highest densities reached in heavy-ion collisions or the densities in
the cores of massive neutron stars Brambilla:2014jmp .

Consequently, the precise structure of the phase diagram of QCD remains
largely unknown at high densities. It is expected to contain various (usually
deconfined) “exotic” phases in this region. Typical examples of such phases
are different color superconducting phases Bailin:1983bm ; Alford:1997zt ;
Alford:2007xm  (including the color-flavor locked phase Alford:1998mk ),
quarkyonic phases McLerran:2007qj , and spatially modulated phases. As an
example of the latter, in Alford:2000ze , a crystalline color superconductor
phase of quark matter was suggested. As a different example, it was shown that
at low temperature and large $N_{c}$, the ground state of QCD is a chiral
density wave Deryagin:1992rw ; Bringoltz:2009ym . Quarkyonic chiral density
waves have also been suggested Kojo:2009ha . Moreover, chiral magnetic wave
Kharzeev:2010gd  and chiral magnetic spiral Basar:2010zd  as well as
(confined) chiral soliton lattice Son:2007ny ; Brauner:2016pko ; Evans:2022hwr
phases can be induced by a magnetic field.

In the absence of precise first-principles methods, modeling and new
techniques can be useful in order to improve the theoretical understanding of
the phases of dense QCD. A possibility is to use the gauge-gravity duality,
which maps strong coupling physics in field theory to problems in classical
gravity in higher dimensions.

The approaches for holographic QCD can be roughly divided into top-down and
bottom-up models. In the first case the models are based on concrete
(typically ten dimensional) setups in string theory for which precise control
of the gauge theory is possible. The bottom-up models are, in contrast,
constructed “by hand” with some inspiration from the top-down models. While
some guidance is provided by the global symmetries of QCD which should be
respected by the duality, a precise control of the correspondence is lost
since the gravity models are not dual to any explicitly known field theory,
but instead involve parameters, which should be adjusted to obtain agreement
with QCD physics. However this also means that there is a great amount of
freedom to do modifications, which may be necessary to model QCD efficiently,
but are difficult to realize in the top-down framework. This is the approach
we will follow in this article.

Within the gauge-gravity duality, several examples of condensed
superconducting phases (see, e.g., Gubser:2008px ; Gubser:2008zu ;
Gubser:2008wv ; Hartnoll:2008vx ; Horowitz:2008bn ) and modulated phases
Domokos:2007kt ; Nakamura:2009tf ; Ooguri:2010xs  are known. In particular,
spatial modulation can arise due to a Chern-Simons (CS) terms on the higher
dimensional gravity side. In the context of QCD, such a CS term reflects the
flavor anomalies of the field theory Wess:1971yu ; Witten:1983tw ;
Witten:1998qj . The presence of spatially modulated ground state typically
implies an instability of the homogeneous state, which can be analyzed by
studying the quasinormal modes (QNMs) of the dual geometry. QNMs are
linearized fluctuation modes of a black hole which assume the plane wave form
$\sim\exp\left(-i\omega t+i\mathbf{q}\cdot\mathbf{x}\right)$, with a discrete
spectrum of complex valued frequencies. In the gauge theory side, QNMs
correspond to the poles of the retarded Green’s function of the dual operator
Son:2002sd ; Nunez:2003eq ; Horowitz:1999jd ; Kovtun:2005ev . For a more
comprehensive information of QNMs, interested readers are referred to detailed
reviews Berti:2009kk ; Konoplya:2011qq ; Kovtun:2005ev . The presence of a
mode with positive imaginary part of the frequency is an instability. If the
instability only appears at nonzero spatial momentum $q=|\mathbf{q}|$, it is
modulated.

In Nakamura:2009tf ; Ooguri:2010kt ; Ooguri:2010xs  it was shown that the
Reissner-Nordstrom black hole in the five dimensional anti–de Sitter space
coupled to the Maxwell theory and with a CS term is indeed unstable when the
CS coupling is sufficiently large. The first instance of this type of
instability was discovered in a chiral symmetry broken confined phase of a
specific bottom-up phenomenological model Domokos:2007kt . In this model, the
CS term mixes vector and axial non-Abelian fields, and the resulting
instability leads to the condensation of corresponding light mesons. This
condensation breaks both translational and rotational invariance of the ground
state at critical densities remarkably close to the nuclear saturation density
$n_{s}\approx 0.16\,\mathrm{fm}^{-3}$. The spatially modulated instability has
also been observed in the top-down Witten-Sakai-Sugimoto (WSS) model, as
discussed in Ooguri:2010xs ; Chuang:2010ku . However, this work explores the
scenario in the chiral symmetric deconfined phase, i.e. the quark-gluon phase
of the WSS model. Through their analysis, the authors propose a spatially
modulated phase transition driven by the CS coupling of Abelian gauge fields.
This phase transition occurs above a critical baryon density of $2.4$ fm-3 for
three flavors (equivalent to about $15n_{s}$) at a temperature of 150 MeV. The
associated inhomogeneous field theory ground state is likely to accommodates
modulated chiral currents. Subsequently, a similar instability in the confined
phase of the WSS model at low temperature and finite axial chemical potential
was identified in Bayona:2011ab .

Spatially modulated instabilities similar to the ones discussed above are
prevalent in dual descriptions of condensed matter systems, such as Fermi
liquids Bergman:2011rf ; Jokela:2012se ; Iizuka:2013ag  and superconductors
Donos:2011ff ; Donos:2012gg ; Donos:2011qt ; Donos:2011pn ; Donos:2012yu  (the
construction of new striped ground states for superconductors can be found in
Donos:2012gg ; Donos:2012wi ; Rozali:2013ama ; Withers:2013kva ; Donos:2013wia
; Iizuka:2012pn ; Rozali:2012es ). Certain factors can contribute to the
stabilization of these ground states in the presence of a CS coupling.
Examples include $R^{2}$ corrections Takeuchi:2011uk  and the inclusion of a
magnetic field Ballon-Bayona:2012qnu ; Jokela:2012vn . Efforts have also been
done to study the Hubbard model, which is used to describe the transition
between conducting and insulating systems: In Fujita:2014mqa  a holographic
construction of a large-$N$ Bose-Hubbard model is presented.

It is also well known that in gravitational solutions dual to $n+1$
dimensional field theories at finite charge density, an
AdS${}_{2}\times\mathbb{R}^{n}$ geometry appears in the extremal limit of
black holes. The AdS2 point is related to the emergence of a quantum critical
region in the dual field theory Liu:2009dm ; Faulkner:2009wj ; Iqbal:2011in ,
with scaling symmetry in time, and controlled by a $(0+1)$-dimensional CFT at
the IR. The IR region is known to be sensitive to instabilities, such as the
superconducting or modulated instabilities discussed above, due to a low
Breitenlohner-Freedman (BF) bound of the AdS2 space Hartnoll:2008vx ;
Amado:2009ts . This kind of quantum critical region also arises in holographic
models for quark matter Alho:2012mh ; Alho:2013hsa .

To describe the long wavelength dynamics of interacting quantum fields one can
usually rely on hydrodynamics. At non-zero temperature quantum critical phases
are found near quantum phase transitions, and in such regions one expects that
transport properties are universal and are governed by scaling properties of
the critical point. The physics of hydrodynamics and near criticality can be
concretely characterized in terms of QNMs. That is, the QNMs with frequencies
vanishing as momentum goes to zero are the hydrodynamic modes and these modes
contain information about all the hydrodynamic transport coefficients. On the
other hand, nonhydrodynamic modes appear as additional poles of the retarded
Green’s function, and provide information beyond the hydrodynamic description.
They play a role in determining the applicability regime of hydrodynamics
Heller:2013fn  and provide a description of the gauge theory plasma at late
times Heller:2013oxa ; Heller:2012km .

Hydrodynamics breaks down at sufficiently short scales set by the local
equilibration time and length, since additional degrees of freedom start to
play a more significant role. In Arean:2020eus  the authors study the
breakdown of hydrodynamics in certain low temperature states dual to black
holes with nearly-extremal AdS${}_{2}\times\mathbb{R}^{2}$ near-horizon
metrics. The authors found that the breakdown of hydrodynamics is caused by
modes associated to the AdS2 region of the geometry, and as a consequence the
critical frequency is set by data coming from the IR. The breakdown is
characterized by a collision, parametrically close to the imaginary $\omega$
axis, between the lowest nonhydrodynamic mode and the hydrodynamic mode. It
was proven in Grozdanov:2019kge ; Grozdanov:2019uhi  that the radii of
convergence of the hydrodynamic gradient series expansions are determined by
the critical points of the associated complex spectral curves. For theories
with a dual gravitational description through holography, this critical points
correspond to level-crossings in the quasinormal spectrum of the dual black
hole.

In this article, we study both the modulated instabilities and physics near
the quantum critical AdS2 point in a holographic bottom-up model which has
been constructed by a careful comparison with QCD data, the V-QCD model
Jarvinen:2011qe . To this end, we carry out a comprehensive fluctuation
analysis of the black hole solutions in this model. We identify and analyze
the QNMs in hot and dense holographic QCD plasma, focusing in sectors where
most complex and interesting phenomena are expected to arise. Our work
significantly extends earlier analysis, where basic transport coefficients
(viscosities and conductivities) were computed in hot and dense V-QCD
Hoyos:2020hmq ; Hoyos:2021njg , and fluctuations at zero density were studied
in the same class of models Gursoy:2013zxa ; Ishii:2015gia ; Demircik:2016nhr
; Alho:2020gwl . This work is also an additional step in the wider program
(see, e.g., Alho:2013hsa ; Jokela:2020piw ; Demircik:2021zll ;
Demircik:2023lsn ; CruzRojas:2023ugm ; CruzRojas:2024etx ; Demircik:2024bxd )
which aims at establishing and improving predictions for the equation of state
and transport properties for dense QCD, based on this model. These predictions
can then be used to analyze heavy-ion collisions and neutron star mergers
Ecker:2019xrw ; Tootle:2022pvd ; Ecker:2024kzs ; Demircik:2022uol .

The rest of the article is organized as follows. In the rest of the
introduction, we give a summary of our results and discuss future directions.
In Sec. 2, we briefly review the holographic model. We go on discussing the
fluctuations in Sec. 3, including their classification and the derivation of
the fluctuation equations. In Sec. 4, we show our results for the quasinormal
modes at zero momentum, and derive analytic results for the modes related to
the AdS2 geometry. These results are generalized to finite momentum in Sec. 5,
where we focus on the modulated instability and discuss relation to the
breaking of hydrodynamics near AdS2 points. The Appendix contains technical
details of our analysis.

###  1.1 Summary and outlook

In this article, we carry out a systematic fluctuation analysis of the
chirally symmetric charged black hole solutions in V-QCD. We classify all
fluctuations, including modes both at nonzero frequency and nonzero momentum.
We solve the QNMs found by solving the fluctuation equations numerically,
focusing on the most complex sectors of the theory. The main results of the
article are the following:

  * •

We write down the complete set of fluctuation equations for the model in terms
of gauge-invariant fluctuation wave functions.

  * •

We analyze the fluctuations at the AdS2 IR point, which realizes the quantum
critical line of the dual theory at zero temperature and finite density. We
compute the dimensions of the fluctuations in the corresponding one-
dimensional IR CFT, and showed how the (purely imaginary) QNMs of the black
hole phase map to these AdS2 modes as the temperature approaches zero. In
particular, we point out an “anomalous” behavior of one of the modes, related
to the fluctuations of the dilaton field.

  * •

We study the possibility of a modulated instability of gauge fields at finite
density, induced by the CS terms which implement the flavor anomaly structure
of QCD in the holographic model. We find that the CS terms are strong enough
to drive an instability at low temperatures and high densities, as expected
from earlier studies Domokos:2007kt ; Nakamura:2009tf ; Ooguri:2010kt ;
Ooguri:2010xs . The instability is also present for “non-Abelian” fluctuations
which transform nontrivially under the flavor group, and the non-Abelian
instability is slightly stronger than the Abelian (flavor singlet)
instability.

  * •

We point out that the region of unstable black holes extends to surprisingly
low densities and high temperatures. This finding will be further analyzed in
the companion article Demircik:2023xxx .

  * •

We analyze the interplay of the hydrodynamic modes with the AdS2 modes. We
find similar structure as in Arean:2020eus  at low temperature: the
convergence of the hydrodynamic dispersion relations for diffusion modes is
determined by the collision of the hydrodynamic mode with the leading AdS2
mode. However in our case this happens in a phase which suffers from a
modulated instability.

This article represents a step towards a realistic holographic model of
various “exotic” phases that are expected to appear in QCD at high densities.
Here we showed that the standard quark matter description is indeed unstable
towards an inhomogeneous ground state, in a holographic setup that has been
carefully compared to QCD data, in particular to lattice data for
thermodynamics.

There are several future directions to explore:

  * •

An obvious task would be the analysis and construction of the end point of the
instability, i.e., the modulated ground state. While the simplest possibility
is a striped state, where translation invariance is only broken in one
direction, more complicated configurations are possible Withers:2014sja . One
may try to numerically construct the ground state, or study its nature by
carrying out higher order nonlinear fluctuation analysis. This construction is
however complicated by the fact that the ground state is likely to be
inherently non-Abelian: we found here the instability to be at its strongest
in the non-Abelian sector. Notice that the DBI action which we used here is
not known for general non-Abelian field configurations, so one needs to first
choose a prescription to pin down the action, and the results may depend on
the prescription.

  * •

Another task is to figure out how model dependent our predictions for the
range of the instability are. This will be discussed in part in the companion
article Demircik:2023xxx . This is important as the computation here suggests
that instability extends down to low density and high temperature, even to the
region which is directly accessible on the lattice.

  * •

Apart from the instability towards a modulated state, there might be other
kinds of instabilities, in particular those related to various paired phases
involving color superconductivity. Such phases are known to be tricky in
holography as color superconductivity requires breaking of the gauge symmetry,
while the holographic method is normally restricted to color singlets. First
attempt could be to construct a simple phenomenological model following the
ideas of BitaghsirFadafan:2018iqr ; Nam:2021ufk , for example (extended to
higher order corrections in the background in Fadafan:2021xcw  and to the
imaginary chemical potential region in Ghoroku:2020fkv ). Inspiration can also
be obtained from top-down approaches to exotic phases, see, e.g. Faedo:2017aoe
; Faedo:2018fjw ; Faedo:2019jlp ; Henriksson:2019zph ; Henriksson:2021zei .
The possible color superconducting instability would compete with the
modulated instability, possibly leading to a rich phase diagram.

  * •

An additional important task, which may be related to the extent of the
instability, is to develop the flavor dependence of the model and investigate
its effect on the modulated and possibly on color superconducting
instabilities. This would mean to study the effects of isospin breaking,
analyze the symmetry energy in quark matter, and include the strange quark
mass in the model.

  * •

A different direction would be to extend the current analysis to include the
residues of the QNMs, which might be particularly interesting near the AdS2
region where interesting patterns are found, also to compute numerically the
full two-point correlators. In the regime of dense matter, the two point
correlators are relevant, for example to analyze neutrino transport
Jarvinen:2023xrx .

We wish to study all these issues in future work.

##  2 The V-QCD model

The V-QCD model Jarvinen:2011qe  is a holographic bottom-up model for QCD
obtained by combining two building blocks: improved holographic QCD (IHQCD)
Gursoy:2007cb ; Gursoy:2007er , which models the gluon sector, and a setup for
adding flavors through tachyonic Dirac-Born-Infeld (TDBI) actions and the
relevant CS terms Bigazzi:2005md ; Casero:2007ae . This model is inspired by
five-dimensional noncritical string theory but does not have a precise top-
down derivation: when the expressions from string theory fail to produce
desired phenomenology, we will switch to a bottom-up approach by modifying and
generalizing the action, aiming at a form that can be fitted to QCD data. We
will go though the basic contents of the model briefly here; for a thorough
discussion, see the review Jarvinen:2021jbd .

We also remark that while we are using the V-QCD model in this article, many
results are not sensitive to the details of the action, and are expected to
hold at qualitative level for a larger class of models. In particular, we
shall discuss below the physics near the quantum critical AdS2 fixed point
which shows some universal features.

###  2.1 Action and EoMs

We wish to model a slightly generalized version of QCD, i.e., a $SU(N_{c})$
gauge theory with $N_{c}$ colors, and $N_{f}$ quark flavors in the fundamental
representation of the gauge group. The full field content of the model in the
bulk contains duals to the most important, relevant and marginal operators of
the theory Gursoy:2007cb ; Gursoy:2007er ; Jarvinen:2011qe :

  * •

The dilaton $\phi$ is dual to the $F^{2}$ operator, where $F$ is the gluon
field in QCD. The corresponding source is the coupling of QCD.

  * •

The (bulk) metric is dual to the energy-momentum tensor $T_{\mu\nu}$ of QCD.
The source is the metric of the field theory.

  * •

The axion $a$ is dual to $F\wedge F$, and the source is the $\theta$-angle.

  * •

The tachyon $T^{ij}$, where $i,j=1\ldots N_{f}$ are the flavor indices, is
dual to the quark bilinear $\bar{\psi}^{i}\psi^{j}$. The source is therefore
the quark mass matrix.

  * •

The left and right handed gauge fields $A_{L/R\,\mu}^{ij}$ are dual to the
chiral currents $\bar{\psi}^{i}(1\pm\gamma_{5})\gamma_{\mu}\psi^{j}$. The
sources are various chemical potentials (for temporal components) and external
fields (for spatial components).

Here the first three (last two) fields arise from the IHQCD (TDBI) sector.

The action of the model separates to four terms,

$S=S_{g}+S_{f}+S_{a}+S_{\mathrm{CS}}\ ,$ (2.1)

which describe the gluon sector, the flavor sector, the CP-odd sector linked
to the axial anomaly, and a CS sector linked to flavor anomalies in QCD,
respectively. The glue action is given by five dimensional dilaton gravity:

$S_{g}=M_{p}^{3}N_{c}^{2}\int
d^{5}x\sqrt{-g}\left[R-\frac{4}{3}g^{MN}\partial_{M}\phi\partial_{N}\phi+V_{g}(\phi)\right]\
,$ (2.2)

where $M_{p}$ is the five-dimensional Planck mass. With an appropriate choice
of the dilaton potential $V_{g}$ this part of the action is that of the IHQCD
model, except for the axion field which is included in $S_{a}$ and is
discussed below.

The flavor action $S_{f}$ is the generalized TDBI action Casero:2007ae ;
Iatrakis:2010jb ; Iatrakis:2010zf ,

$S_{f}=-\frac{1}{2}M_{p}^{3}N_{c}\ \mathrm{Tr}\int
d^{4}x\,dr\,\left(V_{f}(\phi,T^{\dagger}T)\sqrt{-\det{\bf
A}_{L}}+V_{f}(\phi,TT^{\dagger})\sqrt{-\det{\bf A}_{R}}\right)\ .$ (2.3)

Here the trace $\mathrm{Tr}$ is over flavor indices, the determinants are over
Lorenz indices, and we have denoted

$\displaystyle{\bf A}_{L\,MN}$
$\displaystyle=g_{MN}+w(\phi)F^{(L)}_{MN}+\frac{\kappa(\phi)}{2}\left[(D_{M}T)^{\dagger}(D_{N}T)+(D_{N}T)^{\dagger}(D_{M}T)\right]\
,$ $\displaystyle{\bf A}_{R\,MN}$
$\displaystyle=g_{MN}+w(\phi)F^{(R)}_{MN}+\frac{\kappa(\phi)}{2}\left[(D_{M}T)(D_{N}T)^{\dagger}+(D_{N}T)(D_{M}T)^{\dagger}\right]\
,$ (2.4)

where the covariant derivative is defined as

$D_{M}T=\partial_{M}T+iTA_{M}^{L}-iA_{M}^{R}T\ .$ (2.5)

Recall that the fields $A_{L}$, $A_{R}$ and $T$ are $N_{f}\times N_{f}$
matrices in flavor space. We also denote

$x\equiv\frac{N_{f}}{N_{c}}\ .$ (2.6)

The string theory inspiration of the model is obtained by considering the
Veneziano limit Veneziano:1976wm , where one takes $N_{c}$ and $N_{f}$ to
infinity keeping $x$ fixed Jarvinen:2011qe . In this limit the flavor sector
(2.3) is fully backreacted to the glue sector (2.2). The choice of the
potentials $V_{f}$, $\kappa$, and $w$ will be discussed below.

The determinants over the Lorentz indices in (2.3) are ambiguous when the
elements contain non-Abelian matrices. For the purposes of this article,
however, a precise description is not necessary. All fields of the background
solution, which we will soon discuss, will be proportional to the unit matrix
$\mathbb{I}_{N_{f}}$. In particular, we will assume that the quarks are all
massless. Therefore non-Abelian fields will only appear when we fluctuate the
action around the background solution. We will consider the fluctuated action
up to quadratic order, in which case the ambiguity in the ordering of the non-
Abelian elements is absent due to the cyclicity of the trace.

We focus on the chirally symmetric phase (also implying that the quark mass is
zero) where the background value of the tachyon vanishes. Then, because the
action is quadratic in the tachyon, the tachyon fluctuations will decouple
from the rest and can be treated separately. All such quadratic contributions
will arise from the DBI action given above. Therefore we can effectively
simply set the tachyon to zero in the additional terms of the action that we
discuss next. Setting the tachyon to zero, the flavor action simplifies to

$\displaystyle S_{f0}$ $\displaystyle=-\frac{1}{2}M_{p}^{3}N_{c}\int
d^{5}x\,V_{f0}(\phi)\times$ (2.7)
$\displaystyle\qquad\times\mathrm{Tr}\left(\sqrt{-\det(g_{MN}+w(\phi)F^{(L)}_{MN})}+\sqrt{-\det(g_{MN}+w(\phi)F^{(R)}_{MN})}\right)\
$

where $V_{f0}(\phi)=V_{f}(\phi,T=0)$.

The axion action $S_{a}$ arises as a combination of terms from the closed and
open string sectors Arean:2013tja ; Arean:2016hcs . At zero tachyon, it reads

$S_{a}=-\frac{M_{p}^{3}}{2}\int d^{5}x\sqrt{-\det
g}\,Z(\phi)\left[N_{c}\partial_{M}\hat{a}-\mathrm{Tr}\left(A^{L}_{M}-A^{R}_{M}\right)\right]^{2}$
(2.8)

where $\hat{a}=a/N_{c}$. The choice for the potential $Z(\phi)$ will be
analyzed below. This term is responsible for the holographic implementation of
the axial $U(1)_{A}$ anomaly, and the relative normalization between the gauge
field and axion is determined by the anomaly.

When the tachyon vanishes, the (standard) CS term can be written as
Witten:1998qj ; Casero:2007ae

$\displaystyle S_{\mathrm{CS}}$
$\displaystyle=\frac{iN_{c}}{24\pi^{2}}\int\mathrm{Tr}\bigg{[}-iA_{L}\wedge
F_{L}\wedge F_{L}+\frac{1}{2}A_{L}\wedge A_{L}\wedge A_{L}\wedge F_{L}+$
$\displaystyle+\frac{i}{10}A_{L}\wedge A_{L}\wedge A_{L}\wedge A_{L}\wedge
A_{L}+iA_{R}\wedge F_{R}\wedge F_{R}-\frac{1}{2}A_{R}\wedge A_{R}\wedge
A_{R}\wedge F_{R}-$ $\displaystyle-\frac{i}{10}A_{R}\wedge A_{R}\wedge
A_{R}\wedge A_{R}\wedge A_{R}\bigg{]}\ .$ (2.9)

Our sign conventions are such that $F_{L/R}=dA_{L/R}-iA_{L/R}\wedge A_{L/R}$.
The tachyon dependence of the CS term has been analyzed recently in
Jarvinen:2022mys .

###  2.2 Background

We then briefly discuss the background we are using in the fluctuation
analysis (see Alho:2012mh ; Alho:2013hsa ). We set the quark masses and the
$\theta$-angle to zero, and consider the chirally symmetric phase. Then the
tachyon and axion vanish identically. The only nonzero component of the gauge
fields in the background is the temporal component of the vectorial field,
$V_{t}(r)=\Phi(r)$. We follow a normalization convention where

$A_{L}=V+A\ ,\qquad A_{R}=V-A\ .$ (2.10)

The flavor Lagrangian for the background is therefore

$\mathcal{L}_{f0}=-M_{p}^{3}N_{c}N_{f}V_{f0}(\phi)\sqrt{-\operatorname{det}\left[g_{MN}+w(\phi)F_{MN}\right]}$
(2.11)

where $F_{MN}$ is the field strength tensor for the vectorial gauge field.

The metric Ansatz is

$\mathrm{d}s^{2}=e^{2A(r)}\left[-f(r)\mathrm{d}t^{2}+\mathrm{d}\mathbf{x}^{2}+\frac{\mathrm{d}r^{2}}{f(r)}\right],\quad
f(0)=1.$ (2.12)

The nonzero functions describing the background are then the fields $A$ and
$f$ in the metric, the dilaton $\phi$, and the gauge field $\Phi$, which all
are assumed to depend only on the holographic coordinate $r$. We will only be
studying black hole solutions in this article so that $r$ runs from zero at
the boundary to a finite value $r_{h}$ at the black hole horizon, where
$f(r_{h})=0$.

With the above assumptions, the background action only depends on $\Phi$
through its $r$-derivative. Therefore the equation of motion for $\Phi$
implies that we have a constant of integration $\hat{n}$ Alho:2013hsa :

$\frac{1}{M_{p}^{3}N_{c}N_{f}}\frac{\partial\mathcal{L}_{f}}{\partial\Phi^{\prime}}=-\frac{e^{A}V_{f0}(\phi)w(\phi)^{2}\Phi^{\prime}}{\sqrt{1-e^{-4A}w(\phi)^{2}(\Phi^{\prime})^{2}}}\equiv\hat{n}=\frac{1}{M_{p}^{3}N_{c}N_{f}}n\
,$ (2.13)

where $n$ is the number density of quarks. From this we can solve to find
$\Phi^{\prime}$ and integrate to determine the quark number chemical potential
$\mu$:

$\mu=-\int_{0}^{r_{h}}dr\,\Phi^{\prime}(r)=\int_{0}^{r_{h}}dr\frac{\hat{n}}{e^{A}V_{f0}(\phi)w(\phi)^{2}\sqrt{1+\frac{\hat{n}^{2}}{e^{6A}w(\phi)^{2}V_{f0}(\phi)^{2}}}}\
.$ (2.14)

It is important that the weak coupling behavior of the various fields agrees
with QCD in order to have the best possible “boundary conditions” for the
strong coupling behavior of the model.

We then discuss the background in the weak-coupling region, near the
ultraviolet (UV) boundary. Usually in holographic bottom-up models of QCD one
chooses the leading behavior of the bulk fields near the boundary such that it
agrees with the leading UV dimensions of the dual operators. Here we also
require that the first few quantum corrections in field theory, and the
renormalization group (RG) flow imposed by them, agrees with the near-boundary
holographic RG flow of the bulk fields. We demonstrate this here for the
coupling Gursoy:2007cb , but the flow of the quark mass and the chiral
condensate can be analyzed similarly Jarvinen:2011qe .

The boundary conditions for the tachyon are such that it vanishes near the
boundary. It also turns out that the gauge field is irrelevant for the
boundary behavior of the metric Alho:2013hsa . Setting $\tau=0$ in the action
(2.1) we see that the geometry is determined by the effective potential

$V_{\mathrm{eff}}({\lambda})=V_{g}(\log{\lambda})-xV_{f0}(\log{\lambda})$
(2.15)

where ${\lambda}=e^{\phi}$ is identified as the ’t Hooft coupling near the
boundary. For the geometry to be asymptotically AdS5 at the boundary,
$V_{\mathrm{eff}}$ needs to go to a constant at small coupling:

$V_{\mathrm{eff}}(\lambda)\rightarrow\frac{12}{\ell^{2}}\quad\text{ as
}\quad\lambda\rightarrow 0$ (2.16)

where $\ell$ is the $\mathrm{AdS}_{5}$ radius for the UV geometry. It is
natural to assume a Taylor series around $\lambda=0$, i.e.,

$V_{\mathrm{eff}}(\lambda)=\frac{12}{\ell^{2}}\left[1+v_{1}\frac{\lambda}{\lambda_{0}}+v_{2}\left(\frac{\lambda}{\lambda_{0}}\right)^{2}+\mathcal{O}\left(\lambda^{3}\right)\right]$
(2.17)

where the constant $\lambda_{0}=8\pi^{2}$ was introduced for convenience. Then
the near boundary asymptotics of the geometry is $\mathrm{AdS}_{5}$ with
logarithmic corrections :

$\displaystyle A(r)$
$\displaystyle=-\log\frac{r}{\ell_{0}}+\frac{4}{9\log(r\Lambda)}$ (2.18)
$\displaystyle\quad+\frac{\left(\frac{95}{162}-\frac{32v_{2}}{81v_{1}^{2}}\right)+\left(-\frac{23}{81}+\frac{64v_{2}}{81v_{1}^{2}}\right)\log(-\log(r\Lambda))}{(\log(r\Lambda))^{2}}+\mathcal{O}\left(\frac{1}{(\log(r\Lambda))^{3}}\right)\
,$ $\displaystyle\frac{v_{1}\lambda(r)}{\lambda_{0}}$
$\displaystyle=-\frac{8}{9\log(r\Lambda)}+\frac{\left(\frac{46}{81}-\frac{128v_{2}}{81v_{1}^{2}}\right)\log(-\log(r\Lambda))}{(\log(r\Lambda))^{2}}+\mathcal{O}\left(\frac{1}{(\log(r\Lambda))^{3}}\right)\
.$ (2.19)

The flow behaves as expected: the source term of the dilaton has become
logarithmically flowing instead of a constant, and the value of the source is
now identified with the scale $\Lambda=\Lambda_{\mathrm{UV}}$. This scale
defines the units for all dimensionful quantities in our analysis. The field
$A(r)$ can be interpreted as the logarithm of the energy scale in field
theory. This given a mapping between the series coefficients $v_{i}$ of the
effective potential and the $\beta$-function in QCD Gursoy:2007cb ;
Gursoy:2007er .

###  2.3 Choice of potentials

The action (2.1) is parametrized in terms of the potentials
$V_{g}(\phi),V_{f}(\phi,\tau),\kappa(\phi)$, $w(\phi)$, and $Z(\phi)$ which
are chosen to satisfy two basic requirements. First, in the ultraviolet, the
model should reproduce the known perturbative behaviors of the corresponding
field theory, as we demonstrated for the coupling of QCD above. Second, in the
deep infrared the model should lead to the generation of a dynamical wall
shielding the singular behavior as $\phi\rightarrow\infty$, which is
responsible for confinement in the absence of the tachyon. The details IR wall
should be such that it produces among other things, correct behavior of meson
masses at large excitation numbers and at finite quark mass, as well as
correct kind of phase diagram at finite density. These requirements set tight
constraints to the asymptotics of the potentials both at small coupling
($\phi\to-\infty$) Gursoy:2007cb ; Jarvinen:2011qe  and at large coupling
($\phi\to\infty$) Gursoy:2007er ; Jarvinen:2011qe ; Arean:2013tja ;
Arean:2016hcs ; Jarvinen:2015ofa ; Ishii:2019gta .

After the asymptotics of the potentials has been determined, their behavior at
intermediate values of $\phi$ needs to be fixed by comparing to QCD data. It
is convenient to first fit the functions $V_{g}(\phi)$ and $Z(\phi)$ in the
IHQCD sector to lattice data on the thermodynamics, glueball spectrum, and
topological susceptibility of the Yang-Mills theory Gursoy:2009jd ;
Gursoy:2012bt ; Jokela:2018ers . The most challenging step is then to fit the
functions $V_{f}(\phi,\tau),\kappa(\phi)$, and $w(\phi)$ to data for full QCD.
This has been carried out by using different strategies: by fitting only the
lattice data for QCD thermodynamics Jokela:2018ers , by fitting the meson
spectrum (including highly excited states) Amorim:2021gat , and recently by
fitting both the thermodynamics and spectrum data Jarvinen:2022gcc . In this
article, we choose to use the first fit in Jokela:2018ers  (in the slightly
modified form published in Ishii:2019gta ). This fit reproduces the lattice
data for the QCD thermodynamics in the deconfined phase to the highest
precision, and in this article we will be focusing in this phase.

In Jokela:2018ers , several sets of potentials were determined. We will be
using the sets 5b, 7a, and 8b in this article. The different choices reflect
the freedom in the parameterization of the potentials which cannot be
determined through fitting to lattice data. Most of our analysis will use the
intermediate choice, 7a.

Apart from the potentials fitted to thermodynamics in Jokela:2018ers ;
Ishii:2019gta , we need to specify the function $Z(\phi)$ appearing in the
axion term (2.8). We use an Ansatz introduced in Gursoy:2012bt ,

$Z(\phi)=Z_{0}\left[1+c_{a}\,\frac{e^{\phi}}{8\pi^{2}}+d_{a}\left(\frac{e^{\phi}}{8\pi^{2}}\right)^{4}\right]=Z_{0}\left[1+c_{a}\,\frac{{\lambda}}{{\lambda}_{0}}+d_{a}\left(\frac{{\lambda}}{{\lambda}_{0}}\right)^{4}\right]\
.$ (2.20)

Here $Z_{0}$, $c_{a}$, and $d_{a}$ can be fitted to data (see Gursoy:2009jd ;
Gursoy:2012bt ). However, it was noticed in Weitz  that a too large value of
$Z_{0}$ can cause a trouble with the backgrounds at finite axial chemical
potentials: regular solutions asymptoting to the AdS5 geometry at the boundary
no longer exists. Since in this work the axial chemical potential is zero and
only the vectorial potential is finite, this problem does not appear, but we
want anyhow to choose the function $Z(\phi)$ such that the model is also
consistent when the axial potential is nonzero. For the choice of potentials
7a, this implies that $Z_{0}\lesssim 0.9060$. Here we follow Weitz  and first
set the constants $c_{a}$ and $d_{a}$ to such values that a small enough
$Z_{0}$ can be obtained. We then determine the value of $Z_{0}$ by comparing
to the lattice results for the topological susceptibility (see Appendix A).
Our choice is given by

$Z_{0}=0.670\ ,\qquad c_{a}=3\ ,\qquad d_{a}=20\ .$ (2.21)

##  3 Fluctuations

In this section, we analyze the fluctuations on top of the V-QCD background at
finite density and temperature which was introduced in the previous Sec. 2. We
classify these fluctuations both when the momentum is zero and nonzero. We
fluctuate all bulk fields as follows:

$\displaystyle g_{MN}(r,t,z)$ $\displaystyle=\bar{g}_{MN}(r)+e^{-i(\omega
t-qz)}\delta g_{MN}(r)\ ,$ (3.1) $\displaystyle A_{L/R\,M}^{ij}(r,t,z)$
$\displaystyle=\bar{A}_{L/R\,M}(r)\delta^{ij}+e^{-i(\omega t-qz)}\delta
A_{L/R\,M}^{ij}(r)\ ,$ $\displaystyle\phi(r,t,z)$
$\displaystyle=\bar{\phi}(r)+e^{-i(\omega t-qz)}\delta\phi(r)\ ,$
$\displaystyle T^{ij}(r,t,z)$ $\displaystyle=e^{-i(\omega t-qz)}\delta
T^{ij}(r)\ ,$ $\displaystyle\hat{a}(r,t,z)$ $\displaystyle=e^{-i(\omega
t-qz)}\delta\hat{a}(r)\ .$

Here the fluctuations are taken in the plane wave form with both finite
frequency $\omega$ and momentum $q$, which we choose to be in the
$z$-direction. Following Sec. 2, only the metric, dilaton, and temporal part
of the Abelian vector gauge field have backgrounds, i.e. $\bar{g}_{MN}(r)$,
$\bar{A}_{L/R,M}(r)=\delta_{M}^{t}\Phi(r)$ and $\bar{\phi}(r)$, respectively.
For notational simplicity, we will not include the bar in the background
solution elsewhere in this article. As there is no background for the tachyon
field, it will decouple from other fluctuations and all the components of the
complex matrix $\delta T^{ij}$ satisfy the same equation. Therefore it is
enough to take the fluctuation matrix to be proportional to the unit matrix
and consider a single wave function:

$\delta T^{ij}(r)=\delta^{ij}\delta\tau(r)\ .$ (3.2)

We switch to the basis of vectorial and axial gauge fields, and decompose the
fluctuations as

$\displaystyle\delta V^{M}(r)$ $\displaystyle=\frac{1}{2}\left[\delta
A_{L}^{M}(r)+\delta A_{R}^{M}(r)\right]=\delta\hat{V}^{M}(r)\mathbb{I}+\delta
V^{M\,a}(r)t^{a}$ (3.3) $\displaystyle\delta A^{M}(r)$
$\displaystyle=\frac{1}{2}\left[\delta A_{L}^{M}(r)-\delta
A_{R}^{M}(r)\right]=\delta\hat{A}^{M}(r)\mathbb{I}+\delta A^{M\,a}(r)t^{a}$

where $\mathbb{I}$ represents the $N_{f}\times N_{f}$ unit matrix in flavor
space, and the $SU(N_{f})$ generators are normalized as
$\mathrm{Tr}(t^{a}t^{b})=\delta^{ab}/2$. From now on, we refer to the flavors
singlet terms $\delta\hat{V}^{M}$ and $\delta\hat{A}^{M}$ as Abelian
fluctuations, and $\delta V^{M\,a}$ and $\delta A^{M\,a}$ as non-Abelian
fluctuations, respectively. The division to these components is natural:
because the background is flavor singlet, flavor symmetry dictates that
fluctuations in the Abelian and non-Abelian sectors are decoupled.

###  3.1 Classification of fluctuations and the gauge invariant combinations

We then discuss how the fluctuations are reduced to the physically relevant
components and how they can be classified (see, e.g., Ammon:2017ded ;
Jansen:2017oag ). In order to do this, we first recall how the various fields
transform under diffeomorphisms and bulk gauge transformations:

$\displaystyle g_{MN}$ $\displaystyle\mapsto
g_{MN}-\nabla_{M}\xi_{N}-\nabla_{N}\xi_{M},$ $\displaystyle\hat{V}_{M}$
$\displaystyle\mapsto\hat{V}_{M}-\partial_{M}\alpha-\xi^{K}\nabla_{K}\hat{V}_{M}-\hat{V}_{K}\nabla_{M}\xi^{K},$
$\displaystyle\hat{A}_{M}$
$\displaystyle\mapsto\hat{A}_{M}-\partial_{M}\beta-\xi^{K}\nabla_{K}\hat{A}_{M}-\hat{A}_{K}\nabla_{M}\xi^{K},$
$\displaystyle V^{a}_{M}$ $\displaystyle\mapsto
V^{a}_{M}-\partial_{M}\alpha^{a}-f^{abc}V_{M}^{b}\alpha^{c}-\xi^{K}\nabla_{K}V_{M}^{a}-V_{K}^{a}\nabla_{M}\xi^{K},$
$\displaystyle A^{a}_{M}$ $\displaystyle\mapsto
A^{a}_{M}-\partial_{M}\beta^{a}-f^{abc}A_{M}^{b}\beta^{c}-\xi^{K}\nabla_{K}A_{M}^{a}-A_{K}^{a}\nabla_{M}\xi^{K},$
$\displaystyle\phi$ $\displaystyle\mapsto\phi-\xi^{K}\nabla_{K}\phi,\qquad\
\tau\mapsto\tau-\xi^{K}\nabla_{K}\tau,\qquad\hat{a}\mapsto\hat{a}-\frac{2N_{f}}{N_{c}}\beta-\xi^{K}\nabla_{K}\hat{a}.$
(3.4)

Here, $\alpha$, $\beta$, are the gauge functions for the Abelian vector and
axial vector fields, $\alpha^{a}$ $\beta^{a}$ are the gauge functions for the
non-Abelian vector and axial fields, and $\xi_{M}$ are the gauge functions for
the diffeomorphisms (infinitesimal variations of the coordinates). In
addition, $f^{abc}$ is the structure constant defined via
$[t^{a},t^{b}]=if^{abc}t^{c}$, and the covariant derivatives are defined by
using the background metric given in equation (2.12).

The total number of degrees of freedom (DOFs) in the fluctuations in the
Abelian sector is 28, which includes 15 from the metric, 10 from the gauge
fields, and 3 scalars. Fixing the 7 Abelian gauge transformations in (3.4),
which we do here by imposing the radial gauge ($\delta g_{Mr}=0$,
$\delta\hat{V}_{r}=0$, and $\delta\hat{A}_{r}=0$), the number of DOFs is
reduced to 21. These include nondynamical DOFs, which are eliminated by
constraint equations (see Kiritsis:2006ua  for a detailed discussion of the
gravity sector). The number of dynamical DOFs is 14, which includes a four
dimensional spin-two graviton (5 DOFs), vector and axial spin-one fields (6
DOFs), and three scalars (3 DOFs).

In the non-Abelian sector the counting of DOFs is somewhat simpler. We ignore
the non-Abelian components of the tachyon that we have already left out of the
analysis. Then the DOFs are the non-Abelian vector and axial fields, which
have in total 10 multiplets in the adjoint representation of the vectorial
$SU(N_{f})$. After imposing the gauge conditions $\delta V^{a}_{r}=0$, $\delta
A^{b}_{r}=0$ and eliminating nondynamical DOFs, we are left with 6 multiplets
which are recognized as non-Abelian the vector and axial spin-one fields on
the boundary.

| spin 2 | spin 1 | spin 1 | spin 0 | spin 1 | spin 1
---|---|---|---|---|---|---
|  | Abelian | Abelian |  | non-Abelian | non-Abelian
| graviton | vector | axial |  | vector | axial
helicity $\pm$2 | 2 | 0 | 0 | 0 | 0 | 0
helicity $\pm$1 | 2 | 2 | 2 | 0 | 2 | 2
helicity 0 | 1 | 1 | 1 | 3 | 1 | 1
Table 1:  The number of degrees of freedom classified according to helicity,
spin and parity in Abelian and Non-Abelian sectors. Notice that here we left
out the dimension of the adjoint representation of $SU(N_{f})$ under which the
non-Abelian fluctuations transform. Each multiplet in this representation is
counted as one in the table.

A summary of the dynamical fluctuations is given in Table 1. Apart from the
properties already discussed above, the fluctuations can be classified
according to helicity or equivalently the $z$-component of the spin, i.e.,
their transformation properties under the $SO(2)$ rotation symmetry around the
$z$-axis, fixed by the direction of the momentum. As we have chosen the
momentum to lie in the $z$-direction, this rotation is an exact symmetry even
at finite momentum. Apart from the decoupling of Abelian and non-Abelian
fluctuations, the fluctuations with different helicities therefore decouple,
as the background is rotation invariant. Notice that fluctuations with
different spins may be coupled. Apart from the fields listed in the table,
there are in principle also scalar and pseudoscalar non-Abelian fields arising
from the fluctuated tachyon, as well as an additional Abelian scalar mode
corresponding to the “imaginary part” of the tachyon fluctuation. However as
we pointed out above, these fields satisfy the same equation as the Abelian
tachyon field, and can therefore be excluded without loss of generality.

Even though we have already fixed the gauge, it is still useful to write the
fluctuation equations in terms of gauge invariant variables (GIVs), i.e.,
variables invariant under the transformations (3.4) for plane wave gauge
functions (for example $\alpha(x_{M})=e^{-i(\omega t-qz)}\alpha(r)$). This is
because if the variables are gauge invariant, it is guaranteed that all
coefficients in the fluctuation equations are gauge independent as well, which
typically leads to the simplest expressions for these equations. Notice that
this means that actually gauge fixing would not be necessary to obtain the
final equations, but it simplifies the steps in their derivations as there are
less functions to keep track of.

As fluctuations with different helicities are decoupled, we discuss the GIVs
for each helicity channel separately. To obtain the fluctuation equations, we
insert the Ansatz in (3.1) to the equations of motion listed in Appendix B,
and express the fluctuations of these equations in terms of the GIVs listed
here. The resulting fluctuation equations are given in Appendix C.

#### Helicity $0$:

The GIVs in the Abelian sector are

$\displaystyle Z_{0}=e^{2A(r)}\left[-q^{2}f(r)\delta g_{tt}+2\omega q\delta
g_{tz}\frac{}{}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\frac{1}{2}\left(q^{2}f(r)-\omega^{2}+\frac{q^{2}f^{\prime}(r)}{2A^{\prime}(r)}\right)(\delta
g_{xx}+\delta g_{yy})+\omega^{2}\delta g_{zz}\right],$
$\displaystyle\hat{E}_{V}=q\delta\hat{V}_{t}+\omega\delta\hat{V}_{z}-\frac{(\delta
g_{xx}+\delta g_{yy})q\Phi^{\prime}}{4A^{\prime}(r)}\
,\qquad\qquad\;\;Z_{\phi}=\delta\phi-\frac{\phi^{\prime}(\delta g_{xx}+\delta
g_{yy})}{4A^{\prime}(r)}\ ,$ $\displaystyle
Z_{a}=\delta\hat{a}-2ix\frac{\omega\delta\hat{A}_{t}/f+q\delta\hat{A}_{z}}{\omega^{2}/f-q^{2}}\
,\qquad\quad\quad\;\qquad\quad\;\;\hat{E}_{A}=q\delta\hat{A}_{t}+\omega\delta\hat{A}_{z}\
,$

and the tachyon fluctuation, $\delta\tau$ above, which is readily
diffeomorphism invariant in the absence of a tachyon background. The non-
Abelian GIVs are

$E_{V}^{a}=q\delta V_{t}^{a}+\omega\delta V_{z}^{a}\
,\qquad\qquad\qquad\qquad\;\;\;\qquad\quad\;\;E_{A}^{a}=q\delta
A_{t}^{a}+\omega\delta A_{z}^{a}\ .$ (3.5)

The abelian sector further decouples into subsectors of
$\\{Z_{0},\hat{E}_{V},Z_{\phi}\\}$ (where metric, Abelian vector and dilaton
fluctuations couple), $\\{Z_{a},\hat{E}_{A}\\}$ (where axion, Abelian axial
fluctuations couple) and $\\{\delta\tau\\}$ (the tachyon is decoupled from
other fields). The coupling of the subsectors comprising vector and axial
Abelian gauge field fluctuations stems from different signatures under parity
transformation. In non-Abelian sector, the GIVs $\\{E_{V}^{a}\\}$ and
$\\{E_{A}^{a}\\}$ decouple, again due to parity, moreover they satisfy the
same equations.

#### Helicity $\pm 1$:

The GIVs in the Abelian sector are

$\displaystyle Z_{1}^{\pm}=e^{2A}\left[q\left(\delta g_{tx}\pm i\delta
g_{ty}\right)+\omega\left(\delta g_{zx}\pm i\delta g_{zy}\right)\right]\ ,$
(3.6) $\displaystyle\delta\hat{V}^{\pm}=\delta\hat{V}_{x}\pm
i\delta\hat{V}_{y}\ ,\qquad\quad\;\delta\hat{A}^{\pm}=\delta\hat{A}_{x}\pm
i\delta\hat{A}_{y}\ ,$

and the GIVs in the non-Abelian sector are

$\delta V^{\pm\,a}=\delta V_{x}^{a}\pm i\delta V_{y}^{a}\ ,\qquad\delta
A^{\pm\,a}=\delta A_{x}^{a}\pm i\delta A_{y}^{a}\ ,$ (3.7)

where the $\pm$ sign denote different helicities. The GIVs with positive and
negative helicity are decoupled by rotation symmetry so it is enough to
consider the fields with positive helicity.

Abelian sector comprises the set
$\\{Z_{1}^{\pm},\delta\hat{V}^{\pm},\delta\hat{A}^{\pm}\\}$. Therefore it
involves coupled fluctuations of the metric, vector and the axial spin-one
fields. In the non-Abelian sector of $\\{\delta V^{\pm a},\delta A^{\pm
a}\\}$, non-Abelian vector and axial fluctuation couple through the CS term.
Actually, the structure of the CS term is such that the left and right handed
non-Abelian fields are decoupled (see Appendix C).

#### Helicity $\pm 2$:

We choose the GIVs to be the helicity eigenstates

$Z_{2}^{\pm}=e^{2A}\left(\delta g_{xx}-\delta g_{yy}\pm 2i\delta
g_{xy}\right)\ ,$ (3.8)

we note that actually any linear combination of $\delta g_{xx}-\delta g_{yy}$
and $\delta g_{xy}$ works equally well because the fields are decoupled and
satisfy the same fluctuation equation.

#### Summary:

The classification of the fluctuations and coupling among GIVs is
recapitulated in Figure 1. In the figure, $\alpha$ is used to denote spatial
components that are transverse to the momentum direction, i.e. $\alpha=x,y$.
The GIVs obey systems of second order differential equations according to the
coupling patterns described above and in Figure 1. Explicit expressions for
the fluctuation equations are given in Appendix C.

Figure 1: Finite momentum classification of fluctuations (represented by boxes
with dashed frames) and the coupling among gauge invariant combinations (GIVs,
depicted by boxes with solid frames) in the different helicity channels.

###  3.2 Limit of zero momentum

In the limit of zero momentum there is no preferred spatial direction and the
full rotation symmetry of the fluctuations is restored. This implies that
fluctuations with different spin are decoupled, and the fluctuations belonging
to the same spin multiplets (given in Table 1) satisfy the same equations.
Actually, all the fluctuations are decoupled in this limit. Thus the
fluctuations are classified into spin $0$, spin $1$ and spin $2$ as follows:

#### Spin $0$:

This channel consists of the GIVs $Z_{\phi}$, $Z_{a}$, $\delta\tau$,
corresponding to the three scalars: the dilaton, the axion, and the tachyon,
respectively, which each satisfy different equations (see Appendix C.4).

#### Spin $1$:

The Abelian vectorial fields $E_{V}$ and $\delta\hat{V}^{\pm}$ form one spin-
one multiplet, and the Abelian axial fields $E_{A}$ and $\delta\hat{A}^{\pm}$
form another. The vector and the axial fluctuation equations differ in their
mass terms. The structure of the non-Abelian fields is similar: the vector
fields ($E_{V}^{a}$ and $\delta V^{\pm\,a}$) form one multiplet and the axial
fields ($E_{A}^{a}$ and $\delta A^{\pm\,a}$) form another. In this case all
fields satisfy the same fluctuation equation.

#### Spin $2$:

The GIVs $Z_{0}$ from helicity $0$, $Z_{1}^{\pm}$ from helicity $1$ and
$Z_{2}$ from helicity $2$ channels form the graviton multiplet and all obey
the same equation.

#### Summary:

The classification into spin channels and the mapping among GIVs are
recapitulated in the schema depicted in Figure 2. The decoupled equations
obeyed by the GIVs in the limit of vanishing momentum are given in Appendix
C.4.

Figure 2: Zero momentum classification into different spin channels. The
equality sign indicates that the modes satisfy the same fluctuation equations.

##  4 QNMs at zero momentum

In this section, we conduct a detailed analysis of QNMs at zero momentum. This
limit is special as it results in the decoupling of all fluctuations. In the
first subsection, we study the quantum critical low-temperature and high-
density regime, where the geometry is controlled by an AdS2 fixed point
Liu:2009dm ; Faulkner:2009wj ; Iqbal:2011in , which was has been analyzed in
the V-QCD model in Alho:2013hsa ; Hoyos:2021njg . We determine the locations
of the QNMs analytically for the AdS2 geometry and compare these results with
the numerical solutions. Subsequently, in the second subsection, we thoroughly
explore the temperature dependence of the QNMs at zero momentum.

###  4.1 Behavior at the AdS2 point

Let us then analyze the behavior at low temperatures, in the region controlled
by the AdS2 point. Before analyzing the fluctuations, it is useful to remind
how the geometry of the solutions in the deconfined phase at finite density,
$\mu\sim\Lambda$, changes as the temperature varies:

  * •

When $T=0$, the geometry is asymptotically AdS5 near the boundary, and
asymptotes to AdS${}_{2}\times\mathbb{R}^{3}$ in the IR.

  * •

When $0<T\ll\Lambda$, the flow is otherwise similar to the zero temperature
solution, but the “pure” AdS geometry is replaced by a AdS2 black hole. This
is a general structure for a near-extremal charged black hole, and it has been
analyzed in detail in Hoyos:2021njg  in the context of V-QCD.

  * •

When $T\sim\Lambda$, the geometry is that of a “regular” charged black hole:
The geometry is asymptotically AdS5 near the boundary, but does not have other
regions where the geometry takes a simple form.

The fixed point AdS${}_{2}\times\mathbb{R}^{3}$ geometry (without flow) is
also an exact solution of the V-QCD action. This solution is found by setting
$A$ and $\phi$ to constants, which we denote by $A_{*}$ and $\phi_{*}$,
respectively. In this case the equations of motion (see (B.9)–(B.12) in
Appendix B) imply Alho:2013hsa

$V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})=0=\frac{\partial
V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}{\partial\phi}\ ,$ (4.1)

where

$V_{\mathrm{eff}}(\phi,A,\hat{n})\equiv
V_{g}(\phi)-xV_{f0}(\phi)\sqrt{1+\frac{\hat{n}^{2}}{e^{6A}V_{f0}(\phi)^{2}w(\phi)^{2}}}\
.$ (4.2)

These equations determine the value of $\phi_{*}$ and the combination

$\tilde{n}_{*}=\frac{\hat{n}}{e^{3A_{*}}}$ (4.3)

which is invariant under the symmetries (B.15) and (B.16) of the equations of
motion, given in Appendix B. This variable is identified as the ratio between
the number density and the entropy Alho:2013hsa . Choosing the boundary of the
AdS2 space (which is interpreted as the extremal horizon of the full five-
dimensional geometry) to lie at $r=0$, the solution for the blackening factor
is obtained from (B.9):

$f(r)=\frac{e^{2A_{*}}}{L_{2}^{2}}r^{2}$ (4.4)

where

$L_{2}=\frac{1}{\sqrt{\frac{1}{6}\frac{\partial
V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}{\partial A}}}$ (4.5)

is the radius of the AdS2 space. The resulting geometry is

$ds^{2}=\frac{L_{2}^{2}dr^{2}}{r^{2}}-\frac{e^{4A_{*}}r^{2}dt^{2}}{L_{2}^{2}}+e^{2A_{*}}d\mathbf{x}^{2}\
.$ (4.6)

Notice that when this geometry is found as the endpoint of the zero
temperature RG flow, the only free parameter is $A_{*}$ while rest are fixed
by the potentials. This parameter is interpreted as (the logarithm of) the IR
energy scale Alho:2013hsa ; Hoyos:2021njg , and solving its relation to the UV
scale $\Lambda$ in (2.19) requires the knowledge of the whole flow. When the
chemical potential is varied keeping $T=0$ (and $\Lambda$) fixed, the flow and
therefore also the value of $A_{*}$ change. However, this parameter will
cancel in the expressions for the dimensions we derive below.

Let us then study the fluctuations in the AdS2 background. The presence of the
AdS factor in the geometry is interpreted to signal the presence of a one
dimensional IR CFT Faulkner:2009wj ; Edalati:2010hk . Even though we do not
have an explicit dictionary between the fluctuations around the fixed point
and the operators they correspond to in the CFT, the dimensions of the
fluctuations (or the corresponding operators in the CFT) are still useful to
classify them. For this purpose, it is enough to look at the fluctuation
equations at zero frequency and momentum (using the general expressions from
Appendix C.4) for the exact AdS${}_{2}\times\mathbb{R}^{3}$ geometry.

Using the expressions from Appendix C.4, we see that the fluctuations of the
metric and non-Abelian gauge fields satisfy the equation

$F^{\prime\prime}+\frac{2}{r}F^{\prime}=0\ ,$ (4.7)

where $F$ stands for any fluctuation wave functions in the metric and non-
Abelian gauge-field sectors. This equation has the solution

$F=C_{1}r^{\Delta_{*}-1}+C_{2}r^{-\Delta_{*}}$ (4.8)

with $\Delta_{*}=1$.

Another set of fluctuations that turns out to obey a simple fluctuation
equation in the AdS2 geometry is the set of Abelian vector fluctuations. Their
fluctuation equation in (C.17) has the additional mass term

$\frac{\hat{n}^{2}xe^{-4A}}{fV_{f0}(\phi)w(\phi)^{2}R}=\frac{1}{3}\frac{e^{2A}}{f}\frac{\partial
V_{\mathrm{eff}}(\phi,A,\hat{n})}{\partial A}$ (4.9)

so at the AdS2 point, and at zero frequency, we find (using (4.4) and (4.5))
that

$F_{V}^{\prime\prime}+\frac{2}{r}F_{V}^{\prime}-\frac{2}{r^{2}}F_{V}=0\ ,$
(4.10)

where $F_{V}$ stands for all the different helicity components of the Abelian
vector fluctuations. This is the equation for a mode with $\Delta_{*}=2$. The
fluctuations of the metric and the gauge field, except for the Abelian axial
field, are therefore “universal” modes with integer $\Delta_{*}$.

The rest of the modes are “nonuniversal”, i.e., they have noninteger
dimensions. First, for the Abelian axial field we find

$F_{A}^{\prime\prime}+\frac{2}{r}F_{A}^{\prime}-\frac{8Z(\phi_{*})}{\tilde{n}_{*}^{2}r^{2}}F_{A}=0\
,$ (4.11)

from which we read

$\Delta_{*}=\frac{1}{2}+\frac{1}{2}\sqrt{1+\frac{32Z(\phi_{*})}{\tilde{n}_{*}^{2}}}\
.$ (4.12)

For the tachyon fluctuation, we find

$\delta\tau^{\prime\prime}+\frac{2}{r}\,\delta\tau+\frac{4w(\phi_{*})^{2}}{x\tilde{n}_{*}^{2}\kappa(\phi_{*})\sqrt{1+\frac{\tilde{n}_{*}^{2}}{V_{f0}(\phi_{*})^{2}w(\phi_{*})^{2}}}}\frac{1}{r^{2}}\,\delta\tau=0\
,$ (4.13)

and

$\Delta_{*}=\frac{1}{2}+\frac{1}{2}\sqrt{1-\frac{16w(\phi_{*})^{2}}{x\tilde{n}_{*}^{2}\kappa(\phi_{*})\sqrt{1+\frac{\tilde{n}_{*}^{2}}{V_{f0}(\phi_{*})^{2}w(\phi_{*})^{2}}}}}\
.$ (4.14)

Moreover, the axion fluctuation equation becomes

$Z_{a}^{\prime\prime}+\frac{2}{r}Z_{a}^{\prime}-\frac{8Z(\phi_{*})}{\tilde{n}_{*}^{2}r^{2}\left(1+\frac{\tilde{n}_{*}^{2}}{V_{f0}(\phi_{*})^{2}w(\phi_{*})^{2}}\right)}Z_{a}=0\
,$ (4.15)

from which we read

$\Delta_{*}=\frac{1}{2}+\frac{1}{2}\sqrt{1+\frac{32Z(\phi_{*})}{\tilde{n}_{*}^{2}\left(1+\frac{\tilde{n}_{*}^{2}}{V_{f0}(\phi_{*})^{2}w(\phi_{*})^{2}}\right)}}\
.$ (4.16)

The analysis of the dilaton equation (C.18) however turns out to be a bit
involved. Naively imposing the conditions (4.1) leads to

$Z_{\phi}^{\prime\prime}+\frac{2}{r}Z_{\phi}^{\prime}-\frac{9\,\partial_{\phi}^{2}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}{4\,\partial_{A}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}\frac{1}{r^{2}}Z_{\phi}=0\
,$ (4.17)

which gives

$\Delta_{*}=\frac{1}{2}\left[1+\sqrt{1-\frac{9\,\partial_{\phi}^{2}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}{\partial_{A}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}}\right]=\frac{1}{2}\left[1+\sqrt{1-\frac{3}{2}L_{2}^{2}\partial_{\phi}^{2}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}\right]\
.$ (4.18)

However, in general the denominators in the mass term of (C.18) have factors
of $A^{\prime}$ which also vanishes at the AdS2 point. This reflects the fact
that $Z_{\phi}$ in (3.1) becomes ill defined for the AdS2 solution. When the
dilaton does not flow, the fluctuation $\delta\phi$ is diffeomorphism
invariant, and should be used instead of $Z_{\phi}$. Working out the
fluctuations directly around the fixed AdS${}_{2}\times\mathbb{R}^{3}$
geometry of Eq. (4.6), one can check that (4.17) is indeed the correct
fluctuation equation for the dilaton in this case.

The situation for the flows ending on the AdS2 point is however more
complicated. The flow solution near the AdS2 point that has been found
explicitly in Hoyos:2021njg , and can be inserted in the general fluctuation
equation (C.18). As it turns out (see Appendix C.4), the equation (4.17) is
obtained when $\Delta_{*}\geq 3/2$ in (4.18), but not when $\Delta_{*}<3/2$.
The general expression for the dimension in this case is:

$\displaystyle\Delta_{*}^{\mathrm{fl}}$
$\displaystyle=\mathrm{max}\left\\{\Delta_{*},3-\Delta_{*}\right\\}$ (4.19)
$\displaystyle=\mathrm{max}\left\\{\frac{1}{2}\left[1+\sqrt{1-\frac{9\,\partial_{\phi}^{2}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}{\partial_{A}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}}\right],\frac{1}{2}\left[5-\sqrt{1-\frac{9\,\partial_{\phi}^{2}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}{\partial_{A}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}}\right]\right\\}\
.$

Field | Dimension $\Delta_{*}$ | 5b | 7a | 8b
---|---|---|---|---
$\phi$ | $\frac{1}{2}\left[1+\sqrt{1-\frac{3}{2}L_{2}^{2}\partial_{\phi}^{2}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}\right]$ | 1.3558 | 1.6566 | 1.4850
| $\Delta_{*}^{\mathrm{fl}}=\mathrm{max}\left\\{\Delta_{*},3-\Delta_{*}\right\\}$ | 1.6442 | 1.6566 | 1.5150
$\tau$ | $\frac{1}{2}+\frac{1}{2}\sqrt{1-\frac{16w(\phi_{*})^{2}}{x\tilde{n}_{*}^{2}\kappa(\phi_{*})\sqrt{1+\frac{\tilde{n}_{*}^{2}}{V_{f0}(\phi_{*})^{2}w(\phi_{*})^{2}}}}}$ | 0.9967 | 0.9946 | 0.9905
$a$ | $\frac{1}{2}+\frac{1}{2}\sqrt{1+\frac{32Z(\phi_{*})}{\tilde{n}_{*}^{2}\left(1+\frac{\tilde{n}_{*}^{2}}{V_{f0}(\phi_{*})^{2}w(\phi_{*})^{2}}\right)}}$ | 1.0008 | 1.0034 | 1.0180
$\hat{A}_{\mu}$ | $\frac{1}{2}+\frac{1}{2}\sqrt{1+\frac{32Z(\phi_{*})}{\tilde{n}_{*}^{2}}}$ | 1.0266 | 1.0382 | 1.0606
Table 2: The nonuniversal dimensions at the AdS2 fixed point. We include both
the analytic expressions and the numerical values for the potential sets used
in this article. Fluctuated fields from top to bottom rows are the dilaton,
the tachyon, the axion, and the Abelian axial gauge field.

The results for the nonuniversal modes are collected in Table 2. Note that
apart from the dimension of the dilaton, the anomalous dimensions are small:
the dimensions at the fixed point deviate from one by at most a couple of per
cent. The value of the charge $\tilde{n}_{*}$ at the fixed point, which is
controlled by the fit to the lattice data, turns out to be large as compared
to the values of the potentials. Typically we find that
$\tilde{n}_{*}=\mathcal{O}(10)$, while $w(\phi)=\mathcal{O}(1)$,
$\kappa(\phi)=\mathcal{O}(1)$ and $Z(\phi)=\mathcal{O}(1)$. The value of
$V_{f0}(\phi_{*})$ might be a bit larger than other potentials, but still less
than $\hat{n}_{*}$. Because in $n_{*}$ appears in the denominators in the
terms in the square root expressions that cause the dimensions of Table 2 to
deviate from one, the deviations are small.

Let us then recall what these results for the AdS2 dimensions mean for the
QNMs. At nonzero but small temperature, where the geometry approaches the AdS2
form in the IR, we expect that a sequence of “AdS2 modes” is found in each
sector, with frequencies given by the formula Faulkner:2010tq ; Arean:2020eus

$\frac{\omega}{2\pi T}=-i\left(\Delta_{*}+n\right)$ (4.20)

where $n=0,$ $1$, $2$, …. We will verify this explicitly below for the $n=0$
modes. In particular, we find that the dilaton QNMs converge to the value
given by (4.19) when it is distinct from (4.18).

Figure 3: Temperature dependence of QNMs at fixed chemical potential
($\mu=450$ MeV). The left column illustrates the temperature-dependent
trajectories of the lowest two complex modes for each type of fluctuation. In
the right column, the solid curves represent the temperature dependence of
purely imaginary modes, while the dotdashed curves depict the imaginary part
of the complex modes. The colors refer to fluctuations in different sectors as
indicated in the legend, where NA stands for non-Abelian. Lighter (darker)
shade shown the lowest (second to lowest) mode.

Figure 4: Temperature dependence of QNMs at fixed chemical potential
($\mu=600$ MeV). Notation as in Fig. 3.

###  4.2 Numerical analysis: temperature dependence of QNMs

In this subsection, we present the trajectories of QNMs obtained by
numerically solving the fluctuation equations for each sector at zero momentum
while varying the temperature. The numerical solutions are found by using a
variant of pseudospectral methods, which is described in Appendix D. The
chemical potential is held fixed, and we investigate the temperature
dependence for two different values: $\mu=450$ MeV and $\mu=600$ MeV. These
values were chosen to lie above the critical chemical potential of the
confinement-deconfinement transition, which is at $\mu_{c}\approx 406$ MeV for
potentials 7a, but quite close to the transition in order to avoid the weakly
coupled region at asymptotically high $\mu$ where holography is not expected
to work.

As previously mentioned, at vanishing momentum, all modes decouple from each
other and are classified according to spin rather than helicity. The results
for $\mu=450$ MeV and $\mu=600$ MeV are shown in Figure 3 and Figure 4,
respectively. For these plots, we used potentials 7a discussed in Sec. 2.3. In
the left panels, we show the trajectories of the lowest two complex modes for
each sector in the complex frequency plane. The curves are parameterized with
temperature, and the locations of the QNMs at specific temperature values
($T=300$ MeV, $T=100$ MeV, and $T=10$ MeV) are marked on each curve. In this
set of plots, both the imaginary and real parts of the QNM frequency are
normalized with the energy scale $\Lambda\sim\Lambda_{QCD}$ to emphasize the
temperature dependence of the trajectories. All the QNMs have similar
temperature dependence. The detailed shape of the curves does however slightly
depend on the mode, on the temperature, and on the chemical potential. The
modes at higher $\mu$ (Fig. 4) are typically further away from the real axis,
in units of $\Lambda$, as the modes at lower $\mu$.

In the right panels of Figure 3 and Figure 4, we show the temperature
dependence of the purely imaginary modes with solid curves. Additionally, the
imaginary parts of the complex modes are shown in the plots via dotdashed
curves. In contrast to the right-hand side plots, the frequencies are
normalized with $(\pi T)$ in these plots so that the temperature dependence of
the imaginary modes is roughly divided out. The curves of some of the
imaginary modes end abruptly before reaching $T=0$ because we cut out parts of
the curves with significant numerical issues.

As one can see from the plots, pure imaginary modes and imaginary parts of the
complex modes show the behavior expected for charge black holes: the
frequencies of the imaginary modes are roughly proportional to the
temperature, whereas the complex modes are roughly independent of the
temperature Horowitz:1999jd . Due to this behavior, the complex modes give
highly suppressed corrections to the dynamics at low temperatures. The lowest
mode, which controls the equilibration process, remains purely imaginary
within the temperature range considered. These modes are gravity, non-Abelian
gauge, and axion modes for the spin two, spin one, and spin zero sectors,
respectively. Overall, the mode with the longest lifetime is the leading,
imaginary axion mode.

Figure 5: The numerical results at $\mu=450$ MeV for the nonuniversal
dimensions at the AdS2 fixed point. Stars mark the values given by the
analytical expressions in the Table 2. The results demonstrates the
consistency of analytic and numerical calculation in the low-temperature and
high-density regime.

Another observation is that the behavior of the imaginary modes at low
temperatures is consistent with the formula (4.20) and the AdS2 dimensions
computed in Sec. 4.1. From the plots in Figs. 3 and 4 it is however difficult
to read off the low temperature limit precisely due to limited resolution. To
verify the agreement precisely, we therefore zoom into the behavior of QNMs
for vanishing momentum and at low temperatures (down to $T=0.01$ MeV), very
close to the AdS2-critical line, focusing on the nonuniversal modes, i.e., the
modes with noninteger AdS2 dimensions. For this computation, the chemical
potential is fixed to $\mu=450$ MeV.

The results are shown in Fig. 5, where we use the same color code as in Figs.
3 and 4: dilaton, tachyon, axion and Abelian axial gauge field modes are shown
by the green, magenta, purple and yellow curves, respectively. Additionally,
different styles of curves are used to represent different potentials: dashed,
solid, and dotdashed curves correspond to 5b, 7a, and 8b potentials. The stars
in the plots mark the values given by the analytical expressions presented in
Table 2.

In the top-left plot, we observe that the dilaton mode converges to the value
predicted by the equation (4.19) rather than the equation (4.18). The
remaining plots in Figure 5 confirm that other nonuniversal modes also
converge to the analytically computed values presented in Table 2. The curves
were obtained by fitting numerical data with linear combinations of (half)
integer powers of the temperature. This Ansatz gives essentially a perfect fit
to all the data. The mean-field like $\sim\sqrt{T}+\mathrm{const}$ behavior of
the data at low temperature is clear from all other sectors except for the
axion.

##  5 QNMs at finite momentum

In this section, we explore numerically the momentum dependence of QNMs. We
begin by analyzing the helicity one channels, where the CS term contributes
and therefore modulated instabilities can arise Domokos:2007kt ;
Nakamura:2009tf . In Sec. 5.1 and Sec. 5.2, we conduct a comprehensive
analysis of these instabilities. Subsequently, in Sec. 5.3, we examine the
helicity-zero channel in the low-temperature and high-density regime.

Figure 6: The momentum dependence of the QMNs in the non-Abelian (top row) and
the Abelian (bottom row) subsectors of helicity one channel at $\mu=100$ MeV,
$T=130$ MeV (left column) and $\mu=130$ MeV, $T=30$ MeV (right column).
Different styles of the curves represents different situations: solid, dashed,
dotted curves denote the full V-QCD model, the model without the $S_{a}$ term
in (2.8), and the model without both the CS term in (2.1) and the $S_{a}$
term, respectively. The colors of the imaginary modes on the top row refer to
the nature of the mode at zero momentum as indicated in Figs. 3 and 4. The
gray curves are imaginary parts of complex modes. On the bottom mode, the
orange (cyan) curves show the decoupled left (right) handed modes or vice
versa, depending on the sign of helicity. The shade of the specific color
refers to the location on the imaginary frequency axis, such that the mode
with longest lifetime if brightest.

###  5.1 Modulated instability

We start by studying the spatially modulated instabilities in the Abelian and
non-Abelian subsectors of the helicity one channel. Both subsectors are
affected by the CS term. We first analyze these instabilities in two different
points of the phase diagram: the plots in the left and right columns of Fig. 6
correspond to $\mu=100$ MeV, $T=130$ MeV, and $\mu=450$ MeV, $T=30$ MeV,
respectively. The top row illustrates the non-Abelian subsector, while the
bottom row displays the Abelian subsector.

In the non-Abelian subsector (the top row in Fig. 6) the results have slightly
simpler structure than in the Abelian sector (bottom row). Recall that the
left and right handed modes are decoupled, and we use cyan and orange for
these modes. Which of the curves is the left or right handed mode depends on
the signature of the helicity.

The most striking feature in these plots is the strong modulated instability:
the lowest mode crosses to positive $\mathrm{Im}\,\omega$ at finite momentum.
As the non-Abelian subsector only contains gauge field fluctuations, the
dynamics is analogous to that of the Einstein-Maxwell-dilaton model where this
effect was originally observed Nakamura:2009tf ; Ooguri:2010kt . The details
however depend on the location $(\mu,T)$ on the phase diagram: this is why we
chose two different points for the plots in the left and right columns in Fig.
6. In the bottom left plot, the cyan modes are at such high negative
$\mathrm{Im}\,\omega$ that they are not visible. We also note that if the CS
term is turned off by hand (dotted curves in the bottom right plot) the
instability disappears, as expected.

The main difference between the Abelian and non-Abelian sectors is the mixing
of the gauge fields with the fluctuations of the metric in the Abelian sector.
The gravity modes in this sector include the hydrodynamical shear mode, for
which $\omega\to 0$ as $q\to 0$ (see the top row plots in Fig. 6).
Consequently this mode is the most important mode at low momentum.

In the Abelian sector (the bottom row in Fig. 6), a qualitatively similar
instability is observed as in the non-Abelian sector. In these plots, the
colors indicate the classification of the modes in the limit of zero momentum,
$q\to 0$, in the notation of Figs. 3 and 4. We stress that as $q$ grows and
the modes start to mix, the colors do not necessarily have anything to do with
the nature of the mode, and have been added just to guide the eye. The
similarity of the dispersion relations of the unstable modes between the
Abelian and non-Abelian sectors however suggests that the unstable modes are
dominantly left or right handed gauge fields also in the Abelian sector.

The dashed curves on the bottom row show the results in the absence of the
$S_{a}$ term of (2.8), which implements the physics of the axial U(1) anomaly.
The presence or absence of $S_{a}$ leads to a small effect, and there is no
qualitative differences between the curves with or without $S_{a}$. Turning
off the term slightly enhances the instability. We note that again that also
turning off the CS term (2.1) (dotted curves) removes the instability, as
expected.

Figure 7: The phase diagrams of V-QCD, illustrating the instability analysis
for different potential sets: 5b, 7a, and 8b, are presented in the top,
middle, and bottom plots, respectively. The shaded region below the dashed
(dot-dashed) curves indicates the region where the Abelian (non-Abelian)
sector accommodates spatially modulated instabilities. The red dotted curves
and stars denote the nuclear to quark matter transition and its critical
endpoint estimated using a different approach in Demircik:2021zll .

In this rest of this subsection, we investigate the extent of the
instabilities using three different V-QCD potentials, namely 5b, 7a, and 8b.
The results are presented in Fig. 7, with the top, middle, and bottom plots
corresponding to potentials 5b, 7a, and 8b, respectively. In each plot, the
solid blue curves shows the boundary between the deconfined and confined
phases. The shaded region in the deconfined phase below the dashed (dot-
dashed) curves indicates the values of temperature and quark chemical
potential at which the Abelian (non-Abelian) sector exhibits the instability.

The shape of the region where the instability appears is somewhat surprising:
an instability at low temperature near the AdS2 line (the dashed green curve
along the horizontal axis) is expected, but the extent of the instability to
low chemical potentials and relatively high temperatures is not. Recall that
in the Reissner-Nordström background the onset of the instability only depends
on the ratio $\mu/T$ Nakamura:2009tf . Many features are common to all three
potentials. Firstly, the instability curves peak at $T\sim 150$ MeV in the
low-density regime and decrease towards higher chemical potential values.
Secondly, the instability region of the non-Abelian sector is consistently
larger than that of the Abelian sector. The precise extent of the instability,
in particular at high chemical potentials, however depends on the potentials:
the region with instability is at its largest (smallest) for potentials 8b
(5b).

As it turns out, the surprising shape of the region with instability is
closely related to our choice for the function $w(\phi)$, i.e., the coupling
of the gauge fields in the DBI action 2.3(see (2.4)). This can be read off
from the fluctuation equation for the non-Abelian fluctuations
(LABEL:eq:LRinstabilityeq): Here the last term, which arises from the CS term,
provides an effective negative mass squared contribution (for negative
helicity) and drives the instability. Notably, this term is proportional to
$w(\phi)^{-4}$, and consequently sensitive to the functional form of
$w(\phi)$. We use here the function $w(\phi)$ fitted to the lattice data in
Jokela:2018ers . More precisely, it is the baryon number susceptibility which
determines the functional form of $w(\phi)$. Lattice studies indicate (see,
e.g., Borsanyi:2011sw ) that the susceptibility decreases significantly with
decreasing temperature, which implies that the fitted $w(\phi)$ decreases fast
with $\phi$. Thus, for such solutions where large values of $\phi$ are
reached, the instability is strongly enhanced. Since the value of the dilaton
typically grows fast towards the IR, this means small black holes. Actually,
the black hole solutions have been explored as a function of the horizon value
of the dilaton in Alho:2012mh ; Alho:2013hsa ; Hoyos:2021njg , and it is the
black holes with small chemical potentials and temperatures close to the
critical temperature, for which highest horizon values of the dilaton are
reached.

Apart from $w(\phi)$, the term driving the instability in
(LABEL:eq:LRinstabilityeq) is proportional to the charge $\hat{n}$, so it
vanishes at zero chemical potential where the charge is also zero. The shape
of the region with instability in Fig. 7 can therefore be understood to arise
as the combination of these two effects: enhancement of the instability with
growing charge and with growing dilaton values at the horizon.

We also compare our results to the transition line between the nuclear matter
(NM) and quark matter (QM) phases, as computed in Demircik:2021zll , which is
shown as the red dotted curve. It is important to note that this transition
line is calculated for the extended version of the model, which includes the
NM phase and the pressure of electrons, and is included here solely for the
purpose of comparison. The gray stars represent the critical end points from
the same study. Interestingly, our results suggest that the critical point is
“cloaked” by the instability independently of the details of the potentials.

The value of momentum (not shown in the plots) $q_{c}$ where the instability
is at its strongest, i.e., $\mathrm{Im}\,\omega$ reaches its maximum, is of
the order of $\Lambda$ for all values of $\mu$ and $T$ where the instability
is found. The precise value depends on the location in the phase diagram but
the dependence on $T$ and $\mu$ is weak and the value of the momentum is
always comparable to $\Lambda$. For example, in the low-density region (left
panels of Fig. 6) we find that $q_{c}\approx 3.5$ GeV which corresponds to the
wavelength $\sim 0.35$ fm for the modulation. In this region, the maximal
value of $\mathrm{Im}\,\omega$ is $\sim 1.1$ GeV, corresponding to a time
scale $1/(\mathrm{Im}\,\omega)$ of $\sim 0.2$ fm/$c$.

Figure 8: Momentum dependence of the modes in the Abelian subsector of
helicity one channel at $\mu=450$ MeV and $T=100$ MeV, $T=30$ MeV, $T=3$ MeV,
$T=0.3$ MeV. The color code shows the nature of the modes at $q=0$ as in the
bottom row of Fig. 6.

###  5.2 Low temperature behavior in the helicity one sector

In this subsection, we focus on the Abelian subsector of the helicity one
channel, and the behavior as we lower the temperature, approaching the AdS2
line. In particular, we study how the instability affects the behavior of the
hydrodynamic shear diffusion mode near the AdS2 region Arean:2020eus . To this
end, we analyze the momentum dependence at a fixed chemical potential slice of
$\mu=450$ MeV while varying the temperature through four different values:
$T=100$ MeV, $T=30$ MeV, $T=3$ MeV, and $T=0.3$ MeV. We use here potentials
7a.

The results are shown in Fig. 8, employing the same color code as section 4 as
a function of the nature of the modes at zero momentum: red, blue and yellow
represent the gravity, Abelian vector and Abelian axial modes respectively,
with shade varying according to their location on the imaginary axis. We
remind that this classification works only in the limit $q\to 0$. As $q$ grows
past the value where clear mixing effect set in, the colors are only useful to
guide the eye.

First we note that the behavior of the instability in Fig. 8 is in agreement
with Fig. 7: At $T=100$ MeV there is no instability, but as the temperature is
lowered, the effect of the CS term becomes more prominent, and the instability
sets in. The unstable mode is not shown in the bottom row of Fig. 8 because it
is at considerably higher $\mathrm{Im}\,\omega$ than the scale of the
imaginary AdS2 modes. The maximal frequency of the unstable mode is roughly
independent of the temperature, $\mathrm{Im}\,\omega\sim T^{0}$, as $T\to 0$,
while the scale of the frequency of the AdS modes is set by the temperature,
in agreement with Eq. (4.20). Similarly, the characteristic momentum of the
instability (the value of $q$ at which $\mathrm{Im}\,\omega$ reaches its
maximum) is roughly independent of the temperature. Notice that the scale of
the plotted $q/(2\pi T)$ varies between the plots in Fig. 8.

At sufficiently low temperatures, as displayed in the plots in the bottom row
of Figure 8, the system enters the AdS2 region and starts exhibiting
“universal” behavior at low momentum Arean:2020eus . First, in agreement with
the low temperature behavior in Figs. 3, 4, and 5, the ratios
$\mathrm{Im}\,\omega/(2\pi T)$ become integers up to small corrections. That
is, the behavior follows the relation of Eq. (4.20). This appears to hold true
even for the nonuniversal axial mode (yellow curves as $q\to 0$) which has
noninteger $\Delta_{*}$, because the numerical value $\Delta_{*}\approx
1.0382$ is so close to one, the the difference cannot separated from
temperature-dependent corrections to this formula. Second, the shear modes
follows the standard dispersion relation $\omega\approx-iq^{2}$ in the
relevant region as $T\to 0$, where the diffusion constant $D\sim 1/T$.

In the absence of the instability, the collision of the shear mode with the
lowest AdS2 modes (which occurs at complex momentum values) would set the
radius of convergence of the hydrodynamic dispersion relation
Grozdanov:2019kge ; Grozdanov:2019uhi , and determine the characteristic
equilibration time of the system. In the limit of $T\to 0$ the result is
universal Arean:2020eus : it is determined in terms of the diffusion constant
and the dimension of the leading AdS2 mode. This happens because, as one can
see from Fig. 8, the scale of the momentum dependence of the AdS2 modes is
$\sim T^{0}$. Therefore the AdS2 modes have not yet started to move at the
momentum $q\sim T$ where the collision takes place, as we take $T\to 0$. The
higher order corrections in $q$ to the dispersion relation of the shear mode
are also suppressed for $q\sim T$. This leads to the result
$\omega_{\mathrm{coll}}\approx 2\pi T$, $q_{\mathrm{coll}}\approx\sqrt{2\pi
T}$ for the location of the collision. In particular, the collision happens at
almost real momentum. Moreover, the AdS2 modes and the shear mode are
essentially decoupled as $T\to 0$ as one can see from the bottom right plot in
Fig. 8: the shear mode passes the AdS2 modes with suppressed interaction. This
agrees with the observation in Davison:2013bxa  that unlike suggested by the
location of the collision, the behavior of hydrodynamics in the region is
controlled by the chemical potential rather than temperature.

We however stress that in the presence of a perturbative instability, such as
in the V-QCD model, the discussion of the features of hydrodynamics is
academic. For the collision of the modes to have a clear physical meaning, we
would need to turn off the CS term by hand. This would indeed remove the
instability, whereas the low momentum structure of the bottom right plot in
Fig. 8 would still remain. Nevertheless, it is interesting to see that the
low-temperature behavior of the QNMs in V-QCD is essentially the same as in
the neutral translation-breaking AdS4 and AdS4-Reissner-Nordström backgrounds
studied in Arean:2020eus . That is, the current model is much more complex
than in the earlier study, and even includes the instability driven by the CS
term, but the picture at low momentum is unaffected by this.

Figure 9: The plot for momentum dependence of the Abelian subsector of
helicity-zero channel at $\mu=450$ MeV, $T=3$ MeV. While solid curves
represent the purely imaginary modes, dashed (dotdashed) depicts imaginary
(real) parts of the complex mode. The colors show the nature of the modes at
$q=0$ following the notation in Figs. 3 and 6.

###  5.3 Low temperature behavior in the helicity zero sector

In this subsection, we conduct an analysis of the momentum dependence of the
QNM spectrum in the helicity zero channel. We focus on the
$\\{Z_{0},\hat{E}_{V},Z_{\phi}\\}$-subsector of Fig. 1 as it involves coupling
between metric, Abelian vector gauge field and dilaton fluctuations, and
therefore accommodates the sound and diffusion hydrodynamic modes.

We specifically investigate the situation in the AdS2 region, i.e., low
temperature and high density. The momentum dependence of the spectrum in this
region is shown in Figure 9 at $T=3$ MeV and $\mu=450$ MeV. As above, the QNMs
are denoted using a color code referring to their nature at low momentum: red,
blue, and green represent modes identified as gravity, Abelian vector and
dilaton modes at $q\to 0$, respectively. The shade of the curves corresponds
to their location on the complex axis. The representation of modes is such
that solid curves are used for purely imaginary modes, while the imaginary
(real) parts of the complex modes are depicted by dashed (dot-dashed) curves.

In Fig. 9, the sound and charge diffusion hydrodynamic modes are represented
by the light red dashed and light blue curves, respectively. For these modes
$\omega\rightarrow 0$ as $q\rightarrow 0$. As expected they are the most
dominant modes in the low $q$-region and they have the expected momentum
dependence: The real part of the sound mode is given by the red dot-dashed
curves, having a linear dependence, while the imaginary part, as well as the
purely imaginary diffusion modes, display a $q^{2}$ behavior as $q\rightarrow
0$. Notably, the imaginary part of the frequency of the sound mode is smaller
in magnitude than that of the diffusion mode. This indicates that model has
low bulk viscosity.

The modes identified as the metric and Abelian vector fluctuations as $q\to 0$
are represented by the darker red and darker blue solid curves, respectively.
These modes are located along the imaginary axis near the values expected for
universal AdS2 modes when $q\to 0$, i.e., at integer
$\mathrm{Im}\,\omega/(2\pi T)$. This observation confirms that the temperature
(chemical potential) value that is considered is sufficiently low (high), such
that the effects of the AdS2 region can be observed. Moreover, the mode
identified as the dilaton mode in the limit $q\to 0$ approaches a position
consistent with the AdS2 dimension calculated in Sec. 4.1.

As it happens for the shear hydrodynamic mode in the Sec. 5.2 at low
temperatures, the hydrodynamic diffusion mode interacts with the leading
imaginary AdS2 mode. The picture is again similar to the findings of
Arean:2020eus  for neutral translation-breaking AdS4 and AdS4-Reissner-
Nordström geometries. In this case, the modes collide at real momentum, while
for the shear mode and for the AdS4 analysis of Arean:2020eus  the collision
takes place at a momentum value with small but nonzero imaginary part. We
remark that the AdS2 modes depicted in Fig. 9 with dark blue and green solid
curves also undergo a collision, resulting in the emergence of a pair of
complex modes at a relatively low momentum.

Notice that the value of the temperature ($3$ MeV) is significantly higher
than the lowest value in Figure 8. Because of this, the decoupling of the
hydrodynamic diffusion mode from the AdS2 modes is less clear. We expect that
at even lower temperatures (which we do not explore here due to numerical
difficulties) the decoupling of these modes becomes more evident, leading to
the momentum of the collision to be determined by the universal value as
pointed out in Arean:2020eus . Recall however that the low temperature region
is perturbatively unstable due to the instability in the helicity-one sector,
so this universal behavior seems to lack a physically meaningful
interpretation in our case, unless one turns off the CS term giving rise to
the instability by hand. Nevertheless it is again interesting that the
crossing structure of the modes in the relatively complex V-QCD model is
similar to what was found in the somewhat simple AdS4 black holes in
Arean:2020eus .

###### Acknowledgements.

We thank for discussions D. Arean, C. Ecker, U. Gürsoy, T. Ishii, N. Jokela,
M. Kaminski, K.-S. Kim, E. Kiritsis, S. Nakamura, F. Nitti, F. Peña-Benítez,
A. Piispa, E. Préau, M. Roberts, D. Schaich, and A. Schmitt. T.D. acknowledges
the support of the Narodowe Centrum Nauki (NCN) Sonata Bis Grant No.
2019/34/E/ST3/00405. J. C. R. and M. J. have been supported by an appointment
to the JRG Program at the APCTP through the Science and Technology Promotion
Fund and Lottery Fund of the Korean Government. J. C. R. and M. J. have also
been supported by the Korean Local Governments – Gyeongsangbuk-do Province and
Pohang City – and by the National Research Foundation of Korea (NRF) funded by
the Korean government (MSIT) (grant number 2021R1A2C1010834). J. C. R. was
additionally supported by a DGAPA-UNAM postdoctoral fellowship.

##  Appendix A The choice of the function $Z(\phi)$

As discussed in the main text, we choose an Ansatz for the function $Z(\phi)$,
appearing in the axion term (2.8), as follows Gursoy:2012bt :

$Z(\phi)=Z_{0}\left[1+c_{a}\,\frac{e^{\phi}}{8\pi^{2}}+d_{a}\left(\frac{e^{\phi}}{8\pi^{2}}\right)^{4}\right]\
.$ (A.1)

As it turns out, a nonzero value of $Z_{0}$ gives rise to an anomalous
dimension for the axial current and axial chemical potential, given by
Arean:2016hcs ; Weitz

$\gamma_{A}=-1+\sqrt{1+\frac{4N_{f}Z(-\infty)\ell^{2}}{N_{c}V_{f0}(-\infty)w(-\infty)^{2}}}\
,$ (A.2)

where $Z(-\infty)=Z_{0}$ and $\ell$ is the UV AdS radius which satisfies

$\ell^{2}=\frac{12}{V_{g}(-\infty)-\frac{N_{f}}{N_{c}}V_{f0}(-\infty)}\ .$
(A.3)

As pointed out in Weitz , if $\gamma_{A}>1$, the axial current will be an
irrelevant operator. In this case there is no solution of the geometry that
would be asymptotically AdS5 at finite axial chemical potential. To avoid this
issue, we should require $\gamma_{A}<1$. This implies that

$Z_{0}<\frac{3N_{c}V_{f0}(-\infty)w(-\infty)^{2}}{4N_{f}\ell^{2}}\ .$ (A.4)

For the potential sets 5b, 7a, and 8b that we use in this article, this bound
evaluates to (see Jokela:2018ers ; Ishii:2019gta  for the precise definitions
of the potentials)

$Z_{0}<2.116\quad(5b)\ ,\qquad Z_{0}<0.906\quad(7a)\ ,\qquad
Z_{0}<1.893\quad(8b)\ .$ (A.5)

The function $Z(\phi)$ also determines the value of the (Yang-Mills)
topological susceptibility, which is given as Gursoy:2007er ; Gursoy:2009jd ;
Gursoy:2012bt

$\chi_{\mathrm{YM}}^{-1}=M_{p}^{-3}\int_{0}^{\infty}\frac{dr}{e^{3A_{\mathrm{YM}}(r)}Z(\phi_{\mathrm{YM}})}\
,$ (A.6)

where $A_{\mathrm{YM}}$ and $\phi_{\mathrm{YM}}$ are the background functions
for the pure glue theory, i.e., the model in the absence of flavors
($N_{f}=0$). The value of the topological susceptibility and its dependence on
$N_{c}$ has been studied extensively on the lattice Lucini:2001ej ;
Bonati:2016tvi ; Bonanno:2020hht ; Athenodorou:2021qvs ; Bennett:2022gdz .
Here we use the result $\chi_{\mathrm{YM}}/\sigma^{2}=0.01937(136)$ from
Bennett:2022gdz  extrapolated to $N_{c}=\infty$, where $\sigma$ is the string
tension. Instead of computing the string tension in the holographic model, we
study the dimensionless ratio $\chi_{\mathrm{YM}}/T_{c}^{4}$. The Yang-Mills
deconfinement temperature $T_{c}$ can be computed in the model by determining
the entropy and the temperature using black hole thermodynamics and
integrating to obtain the free energy difference between the confined and
deconfined phases Gursoy:2008bu ; Gursoy:2008za . For the lattice value of the
critical temperature, we use the $N_{c}=\infty$ result
$T_{c}/\sqrt{\sigma}=0.5970(38)$ from Lucini:2005vg , so that
$\chi_{\mathrm{YM}}/T_{c}^{4}\approx 0.152$.

Setting the coefficients $c_{a}$ and/or $d_{a}$ to small values (i.e., values
less than one) and requiring agreement with the lattice results typically
leads to values of $Z_{0}$ that violate the bounds (A.5) roughly by an order
of magnitude. However, as pointed out in Weitz , when these coefficients are
increased, the value of $Z_{0}$ decreases rapidly. Here we choose the
following values: $c_{a}=3$ and $d_{a}=20$, which were obtained by trial and
error.111Alternatively, one could also impose an Ansatz for which
$Z(-\infty)=0$, in which case the UV anomalous dimension vanishes and the
bound of (A.5) is absent. For this choice, fitting to the lattice value of
$\chi_{\mathrm{YM}}/T_{c}^{4}$ gives $Z_{0}\approx 0.670$, which is below the
bound for all potentials sets 5b, 7a, and 8b. Notice that the fitted value is
the same for all the potential sets, because it only depends on the Yang-Mills
sector, i.e., the function $V_{g}(\phi)$, which is the same in all these sets.

##  Appendix B Equations of motion

###  B.1 General equations

In this appendix we present the full equations of motion arising from the
action (2.1) in the case of vanishing tachyon, and discuss some of their
consequences.

The equations for the right and left handed gauge fields are given by

$\displaystyle-\frac{M_{p}^{3}N_{c}}{4}\partial_{N}\left(V_{f}(\phi(r))w(\phi(r))\sqrt{-\textrm{det}({\bf
A}_{R})}\left(({\bf A}_{R}^{-1})^{NM}-({\bf A}_{R}^{-1})^{MN}\right)\right)$
(B.1)
$\displaystyle-\frac{1}{24\pi^{2}M_{p}^{3}}\Big{[}\partial_{N}\Big{(}A^{R}_{P}F^{R}_{QS}+\frac{i}{2}A^{R}_{P}A^{R}_{Q}A^{R}_{S}\Big{)}+\frac{1}{2}A^{R}_{P}A^{R}_{Q}A^{R}_{S}A^{R}_{N}$
$\displaystyle-\frac{1}{4}F^{R}_{PQ}F^{R}_{SN}-\frac{3i}{4}A^{R}_{P}A^{R}_{Q}F^{R}_{SN}]\epsilon^{PNMQS}$
$\displaystyle-M_{p}^{3}\sqrt{-\det
g}\,Z(\phi)g^{MN}\left(N_{c}\partial_{N}\hat{a}-\mathrm{Tr}\left(A^{L}_{N}-A^{R}_{N}\right)\right)=0$

and

$\displaystyle-\frac{M_{p}^{3}N_{c}}{4}\partial_{N}\left(V_{f}(\phi(r))w(\phi(r))\sqrt{-\textrm{det}({\bf
A}_{L})}\left(({\bf A}_{L}^{-1})^{NM}-({\bf A}_{L}^{-1})^{MN}\right)\right)$
(B.2)
$\displaystyle+\frac{1}{24\pi^{2}M_{p}^{3}}\Big{[}\partial_{N}\Big{(}A^{L}_{P}F^{L}_{QS}+\frac{i}{2}A^{L}_{P}A^{L}_{Q}A^{L}_{S}\Big{)}+\frac{1}{2}A^{L}_{P}A^{L}_{Q}A^{L}_{S}A^{R}_{N}$
$\displaystyle-\frac{1}{4}F^{L}_{PQ}F^{L}_{SN}-\frac{3i}{4}A^{L}_{P}A^{L}_{Q}F^{L}_{SN}]\epsilon^{PNMQS}$
$\displaystyle+M_{p}^{3}\sqrt{-\det
g}\,Z(\phi)g^{MN}\left(N_{c}\partial_{N}\hat{a}-\mathrm{Tr}\left(A^{L}_{N}-A^{R}_{N}\right)\right)=0\
,$

respectively. The axion equation is given as

$\partial_{M}\left[g^{MN}\sqrt{-\det
g}\,Z(\phi)\left(N_{c}\partial_{N}\hat{a}-\mathrm{Tr}\left(A^{L}_{N}-A^{R}_{N}\right)\right)\right]=0$
(B.3)

where as the dilaton equation reads

$\displaystyle\frac{8}{3\sqrt{-\det(g)}}\partial_{P}\left(\sqrt{-\det(g)}g^{PQ}\partial_{Q}\phi\right)+\frac{\partial
V_{g}(\phi)}{\partial\phi}$ (B.4)
$\displaystyle-\frac{\partial\left(xV_{f0}(\phi)\sqrt{-\det({\bf
A}_{R})}\right)}{\sqrt{-\det(g)}\partial\phi}-\frac{\partial\left(xV_{f0}(\phi)\sqrt{-\det({\bf
A}_{L})}\right)}{\sqrt{-\det(g)}\partial\phi}=0\ .$

The Einstein equations take the form

$R_{MN}-\frac{1}{2}g_{MN}R=T_{MN}^{g}+T_{MN}^{f}+T_{MN}^{a}$ (B.5)

where the various contribution to the bulk energy-momentum tensor are given by

$\displaystyle
T_{MN}^{g}=\frac{-4}{3}\partial_{M}\phi\partial_{N}\phi-\frac{1}{2}g_{MN}V_{g}(\phi)+\frac{2}{3}g_{MN}\partial_{P}\phi\partial^{P}\phi\
$ (B.6) $\displaystyle
T_{MN}^{f}=\frac{1}{4}xV_{f0}(\phi)\frac{\sqrt{-\det({\bf
A}_{R\,MN})}}{\sqrt{-\det(g_{MN})}}g_{MP}(({\bf A}^{-1}_{R})^{PQ}+({\bf
A}^{-1}_{R})^{QP})g_{QN}+\left(R\leftrightarrow L\right),\ $ (B.7)
$\displaystyle
T_{MN}^{a}=Z(\phi)\Big{[}\frac{1}{2}\left(\partial_{P}\hat{a}-\mathrm{Tr}\left(A^{L}_{P}-A^{R}_{P}\right)\right)^{2}g_{MN}\
$
$\displaystyle-\left(\partial_{M}\hat{a}-\mathrm{Tr}\left(A^{L}_{M}-A^{R}_{M}\right)\right)\left(\partial_{N}\hat{a}-\mathrm{Tr}\left(A^{L}_{N}-A^{R}_{N}\right)\right)\Big{]}\
.$ (B.8)

###  B.2 Equations of motion for the background

We then write down the equations of motion for the background. We use the
metric in (2.12), eliminate the Abelian vectorial field using (2.13), and set
other gauge fields, the tachyon, and the axion to zero. The Einstein (B.5) and
dilaton (B.4) equations imply that

$\displaystyle f^{\prime\prime}+3A^{\prime}f^{\prime}$
$\displaystyle=\frac{1}{3}e^{2A}\frac{\partial
V_{\mathrm{eff}}(\phi,A,\hat{n})}{\partial
A}=\frac{x\hat{n}^{2}e^{-4A}}{V_{f0}(\phi)w(\phi)^{2}R(\phi,A,\hat{n})}\ ,$
(B.9) $\displaystyle
A^{\prime\prime}-\left(A^{\prime}\right)^{2}+\frac{4}{9}\left(\phi^{\prime}\right)^{2}$
$\displaystyle=0\ ,$ (B.10)
$\displaystyle\frac{3A^{\prime}f^{\prime}}{f}+12\left(A^{\prime}\right)^{2}-\frac{4}{3}\left(\phi^{\prime}\right)^{2}$
$\displaystyle=\frac{e^{2A}V_{\mathrm{eff}}(\phi,A,\hat{n})}{f}=\frac{e^{2A}}{f}\left[V_{g}(\phi)-xV_{f0}(\phi)R(\phi,A,\hat{n})\right]\
,$ (B.11)
$\displaystyle\phi^{\prime\prime}+3A^{\prime}\phi^{\prime}+\frac{f^{\prime}\phi^{\prime}}{f}$
$\displaystyle=-\frac{3e^{2A}}{8f}\frac{\partial
V_{\mathrm{eff}}(\phi,A,\hat{n})}{\partial\phi}=$ (B.12)
$\displaystyle=-\frac{3e^{2A}}{8f}\left[V_{g}^{\prime}(\phi)-\frac{xV_{f0}^{\prime}(\phi)}{R(\phi,A,\hat{n})}+\frac{x\hat{n}^{2}e^{-6A}w^{\prime}(\phi)}{V_{f0}(\phi)w(\phi)^{3}R(\phi,A,\hat{n})}\right]$

where the dependence on the potentials only appears through

$\displaystyle V_{\mathrm{eff}}(\phi,A,\hat{n})$ $\displaystyle\equiv
V_{g}(\phi)-xV_{f0}(\phi)R(\phi,A,\hat{n})\ ,$ (B.13) $\displaystyle
R(\phi,A,\hat{n})$
$\displaystyle\equiv\sqrt{1+\frac{\hat{n}^{2}}{e^{6A}V_{f0}(\phi)^{2}w(\phi)^{2}}}\
.$ (B.14)

This expression generalizes the effective potential of (2.15) to finite
density. Notice that these equations are not independent: for example the
dilaton equation (B.12) can be derived by using the other equations.

Apart from shifts in $r$, the equations are invariant under the following
scale transformations:

$\displaystyle r$ $\displaystyle\mapsto\Lambda_{r}^{-1}r\ ,$
$\displaystyle\quad A$ $\displaystyle\mapsto A+\log\Lambda_{r}\ ,$
$\displaystyle\quad\hat{n}$ $\displaystyle\mapsto\Lambda_{r}^{3}\hat{n}\ ;$
(B.15) $\displaystyle f$ $\displaystyle\mapsto\Lambda_{f}^{2}f\ ,$
$\displaystyle\qquad A$ $\displaystyle\mapsto A+\log\Lambda_{f}\ ,$
$\displaystyle\quad\hat{n}$ $\displaystyle\mapsto\Lambda_{f}^{3}\hat{n}\ .$
(B.16)

##  Appendix C Fluctuation equations

In this Appendix, we provide the linearized fluctuation equations for the
GIVs. The equations build a system of second order coupled (decoupled)
ordinary differential equations at finite (zero) momentum. The classification
of fluctuations and coupling among GIVs are discussed in Sec. 3. Since the
equations are complicated and their derivation is rather tedious, it is
important to carry out consistency checks in order to make sure that the
expressions are correct:

  * •

After expressing the fluctuation equations in terms of GIVs, all non-gauge-
invariant terms cancel out in the final equations.

  * •

All terms in the equations have consistent dimensions, and in more general,
respect the symmetries (B.15)–(B.16).

  * •

In the small charge limit, the equations reduce to the fluctuation equations
of the Einstein-Maxwell-dilaton system, given as a part of the “QNMspectral”
mathematica package Jansen:2017oag .

  * •

The equations reduce to previously known expressions Arean:2013tja ;
Arean:2016hcs  in the limit of zero temperature and density.

  * •

The equation decouple and respect the classification of modes presented in the
main text in the limit of zero momentum.

In the rest of Appendix, we present the system of the equations first at
finite momentum in following three subsections, following the classification
in Fig. 1. We then present the zero momentum limit of the equations (see Fig.
2 for the classification). Finally, we discuss the behavior of the dilaton
fluctuation equation near the AdS2 point in the last subsection.

###  C.1 Helicity two

The helicity two sector only contains fluctuations of the metric which satisfy

$\left(Z_{2}^{\pm}\right)^{\prime\prime}+\left(\frac{f^{\prime}}{f}-A^{\prime}\right)\left(Z_{2}^{\pm}\right)^{\prime}+\left(\frac{\omega^{2}}{f^{2}}-\frac{q^{2}}{f}-\frac{2A^{\prime}f^{\prime}}{f}-4A^{\prime
2}+\frac{8}{9}\phi^{\prime 2}\right)Z_{2}^{\pm}=0$ (C.1)

or equivalently

$\left(e^{-2A}Z_{2}^{\pm}\right)^{\prime\prime}+\left(3A^{\prime}+\frac{f^{\prime}}{f}\right)\left(e^{-2A}Z_{2}^{\pm}\right)^{\prime}+\left(\frac{\omega^{2}}{f^{2}}-\frac{q^{2}}{f}\right)e^{-2A}Z_{2}^{\pm}=0\
.$ (C.2)

###  C.2 Helicity one

The helicity one sector contains the non-Abelian gauge field equations

$\displaystyle(\delta V^{\pm\,a})^{\prime\prime}+\left[\frac{d}{dr}\log
e^{A}fw(\phi)^{2}V_{f0}(\phi)R\right](\delta V^{\pm\,a})^{\prime}$ (C.3)
$\displaystyle+\left[\frac{\omega^{2}}{f^{2}}-\frac{q^{2}}{fR^{2}}\right]\delta
V^{\pm\,a}\pm\frac{e^{-2A}\hat{n}q}{2\pi^{2}fM^{3}_{p}R^{2}w(\phi)^{4}V_{f0}(\phi)^{2}}\delta
A^{\pm\,a}=0$ $\displaystyle(\delta
A^{\pm\,a})^{\prime\prime}+\left[\frac{d}{dr}\log
e^{A}fw(\phi)^{2}V_{f0}(\phi)R\right](\delta A^{\pm\,a})^{\prime}$ (C.4)
$\displaystyle+\left[\frac{\omega^{2}}{f^{2}}-\frac{q^{2}}{fR^{2}}\right]\delta
A^{\pm\,a}\pm\frac{e^{-2A}\hat{n}q\delta}{2\pi^{2}fM^{3}_{p}R^{2}w(\phi)^{4}V_{f0}(\phi)^{2}}V^{\pm\,a}=0\
,$

which receive a contribution form the CS term. These equations are actually
diagonal in terms of the left and right handed fluctuations:

$\displaystyle(\delta A_{L}^{\pm\,a})^{\prime\prime}+\left[\frac{d}{dr}\log
e^{A}fw(\phi)^{2}V_{f0}(\phi)R\right](\delta A_{L}^{\pm\,a})^{\prime}$ (C.5)
$\displaystyle+\left[\frac{\omega^{2}}{f^{2}}-\frac{q^{2}}{fR^{2}}\right]\delta
A_{L}^{\pm\,a}\pm\frac{e^{-2A}\hat{n}q}{2\pi^{2}fM^{3}_{p}R^{2}w(\phi)^{4}V_{f0}(\phi)^{2}}\delta
A_{L}^{\pm\,a}=0$

and similarly for the right-handed field (but with opposite sign in the CS
contribution). Here $\delta A_{L}^{\pm\,a}=\delta V^{\pm\,a}+\delta
A^{\pm\,a}$.

The fluctuations in the Abelian sectors also couple to metric:

$\displaystyle(Z_{1}^{\pm})^{\prime\prime}+\frac{fA^{\prime}\left(\omega^{2}-fq^{2}\right)-\omega^{2}f^{\prime}}{f\left(fq^{2}-\omega^{2}\right)}(Z_{1}^{\pm})^{\prime}+\left(\frac{2\omega^{2}A^{\prime}f^{\prime}}{f(fq^{2}-\omega^{2})}+\frac{\omega^{2}-fq^{2}}{f^{2}}+\right.$
(C.6) $\displaystyle\left.\frac{8\phi^{\prime}}{9}-4A^{\prime
2}\right)Z_{1}^{\pm}-e^{-A}\hat{n}qx(\delta\hat{V}^{\pm})^{\prime}+\frac{e^{-A}\hat{n}qx\omega^{2}f^{\prime}}{f^{2}q^{2}-f\omega^{2}}\delta\hat{V}^{\pm}=0$
$\displaystyle(\delta\hat{V}^{\pm})^{\prime\prime}+\left[\frac{d}{dr}\log
e^{A}fw(\phi)^{2}V_{f0}(\phi)R\right](\delta\hat{V}^{\pm})^{\prime}$ (C.7)
$\displaystyle+\left[\frac{\omega^{2}}{f^{2}}-\frac{q^{2}}{fR^{2}}-\frac{x\hat{n}^{2}\omega^{2}e^{-4A}}{\left(\omega^{2}-fq^{2}\right)fV_{f0}(\phi)w(\phi)^{2}R}\right]\delta\hat{V}^{\pm}-$
$\displaystyle\frac{\hat{n}q(Z_{1}^{\pm})^{\prime}}{w(\phi)\left(fq^{2}-\omega^{2}\right)e^{3A}Rw(\phi)V_{f0}(\phi)}\pm\frac{e^{-2A}\hat{n}q\delta\hat{A}^{\pm}}{2\pi^{2}fM^{3}_{p}R^{2}w(\phi)^{4}V_{f0}(\phi)^{2}}-$
$\displaystyle\frac{2\hat{n}qA^{\prime}Z_{1}^{\pm}}{w(\phi)\left(fq^{2}-\omega^{2}\right)e^{3A}Rw(\phi)V_{f0}(\phi)}=0$
$\displaystyle(\delta\hat{A}^{\pm})^{\prime\prime}+\left[\frac{d}{dr}\log
e^{A}fw(\phi)^{2}V_{f0}(\phi)R\right](\delta\hat{A}^{\pm})^{\prime}+\left[\frac{\omega^{2}}{f^{2}}-\frac{q^{2}}{fR^{2}}-\frac{4xe^{2A}Z(\phi)}{fV_{f0}(\phi)w(\phi)^{2}R}\right]\delta\hat{A}^{\pm}$
(C.8)
$\displaystyle\pm\frac{e^{-2A}\hat{n}q\delta\hat{V}^{\pm}}{2\pi^{2}fM^{3}_{p}R^{2}w(\phi)^{4}V_{f0}(\phi)^{2}}=0$

where

$R=\sqrt{1+\frac{\hat{n}^{2}}{e^{6A}w(\phi)^{2}V_{f0}(\phi)^{2}}}\ .$ (C.9)

###  C.3 Helicity zero

The helicity zero sector is the most complicated sector. It further divides
into several sets of coupled equations (see Fig. 1) which we discuss
separately.

#### Equations for Abelian vector, helicity zero gravity, and the dilaton
fluctuations.

These three fluctuation wave functions, $\hat{E}_{V}$, $Z_{0}$, and
$Z_{\phi}$, satisfy a coupled system of three ordinary second order
differential equations. We have derived this system but since it is extremely
complicated (several pages) we do not include the expressions here. Note
however that it passes the same consistency checks (listed above) as all the
other equations.

#### Axion and Abelian axial vector equations.

These coupled equations can be written as

$\displaystyle\hat{E}_{A}^{\prime\prime}$
$\displaystyle+\Bigg{[}\frac{4\hat{n}^{2}q^{2}x\omega^{2}e^{-6A}Z(\phi)}{fw(\phi)^{2}V_{f0}(\phi)^{2}\left(q^{2}-\omega^{2}/f\right)\left(q^{2}-R^{2}\omega^{2}/f\right)N_{\mathrm{mix}}}\,\frac{d}{dr}\log\frac{f}{R^{2}}+$
$\displaystyle+\frac{d}{dr}\log\frac{e^{A}w(\phi)^{2}V_{f0}(\phi)R}{\omega^{2}/f-q^{2}/R^{2}}\Bigg{]}\hat{E}_{A}^{\prime}+\Bigg{[}\frac{4q^{2}x\omega^{2}f^{\prime}Z(\phi)}{f^{2}\left(q^{2}-\omega^{2}/f\right)^{2}N_{\mathrm{mix}}}\,\frac{d}{dr}\log\frac{f}{R^{2}}-$
$\displaystyle-\frac{\left(q^{2}-R^{2}\omega^{2}/f\right)\left(4xe^{2A}Z(\phi)+w(\phi)^{2}V_{f0}(\phi)\left(q^{2}-\omega^{2}/f\right)R\right)}{R^{3}fw(\phi)^{2}V_{f0}(\phi)\left(q^{2}-\omega^{2}/f\right)}\Bigg{]}\hat{E}_{A}-$
$\displaystyle-\frac{2iq\omega
Z(\phi)}{N_{\mathrm{mix}}}\,\frac{d}{dr}\log\frac{f}{R^{2}}\,Z_{a}^{\prime}+\frac{2i\hat{n}^{2}q\omega
e^{-4A}Z(\phi)}{fw(\phi)^{4}V_{f0}(\phi)^{3}R^{3}}\,Z_{a}=0$ (C.10)
$\displaystyle Z_{a}^{\prime\prime}$
$\displaystyle+\Bigg{[}\frac{d}{dr}\log\left(e^{3A}fZ(\phi)\right)+\frac{4xZ(\phi)q^{2}}{\left(q^{2}-\omega^{2}/f\right)N_{\mathrm{mix}}}\,\frac{f^{\prime}}{f}-\frac{4xZ(\phi)}{N_{\mathrm{mix}}}\,\frac{d}{dr}\log\frac{e^{5A}fZ(\phi)}{w(\phi)}-$
$\displaystyle-\frac{4e^{6A}x\left(q^{2}+\left(2R^{2}-3\right)\omega^{2}/f\right)w(\phi)^{2}V_{f0}(\phi)^{2}Z(\phi)}{\hat{n}^{2}N_{\mathrm{mix}}\left(q^{2}-\omega^{2}/f\right)}\,\frac{R^{\prime}}{R}\Bigg{]}Z_{a}^{\prime}+$
$\displaystyle+\frac{\left(q^{2}-\omega^{2}/f\right)^{2}w(\phi)^{2}V_{f0}(\phi)R^{3}+4e^{2A}x\left(q^{2}R^{2}-\omega^{2}/f\right)Z(\phi)}{R^{3}\left(\omega^{2}/f-q^{2}\right)fw(\phi)^{2}V_{f0}(\phi)}\,Z_{a}+$
$\displaystyle+\Bigg{[}\frac{2ie^{-2A}xq\omega\left(4e^{2A}xZ(\phi)-R\left(\left(R^{2}-2\right)q^{2}+R^{2}\omega^{2}/f\right)w(\phi)^{2}V_{f0}(\phi)\right)}{f\left(q^{2}-\omega^{2}/f\right)^{2}N_{\mathrm{mix}}}\,\frac{f^{\prime}}{f}+$
$\displaystyle+\frac{2ie^{-2A}xq\omega\left(3R\left(q^{2}-\omega^{2}/f\right)V_{f0}(\phi)w(\phi)^{2}+8e^{2A}xZ(\phi)\right)}{f\left(q^{2}-\omega^{2}/f\right)^{2}N_{\mathrm{mix}}}\,\frac{R^{\prime}}{R}+$
$\displaystyle+\frac{2ie^{-8A}\hat{n}^{2}xq\omega
R}{V_{f0}(\phi)\left(q^{2}f-\omega^{2}\right)N_{\mathrm{mix}}}\,\frac{d}{dr}\log\frac{e^{5A}fZ(\phi)}{w(\phi)}\Bigg{]}\hat{E}_{A}^{\prime}+$
$\displaystyle+\Bigg{[}\frac{2e^{-6A}i\hat{n}^{2}xq\omega\left(RV_{f0}(\phi)\left(q^{2}+e^{2A}RxV_{f0}(\phi)-\omega^{2}/f\right)w(\phi)^{2}+4e^{2A}xZ(\phi)\right)}{R^{3}\left(q^{2}-\omega^{2}/f\right)^{2}f^{2}w(\phi)^{4}V_{f0}(\phi)^{3}}-$
$\displaystyle-\frac{2ixq\omega\left(5w(\phi)A^{\prime}-w^{\prime}(\phi)\phi^{\prime}\right)}{\left(q^{2}-\omega^{2}/f\right)^{2}fw(\phi)}\,\frac{f^{\prime}}{f}+$
$\displaystyle+\frac{4ie^{-2A}q^{3}x\omega\left(R\left(q^{2}-R^{2}\omega^{2}/f\right)V_{f0}(\phi)w(\phi)^{2}+2e^{2A}xZ(\phi)\right)}{f\left(\omega^{2}/f-q^{2}\right)^{3}N_{\mathrm{mix}}}\left(\frac{f^{\prime}}{f}\right)^{2}+$
$\displaystyle+\frac{\left(8ie^{6A}qx^{2}\omega\left(q^{2}+\left(2R^{2}-3\right)\omega^{2}/f\right)w(\phi)^{2}V_{f0}(\phi)^{2}Z(\phi)\right)}{\hat{n}^{2}f\left(\omega^{2}/f-q^{2}\right)^{3}N_{\mathrm{mix}}}\,\frac{f^{\prime}}{f}\,\frac{R^{\prime}}{R}+$
$\displaystyle+\frac{2ie^{-2A}qRx\omega\left(q^{2}-R^{2}\omega^{2}/f\right)w(\phi)^{2}V_{f0}(\phi)}{f\left(q^{2}-\omega^{2}/f\right)^{2}N_{\mathrm{mix}}}\,\frac{f^{\prime}}{f}\,\frac{d}{dr}\log\frac{e^{5A}fZ(\phi)}{w(\phi)}\Bigg{]}\hat{E}_{A}=0$
(C.11)

where222The point where $N_{\mathrm{mix}}$ vanishes gives rise to an apparent
singularity of the fluctuation equations. It was shown in Arean:2016hcs  that
at zero temperature and density this singularity can be removed by a
redefinition of the wave functions. We expect that the same is true here.

$N_{\mathrm{mix}}=4xZ(\phi)+e^{-2A}w(\phi)^{2}V_{f0}(\phi)\left(q^{2}-R^{2}\omega^{2}/f\right)R\
.$ (C.12)

#### The tachyon equation.

The tachyon equation reads

$\delta\tau^{\prime\prime}+\frac{d}{dr}\left[\log
e^{3A}f\kappa(\phi)V_{f0}(\phi)R\right]\delta\tau^{\prime}+\left[\frac{\omega^{2}}{f^{2}}-\frac{q^{2}}{R^{2}f}+\frac{2e^{2A}}{R^{2}f\kappa(\phi)V_{f0}(\phi)}\right]\delta\tau=0\
.$ (C.13)

#### Non-Abelian gauge-field equations.

For the non-Abelian modes of the gauge fields we find the equation

$\left(E_{V}^{a}\right)^{\prime\prime}+\frac{d}{dr}\left[\log\frac{e^{A}V_{f0}(\phi)w(\phi)^{2}R}{\omega^{2}/f-q^{2}/R^{2}}\right]\left(E_{V}^{a}\right)^{\prime}+\left(\frac{\omega^{2}}{f^{2}}-\frac{q^{2}}{fR^{2}}\right)E_{V}^{a}=0$
(C.14)

and the axial field $E_{A}^{a}$ satisfies the same equation.

###  C.4 Equations in the limit $q\to 0$

In the limit of zero momentum, the fluctuations can be classified according to
their spin rather than helicity (or the $z$ component of the spin), as there
is no longer preferred spatial direction, see Fig. 2. All fluctuations are
decoupled.333To be precise, the decoupling is not complete: in the helicity
zero sector the fluctuation equation of $Z_{\phi}$ still includes dependence
on $Z_{2}$ even when $q=0$. However, since these are the only mixing terms,
they do not contribute to the QNMs. One way to see this to redefine
$Z_{2}=\sqrt{q}\,\tilde{Z}_{2}$ after which taking $q\to 0$ leads to complete
decoupling.

#### Spin two.

All fluctuations in the spin two sector satisfy the equation of a massless
probe scalar in the gravity background,

$F^{\prime\prime}+\left(3A^{\prime}+\frac{f^{\prime}}{f}\right)F^{\prime}+\frac{\omega^{2}}{f^{2}}F=0\
,$ (C.15)

where $F=e^{-2A}Z_{2}^{\pm}$, $e^{-2A}Z_{1}^{\pm}$, $e^{-2A}Z_{0}$.

#### Spin one.

The non-Abelian gauge field fluctuations satisfy the equation

$F^{\prime\prime}+\frac{d}{dr}\left[\log
e^{A}fV_{f0}(\phi)w(\phi)^{2}R\right]F^{\prime}+\frac{\omega^{2}}{f^{2}}F=0\
.$ (C.16)

That is, here $F=E_{V}^{a}$, $\delta V^{a\pm}$, $E_{A}^{a}$, $\delta
A^{a\pm}$. The equation for the Abelian vector and axial vector fields include
additional mass terms:

$\displaystyle F_{V}^{\prime\prime}+\frac{d}{dr}\left[\log
e^{A}fV_{f0}(\phi)w(\phi)^{2}R\right]F_{V}^{\prime}+\left(\frac{\omega^{2}}{f^{2}}-\frac{\hat{n}^{2}xe^{-4A}}{fV_{f0}(\phi)w(\phi)^{2}R}\right)F_{V}$
$\displaystyle=0\ ,$ (C.17) $\displaystyle
F_{A}^{\prime\prime}+\frac{d}{dr}\left[\log
e^{A}fV_{f0}(\phi)w(\phi)^{2}R\right]F_{A}^{\prime}+\left(\frac{\omega^{2}}{f^{2}}-\frac{4xe^{2A}Z(\phi)}{fV_{f0}(\phi)w(\phi)^{2}R}\right)F_{A}$
$\displaystyle=0\ ,$

with $F_{V}=\hat{E}_{V}$, $\delta\hat{V}^{\pm}$ and $F_{A}=\hat{E}_{A}$,
$\delta\hat{A}^{\pm}$. The vectorial mass term arises due to the coupling of
the vectorial fluctuations to the vectorial gauge field in the background, and
the axial mass term comes from the axion term in (2.8).

#### Spin zero.

In the spin zero sector, we have the fluctuation equations for the dilaton,
the tachyon, and the axion:

$\displaystyle Z_{\phi}^{\prime\prime}$
$\displaystyle+\left(3A^{\prime}+\frac{f^{\prime}}{f}\right)Z_{\phi}^{\prime}+\Bigg{[}\frac{\omega^{2}}{f^{2}}+$
(C.18)
$\displaystyle+\frac{e^{2A}}{f}\left(\frac{3}{8}\frac{\partial^{2}V_{\mathrm{eff}}(\phi,A,\hat{n})}{\partial\phi^{2}}+\frac{2\phi^{\prime}}{3A^{\prime}}\frac{\partial
V_{\mathrm{eff}}(\phi,A,\hat{n})}{\partial\phi}+\frac{8(\phi^{\prime})^{2}}{27(A^{\prime})^{2}}V_{\mathrm{eff}}(\phi,A,\hat{n})\right)\Bigg{]}Z_{\phi}=0$
$\displaystyle\delta\tau^{\prime\prime}$ $\displaystyle+\frac{d}{dr}\left[\log
e^{3A}f\kappa(\phi)V_{f0}(\phi)R\right]\delta\tau^{\prime}+\left[\frac{\omega^{2}}{f^{2}}+\frac{2e^{2A}}{R^{2}f\kappa(\phi)V_{f0}(\phi)}\right]\delta\tau=0$
(C.19) $\displaystyle Z_{a}^{\prime\prime}$
$\displaystyle+\Bigg{[}\frac{d}{dr}\log\left(e^{3A}fZ(\phi)\right)-\frac{4xZ(\phi)}{N_{\mathrm{mix}}^{(0)}}\,\frac{d}{dr}\log\frac{e^{5A}fZ(\phi)}{w(\phi)}+$
$\displaystyle+\frac{4e^{6A}x\left(2R^{2}-3\right)w(\phi)^{2}V_{f0}(\phi)^{2}Z(\phi)}{\hat{n}^{2}N_{\mathrm{mix}}^{(0)}}\,\frac{R^{\prime}}{R}\Bigg{]}Z_{a}^{\prime}+$
$\displaystyle+\left[\frac{\omega^{2}}{f^{2}}-\frac{4xe^{2A}Z(\phi)}{fw(\phi)^{2}V_{f0}(\phi)R^{3}}\right]\,Z_{a}=0$
(C.20)

where the effective potential $V_{\mathrm{eff}}$ was defined in (B.13), and at
$q=0$, $N_{\mathrm{mix}}$ of (C.12) reduces to

$N_{\mathrm{mix}}^{(0)}=4xZ(\phi)-\omega^{2}e^{-2A}V_{f0}(\phi)w(\phi)^{2}R^{3}/f\
.$ (C.21)

###  C.5 Behavior of the dilaton equation near the AdS2 point

Finally, we discuss how the dilaton equation (C.18) behaves near the endpoint
of the zero temperature flows that end in the AdS2 geometry. The flow is given
as an explicit asymptotic transseries expansion in Hoyos:2021njg . The terms
relevant here can be written as

$A=A_{*}+A_{1}r^{2\alpha}+A_{2}r+\cdots\
,\qquad\phi=\phi_{*}+\phi_{1}r^{\alpha}+\cdots\ ,\qquad
f=f_{0}r^{2}(1+\cdots)\ ,$ (C.22)

where

$\displaystyle\alpha$
$\displaystyle=\Delta_{*}-1=\frac{1}{2}\left[-1+\sqrt{1-\frac{9\,\partial_{\phi}^{2}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}{\partial_{A}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}}\right]\
,$ (C.23) $\displaystyle\frac{e^{2A_{*}}}{f_{0}}$
$\displaystyle=L_{2}^{2}=\frac{6}{\partial_{A}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}\
,\qquad\frac{\phi_{1}^{2}}{A_{1}}=\frac{9(1-2\alpha)}{2\alpha}\ .$ (C.24)

Here we already assumed that $0<\alpha<1$. This parameter is always positive,
and if $\alpha>1$, it is immediate that the equation reduces to that given in
(4.17). For the coefficient of the first order derivative we find for all
values of $\alpha$ that

$3A^{\prime}+\frac{f^{\prime}}{f}\approx\frac{2}{r}\ .$ (C.25)

Let us then assume that $1/2<\alpha<1$. In this case

$\displaystyle\frac{\phi^{\prime}}{A^{\prime}}$
$\displaystyle\approx\frac{\alpha\phi_{1}r^{\alpha-1}}{A_{2}}\
,\qquad\partial_{\phi}V_{\mathrm{eff}}(\phi,A,\hat{n})\approx\phi_{1}r^{\alpha}\,\partial_{\phi}^{2}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})\
,$ (C.26) $\displaystyle V_{\mathrm{eff}}(\phi,A,\hat{n})$
$\displaystyle\approx
A_{2}r\,\partial_{A}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})\ ,$ (C.27)

where we used the fact that both the effective potential and its first
$\phi$-derivative vanish at the fixed point. Inserting these in the mass term
on the second row of (C.18), we see that the first factor dominates, so that
(4.17) is again reproduced.

However, if $0<\alpha<1/2$ we find instead

$\displaystyle\frac{\phi^{\prime}}{A^{\prime}}$
$\displaystyle\approx\frac{\phi_{1}}{A_{1}r^{\alpha}}\
,\qquad\partial_{\phi}V_{\mathrm{eff}}(\phi,A,\hat{n})\approx\phi_{1}r^{\alpha}\,\partial_{\phi}^{2}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})\
,$ (C.28) $\displaystyle V_{\mathrm{eff}}(\phi,A,\hat{n})$
$\displaystyle\approx\frac{1}{2}\phi_{1}^{2}r^{2\alpha}\partial_{\phi}^{2}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})+A_{1}r^{2\alpha}\,\partial_{A}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})\
.$ (C.29)

Now all three factors in the mass term contribute. After some algebra and
using the relations in (C.23) and in (C.24), we find that

$\displaystyle\frac{e^{2A}}{f}\\!\left(\frac{3}{8}\frac{\partial^{2}V_{\mathrm{eff}}(\phi,A,\hat{n})}{\partial\phi^{2}}+\frac{2\phi^{\prime}}{3A^{\prime}}\frac{\partial
V_{\mathrm{eff}}(\phi,A,\hat{n})}{\partial\phi}+\frac{8(\phi^{\prime})^{2}}{27(A^{\prime})^{2}}V_{\mathrm{eff}}(\phi,A,\hat{n})\right)\approx\frac{-\alpha^{2}+3\alpha-2}{r^{2}}\
.$ (C.30)

The solutions of the resulting equation (at zero frequency)

$Z_{\phi}^{\prime\prime}+\frac{2}{r}Z_{\phi}^{\prime}+\frac{-\alpha^{2}+3\alpha-2}{r^{2}}Z_{\phi}=0$
(C.31)

are given by

$Z_{\phi}=C_{1}r^{\Delta_{*}^{\mathrm{fl}}-1}+C_{2}r^{-\Delta_{*}^{\mathrm{fl}}}$
(C.32)

with

$\Delta_{*}^{\mathrm{fl}}=2-\alpha=3-\Delta_{*}=\frac{1}{2}\left[5-\sqrt{1-\frac{9\,\partial_{\phi}^{2}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}{\partial_{A}V_{\mathrm{eff}}(\phi_{*},A_{*},\hat{n})}}\right]\
.$ (C.33)

Recall that this result holds for $0<\alpha<1/2$. Combining this with the
result for the case of $\alpha>1/2$ (so that $\Delta_{*}>3/2$), we recover
(4.19).

##  Appendix D Numerical method

In order to determine QNM spectrum, one needs to solve numerically a coupled
system ordinary differential equations (ODEs) that is described in the
previous Appendix C. We use pseudospectral methods, a well-established
approach frequently employed in gravity Dias:2009iu ; Dias:2010eu ;
Monteiro:2009ke  and in holography Gursoy:2016ggq ; Jansen:2017oag ;
Bosch:2017ccw ; Gursoy:2016ggq ; Buchel:2017map ; Bu:2019mow ; Gursoy:2016tgf
for QNM computations. In this appendix, we provide a concise overview of this
method (see, e.g., boyd01  for more details) and how it can be applied to the
specific equations derived in this article.

We start by discussing the choice of coordinates. Eddington-Finkelstein (EF)
coordinates are well-suited for implementing the horizon boundary conditions
when using pseudospectral methods. In EF coordinates the metric in (2.12) is
written as

$ds^{2}=e^{2A(r)}\left[-f(r)(dt^{\prime})^{2}+2dt^{\prime}dr+d\mathbf{x}^{2}\right],$
(D.1)

where $t^{\prime}$ is EF time coordinate defined through the exact
differential $dt^{\prime}=dt+f(r)^{-1}dr$. This definition directly integrates
to $t^{\prime}=t+G(r)$, where $G^{\prime}(r)=f(r)^{-1}$. Recalling that the
time dependence of the fluctuation is that of a plane wave, $\propto
e^{-i\omega t}$, this implies that the fluctuation wave functions $\psi$
transform as $\psi\mapsto e^{i\omega G(r)}\psi$. Near the horizon, all
fluctuation wave functions can be expressed as a sum of two solutions: an
ingoing solution with a constant leading term, and an outgoing solution that
oscillates rapidly. This is convenient when using the pseudospectral method,
because the method automatically picks the regular solution, i.e., the ingoing
solution, which is the horizon boundary condition dictated by causality.

It is also convenient to carry out a change of coordinates in the holographic
direction, first from the conformal coordinate $r$ to $A$ given in (D.1). In
these coordinates the boundary is located at $A=+\infty$. Therefore we impose
a UV cutoff and only solve the fluctuations in a finite range of $A$. The
fluctuation wave functions typically decrease exponentially with $A$ so a
moderate cutoff at $A=\mathcal{O}(10)$ is enough. As a second coordinate
transformation, we apply a linear mapping from $A$ to a new coordinate,
defined in the interval $[-1,1]$, which is the standard interval for the
pseudospectral method.

Finally, we need to choose the normalizable UV boundary condition for the
fluctuations. In order to implement this, we need to make sure that the
independent nonnormalizable solution diverges at the boundary. If it goes to a
constant or vanishes, we apply a redefinition of the wave function (e.g., by
an exponential factor in $A$) such that the normalizable solution is the only
solution remaining finite at the boundary. After this, the pseudospectral
method, which can only produce regular solutions to differential equations,
automatically excludes the nonnormalizable solution.

The essence of pseudospectral method is the discretization of differential
equations to turn them into matrix equations. This procedure involves
replacing the continuous independent variable with a discrete counterpart. The
domain of this discrete variable commonly referred to as the collocation grid.
Functions can then be approximated by

$\displaystyle f(x)\approx\sum_{i=0}^{N}f(x_{i})C_{i}(x)\ ,$ (D.2)

where $x_{i}$ denotes the collocation grid points $\\{x_{i}|i=0,\dots,N\\}$
and $C_{i}(x)$ are the cardinal functions satisfying
$C_{i}(x_{j})=\delta_{ij}$ so that the approximation (D.2) is exact on the
grid points. This expansion enables one to estimate the derivatives of the
function $f$ by using derivative matrices constructed from the derivatives of
the cardinal functions on the grid points. That is,

$f^{(n)}(x_{i})\approx\sum_{j}D_{ij}^{(n)}f(x_{j})$ (D.3)

where $D_{ij}^{(n)}=C_{j}^{(n)}(x_{i})$. For $n=0$, we take
$D_{ij}^{(0)}=\delta_{ij}$.

The standard choice for the grid, which we also use, is the
Chebyshev–Gauss–Lobatto (CGL) grid. The definition on the interval $[-1,1]$ is

$x_{i}=\cos\left(\frac{\pi i}{N}\right),\qquad i=0,1,\dots,N\ .$ (D.4)

The cardinal functions are constructed from Chebyshev polynomials $T_{n}$ as

$C_{i}(x)=\frac{2}{Np_{i}}\sum_{n=0}^{N}\frac{1}{p_{n}}T_{n}(x_{i})T_{n}(x)\
,$ (D.5)

where $p_{0}=p_{N}=2$ and $p_{j}=1$ for $0<j<N$. One can show that for this
choice of the grid the error in the approximation (D.2) decreases
exponentially with increasing grid size $N$.

Let us consider for simplicity a fluctuation equation for one decoupled wave
function $\psi$ at zero momentum. It can be written as

$M(\omega)\psi=\sum_{I=0}^{p}\omega^{I}M_{I}\psi=0,$ (D.6)

where $p$ is the highest power of the frequency $\omega$ appearing in the
equation, and $M(\omega)$ and $M_{I}$ are differential operators involving
derivatives of the holographic coordinate, and the functions forming the
background solution. The idea is then to evaluate the equation on the
collocation points, and apply the approximation (D.3) to the differential
operators. The resulting $N+1$ equations only depend on $f$ through its values
on the grid points. That is, the operators $M_{I}$ are replaced by the
matrices

$(\widetilde{M}_{I})_{ij}=\sum_{n=0}^{2}c_{nI}(x_{j})D^{(n)}_{ij}\ ,$ (D.7)

where the coefficients $c_{nI}(x_{j})$ are obtained by inserting the chosen
background evaluated at the collocation points. Similarly, the wave function
is replaced by the vector $\tilde{\psi}$, obtained by discretizing the
function as $\tilde{\psi}_{j}=\psi(x_{j})$. Defining
$\widetilde{M}(\omega)=\sum_{I=0}^{p}\omega^{I}\widetilde{M}_{I}$, we can
write the final discretized counterpart of (D.6):

$\widetilde{M}(\omega)\widetilde{\psi}=0\ .$ (D.8)

The QNMs are then approximated by nontrivial solutions to this equation, which
occur when the determinant of the matrix vanishes. In order to locate the
QNMs, one therefore just needs to find numerically the roots of the equation
$\det(\widetilde{M}(\omega))=0$.

We test the convergence of the modes by varying the grid size $N$. We find
that a larger grid is necessary for convergence at low temperature values,
near the AdS2 critical point. At high temperatures $N=50$ is sufficient to
compute the locations of the leading modes, but for the lowest temperatures
considered in this articles, $N=500$ or even more is required for convergence.

Finally, as a nontrivial consistency check, the QNMs behave as expected in the
limit of zero momentum. In particular, the AdS2 modes at low temperatures are
in agreement with analytical computations, and the diffusive, shear, and sound
hydro modes show the expected dispersion relation in the limit $q\rightarrow
0$.

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