aainstitutetext: School of Nuclear Science and Technology, Lanzhou University,
222 South Tianshui Road, Lanzhou 730000, China bbinstitutetext: Department of
Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA

# Constraints on quasinormal modes and bounds for critical points from pole-
skipping

Navid Abbasi b and Matthias Kaminski<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider a holographic thermal state and perturb it by a scalar operator
whose associated real-time Green’s function has only gapped poles. These
gapped poles correspond to the non-hydrodynamic quasinormal modes of a massive
scalar perturbation around a Schwarzschild black brane. Relations between
pole-skipping points, critical points and quasinormal modes in general emerge
when the mass of the scalar and hence the dual operator dimension is varied.
First, this novel analysis reveals a relation between the location of a mode
in the infinite tower of quasinormal modes and the number of pole-skipping
points constraining its dispersion relation at imaginary momenta. Second, for
the first time, we consider the radii of convergence of the derivative
expansions about the gapped quasinormal modes. These convergence radii turn
out to be bounded from above by the set of all pole-skipping points.
Furthermore, a transition between two distinct classes of critical points
occurs at a particular value for the conformal dimension, implying close
relations between critical points and pole-skipping points in one of those two
classes. We show numerically that all of our results are also true for gapped
modes of vector and tensor operators.

##  1 Introduction

Stability and causality of distinct formulations of hydrodynamics have been
discussed over several decades by now Hiscock:1985zz  and are still under
debate Kovtun:2019hdm ; Hoult:2020eho ; Taghinavaz:2020axp ; Bemfica:2017wps ;
Bemfica:2019knx ; Bemfica:2020zjp . One central question is what the
convergence radius of the hydrodynamic derivative expansion is when written in
momentum space (after a Fourier transformation in a translation-invariant
state).111The convergence of the hydrodynamic derivative expansion in position
space has been recently discussed again in Heller:2013fn ; Heller:2015dha ,
and the relation between position space and momentum space convergence has
been discussed in the regime linearized in the hydrodynamic fields
Heller:2020uuy . This question has recently been addressed by considering the
analytic continuation of the hydrodynamic derivative expansion to complex
frequencies as functions of complex momenta Withers:2018srf ;
Grozdanov:2019kge . There it was shown that, at least in systems with a
holographic dual, the radius of convergence is determined by the magnitude of
the momentum associated with the critical points of the spectral curve, those
which are the closest to the origin, which will be defined and discussed in
section 3. Hydrodynamic (eigen)modes can be understood as those momenta and
frequencies satisfying the spectral curve equation.

Hydrodynamic (eigen)modes are relevant for the description of systems out of
but close to equilibrium. Hydrodynamic perturbations are restricted to have
low energies, which translates to small frequencies and small momenta. They
describe the evolution of conserved charges like energy density and electric
charge over large distances in space and over large time intervals. Hence, the
relevance of such an expansion of gapless (eigen)modes about zero frequency
and zero momentum is immediately clear.

In this work, we consider gapped (eigen)modes and the radius of convergence of
derivative expansions about those gapped mode locations. There are several
physical motivations for this study. First, most systems in Nature only
contain gapped (eigen)modes. In that case, the lowest-lying mode, the one with
the smallest imaginary part of its frequency, dominates the time-evolution of
the state. Hence, an expansion about that mode location allows a perturbative
description of the time-dependence of the state. Examples of expansions around
gapped modes are (i) perturbations around the Fermi surface, or (ii)
perturbations around a large mass or energy scale leading to non-relativistic
excitations. Second, even in systems which contain both gapless and gapped
modes, the gapped modes can dominate the evolution of the state. Consider for
example a system which contains a diffusion mode, i.e. a mode with purely
imaginary (quasi-eigen)frequency, $\mathfrak{w}\propto\mathfrak{q}^{2}$, with
the momentum $\mathfrak{q}$ directed along one of the spatial coordinates. At
a particular value of the momentum $\mathfrak{q}$ the lowest non-hydrodynamic
mode has a smaller imaginary part than the hydrodynamic mode, thus becoming
the dominant QNM for larger momenta. Note that this transition from a regime
of hydrodynamic mode dominance to non-hydrodynamic mode dominance has been
pointed out before Amado:2008ji ; Kaminski:2009ce . The same change in
dominance of QNMs can happen when certain parameters of the system are
changed, for example the charge of a Reissner-Nordström black brane
Janiszewski:2015ura . Lastly, our results establish relations between pole-
skipping points Blake:2017ris , the critical points and the general locations
of gapped QNMs which have not been established for the gapped modes until now.

In thermal equilibrium, the spectrum of quasinormal modes (QNMs) is
holographically dual to the poles in the retarded Green’s function of a given
operator. The values of the QNMs determine how fast perturbations of that
operator thermalize. For operators without gapless excitations, the associated
spectrum contains a tower of infinitely many gapped QNMs. The lower a mode is
in the spectrum, the smaller is the imaginary part of its frequency and
consequently it will be swallowed later by the medium.

Recently, it has been shown that the pole-skipping phenomenon, at least in
holographic systems, forces the response functions associated with operators
in the theory to satisfy an infinite number of constraints Blake:2019otz ;
Grozdanov:2019uhi  222The pole-skipping phenomenon was first observed as the
indication for the hydrodynamic origin of quantum chaotic behavior
Grozdanov:2017ajz ; Blake:2017ris ; Blake:2018leo ; Grozdanov:2018kkt . Then
it was investigated in many other examples and with distinct motivations
Wu:2019esr ; Ahn:2019rnq ; Li:2019bgc ; Ceplak:2019ymw ; Das:2019tga ;
Abbasi:2019rhy ; Liu:2020yaf ; Ahn:2020bks ; Moitra:2020dal ;
Grozdanov:2019uhi ; Abbasi:2020ykq ; Jansen:2020hfd ; Grozdanov:2020koi ;
Ramirez:2020qer ; Choi:2020tdj ; Ahn:2020baf ; Natsuume:2020snz ;
Arean:2020eus ; Kim:2020url ; Sil:2020jhr .. These constraints are discrete in
the sense that they occur at discrete pole-skipping frequencies. Recalling
that the spectrum of QNMs is discrete, too, one may be tempted to look for a
possible relation between these two sets of discrete data. That task is what
the first part of this paper is devoted to. In section 2, we will show that
pole-skipping points constrain QNMs in a way more severe than what has been
known so far.

We start with computing the spectrum of gapped QNMs associated with a massive
scalar field in the bulk of AdS5. We will mostly work at imaginary momenta,
since the pole-skipping points in the dual theory exclusively occur at such
values. As one might expect, we find that all pole-skipping points are lying
on the dispersion relations of QNMs. But the non-trivial part of our result is
that we see that the dispersion relation of the $n^{th}$ QNM is constrained by
pole-skipping points with the frequencies $\omega=m(i2\pi T)$ where $m\geq n$.
It simply means that the lowest QNM is constrained by pole-skipping points at
all possible frequencies and consequently is the most constrained one.
However, by every one level going higher in the spectrum, the number of pole-
skipping frequencies constraining the dispersion relation decreases by one.

In order to get more insight into these relations among QNMs, in section 3, we
study the analytic structure of the spectral curve, see Withers:2018srf ;
Grozdanov:2019kge  and Grozdanov:2020koi . We discuss to what extent the
derivative expansion is reliable for probing the dispersion relation of a
given gapped QNM. Our focus lies on computing the radius of convergence of the
derivative expansion about the lowest-lying QNM with mode number $n=1$, as
well as the second QNM with mode number $n=2$. We perform this analysis for
scalar operators of conformal operator dimensions within the range
$2\leq\Delta\leq 8$. This is achieved by finding the critical points of the
spectral curve at complex momenta. It turns out that there exists a transition
value $\Delta_{\text{t}}$ which distinguishes between two types of critical
points. For the values of $\Delta$ lower than $\Delta_{\text{t}}$, the
critical point displays a behavior which we refer to as the lowest-level-
degeneracy of the two lowest-lying complexified QNMs. In this case, the real
part of the critical frequency vanishes, i.e. $\text{Re}\,\omega_{c}=0$. In
other words, within the complex frequency plane the two lowest-lying QNMs
collide with each other on the imaginary frequency axis, as illustrated in the
upper two plots in figure 5. Beyond $\Delta_{\text{t}}$ however, the relevant
critical point of the spectral curve is identified with the level-crossing of
QNMs at complex momentum and complex frequency, as it was described previously
Grozdanov:2019kge ; Grozdanov:2019uhi ; Abbasi:2020ykq ; Jansen:2020hfd ;
Heller:2020uuy ; Baggioli:2020loj . This is illustrated in figure 6. Right at
the transition point, $\Delta=\Delta_{\text{t}}$, separating these two classes
of critical points, there are three critical points which simultaneously
determine the radius of convergence of the derivative expansion about the
lowest-lying QNM, see figure 7.

In section 4 we find that pole-skipping points put a strong constraint on the
position of the aforementioned critical points. We observe that the critical
point is always enclosed within the circle passing through the nearest pole-
skipping point in the $\text{Im}{\mathfrak{w}}-\text{Im}{\mathfrak{q}}^{2}$
plane. In other words, the critical point and hence the radius of convergence
of the derivative expansion about the lowest-lying mode is bounded form above.
This upper bound on the radius of convergence is illustrated in figure 8.

Moreover, we have shown that all the observations we state for scalar
operators do also hold for vector and tensor operators, as discussed in
section 5. In order to relate our numerical results to a simple analytically
solvable model, in section 6, we consider the BTZ black brane which is
asymptotically $AdS_{3}$. In the discussion section 7 we provide a summary of
our results and discuss promising extensions of our work. In particular, we
consider possible implications of our results for quantum chaos and the bounds
on quantum chaos Maldacena:2015waa . More details about the relevant
calculations are provided in the appendices A and B.

##  2 Complexified quasinormal modes and pole-skipping

As our holographic model, we consider the Einstein-Hilbert action given by

$S=-\frac{1}{16\pi
G_{5}}\int_{\mathcal{M}}d^{5}x\,\,\sqrt{-g}\left(R-\frac{12}{L^{2}}\right)+S_{bdy}\,,$
(2.1)

with the $AdS$-radius, $L$. The Einstein equations in the trace-reduced form
are given by

$R_{\mu\nu}+4g_{\mu\nu}=\,0\,.$ (2.2)

We use the usual scaling symmetries of the metric to set $L=1$. Following the
notation of Policastro:2002se , the solution dual to a thermal state is the
Schwarzschild metric

$ds^{2}=\frac{(\pi
T)^{2}}{u}\left(-f(u)dt^{2}+\frac{}{}d\vec{x}^{2}\right)+\frac{1}{4u^{2}f(u)}du^{2}\,,$
(2.3)

with $f(u)=1-u^{2}$ and the Hawking temperature $T=r_{h}/\pi$, where $r_{h}$
is the location of the horizon in Poincaré coordinates.

### Scalar field in the bulk

In order to find the spectrum of long wavelength fluctuations of a boundary
probe operator we study the quasi normal modes of the dual bulk field. We
consider a scalar field of mass $m$ in the bulk. The corresponding equation of
motion reads

$\bigg{(}\partial_{\mu}(\sqrt{-g}\partial^{\mu})-m^{2}\sqrt{-g}\bigg{)}\Phi=\,0\,.$
(2.4)

The field $\Phi(u)$ in the bulk is dual to a boundary operator $\mathcal{O}$
with the following conformal operator dimension Witten:1998qj 333For example
in a $d-$dimensioanl conformal theory, the conformal dimension of the charge
density operator is $\Delta_{J}=d-1$ while that of the energy-momentum tensor
operator is $\Delta_{T}=d$.

$\Delta=2\left(1+\sqrt{1+\mathfrak{m}^{2}}\right)\,.$ (2.5)

With the Fourier (plane wave) ansatz $\Phi(u,x_{\mu})=\int d^{4}k\,e^{-i\omega
t+ikz}\,\phi(u;\omega,k)$, the equation of motion becomes

$\phi^{\prime\prime}+\frac{uf^{\prime}-f}{uf}\,\phi^{\prime}+\frac{u\mathfrak{w}^{2}-(\mathfrak{m}^{2}+\mathfrak{q}^{2}u)f}{u^{2}f^{2}}\,\phi=\,0\,,$
(2.6)

where $\mathfrak{w}=\omega/2\pi T$, $\mathfrak{q}=k/2\pi T$ and
$\mathfrak{m}=\frac{m}{2}$. To find the spectrum of QNMs, we look for a bulk
solution which is ingoing at the horizon. It can be expressed in terms of a
near horizon expansion of the form

$\phi(u)=(1-u)^{-i\mathfrak{w}/2}\sum_{n=0}^{\infty}a_{n}(\mathfrak{w},\mathfrak{q};\mathfrak{m})(1-u)^{n}\,.$
(2.7)

Then the Dirichlet condition at $u=0$ gives the spectral curve of the operator

$\sum_{n=0}^{n}a_{n}(\mathfrak{w},\mathfrak{q};\mathfrak{m})=\,0:=\,\mathcal{S}(\mathfrak{q}^{2},\mathfrak{w})\,.$
(2.8)

Numerically solving the spectral curve gives the spectrum of the QNMs
associated with the operator of the scaling dimension $\Delta$ (2.5)
Nunez:2003eq . The result associated with $\Delta=6$ is shown in figure 1.

Before closing this section, let us comment on the sum in eq. (2.8). According
to Nunez:2003eq , we need a sufficiently large but finite number of terms in
the sum, so that by any change in that number of terms, the frequencies of the
QNMs remain stable up to a specific number of digits. For the purposes of this
work, we have found that $n=50$ is sufficient to find the QNMs at the four
lowest levels, in complete agreement with those of Nunez:2003eq .

### Pole-skipping

In order to find the pole-skipping points of the operator $\mathcal{O}$ dual
to the scalar field in the bulk, it is convenient to work with ingoing
Eddington-Finkelstein coordinates. We first take $u=(\pi T/r)^{2}$ and then
consider

$dv=dt+\frac{dr}{r^{2}f(r)}\,.$ (2.9)

Then the metric (2.3) can be rewritten as

$ds^{2}=-r^{2}f(r)dv^{2}+2dvdr+r^{2}d\vec{x}^{2}\,.$ (2.10)

The strategy is to find a near-horizon solution to the equation (2.4). Let us
choose the Fourier ansatz $\Phi(r)=\int d^{4}k\,e^{-i\omega
v+ikz}\,\phi(r;\omega,k)$ with $\phi(r;\omega,k)$ expanded near the horizon
($r_{h}=\pi T$) as

$\phi(r)=\sum_{n=0}^{\infty}\phi_{n}(r-r_{h})^{n}=\,\phi_{0}+(r-r_{h})\phi_{1}+\cdots\,.$
(2.11)

Then by plugging (2.11) back into (2.4), we find a set of coupled algebraic
equations for the set of coefficients $\phi_{n}$. For instance, the first four
of these equations can be formally written as

$\displaystyle 0$ $\displaystyle=$ $\displaystyle M_{11}\phi_{0}+4(\pi
T)\,(i{\mathfrak{w}}-1)\phi_{1}\,,$ (2.12) $\displaystyle 0$ $\displaystyle=$
$\displaystyle M_{21}\phi_{0}+M_{22}\phi_{1}+8(\pi
T)^{2}\,(i{\mathfrak{w}}-2)\phi_{2}\,,$ (2.13) $\displaystyle 0$
$\displaystyle=$ $\displaystyle
M_{31}\phi_{0}+M_{32}\phi_{1}+M_{33}\phi_{2}+12(\pi
T)^{3}\,(i{\mathfrak{w}}-3)\phi_{3}\,,$ (2.14) $\displaystyle 0$
$\displaystyle=$ $\displaystyle
M_{41}\phi_{0}+M_{42}\phi_{1}+M_{43}\phi_{2}+M_{44}\phi_{3}+\,16(\pi
T)^{4}\,(i{\mathfrak{w}}-4)\phi_{4}\,,$ (2.15)

where the coefficients, $M_{rs}$, are in fact functions of ${\mathfrak{w}}$
and ${\mathfrak{q}}^{2}$. The resulting equations are special in the sense
that at $\ell^{th}$ order of the near-horizonn expansion, one finds a linear
equation relating $\phi_{0}$, $\phi_{1}$, $\cdots$ and $\phi_{\ell+1}$, with
the coefficient of $\phi_{\ell+1}$ vanishing at the pole-skipping frequency
${\mathfrak{w}}=-i(\ell+1)$. Here $T$ is the Hawking temperature of the black
brane solution.

As obvious from the above equations, just at the frequency
${\mathfrak{w}}_{\ell}=-i\ell$, the first $\ell$ equations decouple from the
rest of them and take the following form

$0=\,\mathcal{M}_{\ell\times\ell}({\mathfrak{w}}=-i\ell,{\mathfrak{q}}^{2}_{*})\begin{pmatrix}\phi_{0}\\\
\phi_{1}\\\ .\\\ .\\\ \phi_{\ell-1}\\\ \end{pmatrix}.$ (2.16)

The roots of the equation
$\det\mathcal{M}_{\ell\times\ell}({\mathfrak{w}}=-i\ell,{\mathfrak{q}}^{2}_{*})=0$,
are those wavenumbers at which, for a given UV normalization constant, the
ingoing boundary condition at the horizon is not sufficient to uniquely fix a
solution for $\Phi$ in the bulk. Let us call the roots
${\mathfrak{q}}_{1}^{*},{\mathfrak{q}}_{2}^{*},\cdots,{\mathfrak{q}}_{2\ell}^{*}$.
Let us also recall that $\Phi$ is dual to the boundary operator $\mathcal{O}$.
It is clear that at these $2\ell$ points, the response function of the
boundary operator, namely $G^{R}_{\mathcal{O}\mathcal{O}}(x_{1}-x_{2})$, is
multi-valued. These points are the so-called pole-skipping points
Blake:2017ris ; Blake:2018leo .

One concludes that to every pole-skipping frequency
${\mathfrak{w}}_{\ell}=-i\ell$, $2\ell$ pole-skipping points of the dual
boundary operator correspond:

${\mathfrak{w}}=\,-i\ell:\,\,\,\,\,{\mathfrak{q}}*=\\{\pm{\mathfrak{q}}^{*}_{1,\ell},\pm{\mathfrak{q}}^{*}_{2,\ell},\cdots,\pm{\mathfrak{q}}^{*}_{\ell,\ell}\\}$
(2.17)

For later requirements, it is useful to express the pole-skipping points in
terms of $\Delta$, the conformal dimension of the boundary operator
$\mathcal{O}$. For the pole-skipping points at $\ell=1,2$, by using (2.5), we
find:

$\begin{split}{\mathfrak{w}}=-i:\,\,\,\,\,\,\,{\mathfrak{q}}^{*}_{1,1}=&\frac{1}{2}\,i\,\sqrt{\Delta^{2}-4\Delta+6}\,,\\\
{\mathfrak{w}}=-2i:\,\,\,\,\,\,\,{\mathfrak{q}}^{*}_{j,2}=&\frac{1}{2}\,i\,\sqrt{\Delta^{2}-4\Delta+12+\,(-1)^{j}2\sqrt{2}\sqrt{\Delta^{2}-4\Delta+6}},\,\,\,\,j=1,2\,.\end{split}$
(2.18)

The pole-skipping points at higher frequencies can be similarly computed
through the near horizon analysis (see appendix A), however, their analytic
expressions are lengthy and we do not explicitly write them down here. In the
rest of this section we will discuss the pole-skipping points and specifically
how their behavior changes with $\Delta$.

Figure 1: Left panel: Massive scalar dispersion relations for the lowest four
QNMs evaluated at purely imaginary momenta, i.e. $\text{Re}\mathfrak{q}=0$.
The stars lying on the vertical axis denote the imaginary part of the QNMs at
${\mathfrak{q}}=0$. Colorful dots show the pole-skipping points at frequencies
$\text{Im}\,{\mathfrak{w}}=-1,-2,-3,-4$. Right panel: spectrum of the lowest
four QNMs at ${\mathfrak{q}}=0$. These poles are exactly the poles indicated
by stars in the left panel. We have checked that this same figure arises for
all values of $\Delta$ as well as for massive vector and tensor fields which
will be discussed in section 5.

Let us begin by pointing out a very interesting property of pole-skipping
points which is common for all values of $\Delta$. To that end, we go back and
revisit figure 1. The colorful dots in the left plot are the pole-skipping
points associated with the response function of a boundary operator with
$\Delta=6$. It is observed that all pole-skipping points lie on the dispersion
relations of QNMs at imaginary momenta. Although this result has not been well
known for a theory without gapless excitations, it is in agreement with
previous studies Blake:2019otz ; Grozdanov:2019uhi .

However, the non-trivial feature of our result is revealed when splitting the
pole-skipping points lying on different dispersion relations. This behavior
has not been shown for gapped modes before. As it is obvious in the figure, we
indicate which pole skipping points belong to which QNM dispersion relation by
showing them in the same color. Each color indicates a different QNM
dispersion relation and the corresponding set of pole-skipping points:

$\begin{split}\text{Red
dots:}&\,\,\,(\pm{\mathfrak{q}}^{*}_{1,1},-i),(\pm{\mathfrak{q}}^{*}_{1,2},-2i),(\pm{\mathfrak{q}}^{*}_{1,3},-3i),(\pm{\mathfrak{q}}^{*}_{1,4},-4i),\cdots\,,\\\
\text{Green
dots:}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\pm{\mathfrak{q}}^{*}_{2,2},-2i),(\pm{\mathfrak{q}}^{*}_{2,3},-3i),(\pm{\mathfrak{q}}^{*}_{2,4},-4i),\cdots\,,\\\
\text{Blue
dots:}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\pm{\mathfrak{q}}^{*}_{3,3},-3i),(\pm{\mathfrak{q}}^{*}_{3,4},-4i),\cdots\,,\\\
\text{Purple
dots:}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\pm{\mathfrak{q}}^{*}_{4,4},-4i),\cdots\,.\end{split}$
(2.19)

The location of the red dots can be described by a “one-to-one” map between
the red dots and discrete frequencies $\omega=-m\,i(2\pi T)$ with the pole-
skipping levels labeled by $m=1,2,\cdots$. This means that the lowest QNM with
mode number labeled by $n=1$, namely the one with the dispersion relation
shown in red, is constrained to coincide with the pole-skipping frequencies
for all levels $m\geq 1$ with $m\in\mathbb{N}$. But the dispersion relation of
the second level QNM meets one less constraint; the Green dispersion relation
is constrained by the pole-skipping points with $m\geq 2$. By every step going
one level higher in the tower of QNMs, the number of pole-skipping frequencies
constraining the dispersion relations decreases by one.

Let us emphasize that figure 1 does not show the $\text{Re}\,{\mathfrak{w}}$.
That just shows $\text{Im}\,{\mathfrak{w}}$ at purely imaginary momenta:
${\mathfrak{q}}^{2}=e^{i\theta}|{\mathfrak{q}}^{2}|=\,-|{\mathfrak{q}}^{2}|$.
In other words all points in the figure 1 correspond to $\theta=\pi$. In order
to show why the classification of curves by the colors used in the figure 1 is
correct, we illustrate the position of poles (by poles we mean the stars shown
in the right panel) in the complex ${\mathfrak{w}}$ plane at $\theta=\pi$.

The result is given in figure 2. Each pole is identified with a dot at
$|{\mathfrak{q}}|=1$. Now we start to vary $|{\mathfrak{q}}|$ in the interval
$(1,2)$. When the imaginary ${\mathfrak{q}}$ increases, namely when
$|{\mathfrak{q}}|$ increases, with fixed $\theta=\pi\sout{=\text{constant}}$,
each pole moves along a trajectory and arrives at a circle. It is obvious from
the plot that in the range $|{\mathfrak{q}}|\in(1,2)$ the modes $n=1$ and
$n=2$ do not collide at $\theta=\pi$. The horizontal dashed line in figure 2
shows that at $\text{Im}\,{\mathfrak{q}}=1.50$ (the middle point of the
interval), the green and red stars have the same
$\text{Im}\,{\mathfrak{w}}=-3.49$ (at the locations indicated by the black
crosses). This is exactly what is observed in figure 1. But in figure 2 we see
that the real part, $\text{Re}\,{\mathfrak{w}}$, for the red and green stars
is not the same. This simply reveals that the crossing between the red and
green curves in figure 1 is not a mode collision point.

More importantly, the above discussion makes it clear how we have chosen the
colors in the figure 1. They are selected such that one color, e.g. Red,
indicates the trajectory of one QNM at a particular level, e.g. the red one is
the lowest-lying QNM.

Figure 2: This figure shows the same QNMs as figure 1, however, now in the
complex frequency plane. Along the trajectories, $|\mathfrak{q}^{2}|$ changes
from 1 (dots) to 2 (circles). This range is chosen in order to resolve the
apparent collision between the red and green pole in figure 1 and the apparent
branch point of the red curve in that same figure, both occuring near
$\text{Im}\,{\mathfrak{w}}\approx-3.5$. All modes are shown for the fixed
phase $\theta=\pi$.

This counting also works for the constraints on hydrodynamic modes in a theory
with macroscopic conserved currents. By definition, hydrodynamic modes are
constrained to pass through $({\mathfrak{q}}=0,{\mathfrak{w}}=0)$. For a non-
hydrodynamic QNM, however, ${\mathfrak{w}}({\mathfrak{q}}\rightarrow 0)$ is a
complex frequency with negative imaginary part. In this sense, the
hydrodynamic modes are restricted by all $m\geq 0$, as shown for the diffusion
and shear mode in Blake:2019otz . According to this analysis, non-hydrodynamic
QNMs are not as severely constrained as the hydrodynamic modes at real
${\mathfrak{q}}$. On the other hand, according to the absolute value of
${\mathfrak{w}}({\mathfrak{q}}\rightarrow 0)$, they are classified into
different levels in a tower spectrum, including the lowest-lying level, the
first excited level and so on. Then the behavior observed in figure 1 suggests
that there is a relation between this classification and the constraints on
their dispersion relations at imaginary ${\mathfrak{q}}$’s:
The more pole-skipping points lie on the dispersion relation of a gapped QNM,
the more constrained that mode is at imaginary ${\mathfrak{q}}$’s. Then our
observation implies that the longest-lived mode in the system is the one which
is the most constrained. The latter is actually the lowest-lying QNM. To
summarize, our results indicate that the hierarchy in the life-time of QNMs
originates from the number of constraints induced by the pole-skipping points
at imaginary momenta.

At this point, we emphasize that the results obtained from figure 1 all
consider purely imaginary momenta. It should also be noted that the dispersion
relations for QNMs at imaginary or complex-valued momentum do not directly
tell us anything about the dispersion relations of the same QNMs at real-
valued momentum. Hence we can not derive a rigorous statement for the relation
between the hierarchy of QNM life-times (at real-valued momenta) and the
number of pole-skipping points (at imaginary/complex-valued momentum). Hence
we stress that our statement above is an observation.

##  3 Derivative expansion for gapped modes

There is yet another interesting feature associated with the dispersion
relations shown in figure 1 which deserves attention. Critical points have
been defined as those points at which two QNMs “collide” with each other
Grozdanov:2019kge ; Grozdanov:2019uhi . This means that the real and imaginary
parts of their frequencies agree with each other at a particular value of a
complex momentum. This is illustrated in the upper panel of figure 5, 6 and 7,
which we will discuss further below. Looking at the dispersion relation of
each QNMs shown in figure 1, one type of potential critical points can
immediately be recognized; the points where the dispersion relation of the QNM
splits into two branches. Such points correspond to the collision of two modes
with the same $\text{Im}\,{\mathfrak{q}}$. However, they might be some other
critical points that cannot be specified in Im ${\mathfrak{w}}$ \- Im
${\mathfrak{q}}$ plane.

It should be noted that at a critical point, $\text{Re}\,\mathfrak{w}$,
$\text{Re}\,\mathfrak{q}$, $\text{Im}\mathfrak{w}$ and $\text{Im}\mathfrak{q}$
of the two colliding QNMs have to be identical. We discuss this point further
below.

Very recently, in the context of hydrodynamics, it has been shown that the
critical points of the spectral curves defining a given hydrodynamic QNM may
determine the convergence radius of the derivative expansion of that QNM.
There, the spectral curve is defined to be an implicit function relating the
frequency and momentum, i.e. defining the dispersion relation of the mode.
This definition of critical points utilizing the spectral curve yields the
same values as the definition through the collision of QNMs mentioned above
Grozdanov:2019kge . In other words, QNMs collide at critical points of the
spectral curve.

The scalar field fluctuation we choose to consider now has no hydrodynamic
mode. In other words, the corresponding spectral curve does not include any
dispersion relation passing through the origin. Instead, in this section we
intend to study the analytic structure of the spectral curve with the aim of
establishing a relation between critical points and the gapped QNMs displayed
in figure 1. Eventually, this will lead us to the convergence radii of
derivative expansions for gapped modes.

The spectral curve of a given operator $\mathcal{O}$ can be written in Fourier
space as

$\mathcal{S}(\mathfrak{w},\mathfrak{q}^{2})=\,0$ (3.20)

where both $\mathfrak{w}$ and $\mathfrak{q}^{2}$ are complex numbers. The
solution to this (implicit function) equation is a discrete set of QNMs
${\mathfrak{w}}^{(n)}\equiv{\mathfrak{w}}^{(n)}({\mathfrak{q}}^{2})$. In our
present case:

$\lim_{{\mathfrak{q}}^{2}\rightarrow
0}{\mathfrak{w}}^{(n)}({\mathfrak{q}}^{2})={\mathfrak{w}}^{(n)}_{\text{g}}\,,$
(3.21)

where ${\mathfrak{w}}^{(n)}_{\text{g}}$ is complex with
$\text{Im}\,\mathfrak{w}^{(n)}_{\text{g}}<0$ corresponding to the (generally
complex-valued) gap of the $n^{th}$ mode. Among the QNMs, the one with
smallest $|\text{Im}\,{\mathfrak{w}}_{g}|$ is the longest lived mode in the
system. Obtaining any information about this mode may reveal how fast the
operator is thermalized. In the previous section, we have shown results in
figures 1 and 2 which display information obtained numerically by solving eq.
(3.20) through the standard method of computing QNMs by requiring ingoing
boundary conditions at the horizon and Dirichlet boundary condition at the
conformal boundary. In this section, we want to extract information from the
spectral curve by studying its analyticity properties. To this end, we search
perturbative solutions to (3.20) in the long wavelength limit. Formally, one
writes the solution in a derivative expansion in momentum space444Let us
emphasize that this is not a hydrodynamic derivative expansion; equation
(3.22) does not pass through the origin. This is a derivative expansion about
the $n^{th}$ gapped QNM. In this sense, it differs from what has been
discussed in Withers:2018srf  and Grozdanov:2019kge .

$\mathfrak{w}^{(n)}=\mathfrak{w}_{\text{g}}^{(n)}-i\,\sum_{n=1}^{\infty}a_{k}^{(n)}\mathfrak{q}^{2k}\,.$
(3.22)

The set of coefficients $a_{k}^{(n)}$ corresponds to the branch of the Puiseux
series Grozdanov:2019kge ; Grozdanov:2019uhi  passing through the point
$(\mathfrak{w}_{\text{g}}^{(n)},0)$ 555In holography, we do not know the
structure of $\mathcal{S}(\mathfrak{w},\mathfrak{q}^{2})$ in general. However
in well-known examples, it does not feature any non-analyticity, such as that
of RTA kinetic theory Heller:2020hnq . Thus, according to the implicit
function theorem, one may expect the behavior of
$\mathcal{S}(\mathfrak{w},\mathfrak{q}^{2})$ to be described by a Puiseux
series with a non-zero radius of convergence.. In general, one can
perturbatively find the coefficients $a_{k}^{(n)}$ for each of the gapped
modes in the system. But as we will explain in detail, we limit our study to
the two cases $n=1,\,2$.

Figure 3: The radius of convergence, $|\mathfrak{q}_{c}|$, of the derivative
expansion versus the scaling dimension of the boundary operator, within the
range $2\leq\Delta\leq 8$. The transition between the regime of level-crossing
and the regime of lowest-level-degeneracy occurs at $\Delta_{\text{t}}\approx
5.23$.

More precisely, we would like to investigate to what extent the derivative
expansions associated with ${\mathfrak{w}}^{(1)}$ and ${\mathfrak{w}}^{(2)}$
are convergent. Thus we follow Grozdanov:2019kge  and compute the
corresponding critical points of the spectral curve. For this purpose one
needs to find the set of roots
$\\{\text{Re}\,\mathfrak{w},\,\text{Im}\,\mathfrak{w},\,\text{Re}\,\mathfrak{q}^{2},\,\text{Im}\,\mathfrak{q}^{2}\\}$
satisfying the following two complex-valued equations666It is worth mentioning
that, in general, the radius of convergence about
$\mathfrak{w}_{\text{g}}^{(n)}$ is determined by the singularity closest to it
Heller:2020hnq . In this sense, eqs. (3.23) may admit solutions which do not
necessarily correspond to singularities of
${\mathfrak{w}}^{(n)}({\mathfrak{q}})$ Heller:2020hnq  (see section 6 as an
example illustrating this point). However, in our present case, solving these
equations gives the correct radius of convergence.

$\mathcal{S}(\mathfrak{q}^{2},\mathfrak{w})=\,0,\,\,\,\,\,\,\,\,\,\,\,\,\frac{\partial\mathcal{S}(\mathfrak{q}^{2},\mathfrak{w})}{\partial\mathfrak{w}}=\,0.$
(3.23)

Starting with $\mathfrak{w}_{\text{g}}^{(1)}$, we look for the critical point
obtained from (3.23) which has the smallest distance to
$(0,\mathfrak{w}_{\text{g}}^{(1)})$. Let us call this critical point
$({\mathfrak{q}}_{c}^{(1)},{\mathfrak{w}}_{c}^{(1)})$. Then
$|{\mathfrak{q}}_{c}^{(1)}|$ sets the radius of convergence of the derivative
expansion about $\mathfrak{w}_{\text{g}}^{(1)}$. We can repeat the computation
by starting with $\mathfrak{w}_{\text{g}}^{(2)}$ in order to find
$|{\mathfrak{q}}_{c}^{(2)}|$, the radius of convergence of the derivative
expansion about the second quasinormal mode, $\mathfrak{w}_{\text{g}}^{(2)}$.

We have found $|{\mathfrak{q}}_{c}^{(1)}|$ and $|{\mathfrak{q}}_{c}^{(2)}|$
for operators of weight $2\leq\Delta\leq 8$. The result is shown in figure 3.
As it can be seen in the figure, it turns out that
$|{\mathfrak{q}}_{c}^{(2)}|$ is approximately a linear function of $\Delta$.
In contrast to that, the curve showing $|{\mathfrak{q}}_{c}^{(1)}|$ as a
function of $\Delta$ has a kink at a particular operator dimension
$\Delta=\Delta_{\text{t}}$:

$\Delta_{\text{t}}\,\approx\,5.23\,.$ (3.24)

When $\Delta<\Delta_{\text{t}}$, we find
$|{\mathfrak{q}}_{c}^{(1)}|<|{\mathfrak{q}}_{c}^{(2)}|$, while for
$\Delta>\Delta_{\text{t}}$, the two derivative expansions about the lowest QNM
($n=1$) or the first excited QNM ($n=2$) have the same radius of convergence:
$|{\mathfrak{q}}_{c}^{(1)}|=|{\mathfrak{q}}_{c}^{(2)}|$.

The transition between the two regimes discussed above can also be visualized
in a different way. Since in our model
${\mathfrak{w}}\equiv{\mathfrak{w}}({\mathfrak{q}}^{2})$, we can display the
critical points in the complex ${\mathfrak{q}}^{2}$-plane. We have done so for
various values of $\Delta$ in figure 4. For every $\Delta<\Delta_{\text{t}}$,
the convergence of the derivative expansion about the lowest QNM is identified
with only one collision point which lies on the horizontal axis, i.e. at
$\text{Im}\,\mathfrak{q}^{2}=0$. Such points are indicated by dots in the
figure and occur at purely imaginary momenta. The distance between each dot
and the origin sets the radius of convergence. Within the same range of
$\Delta$, to each value of $\Delta$ two crosses correspond, too. The pair of
crosses are located symmetrically with respect to the
$(\text{Im}\,{\mathfrak{q}}^{2})$-axis and set the domain of convergence of
the derivative expansion about the second QNMs. The latter is actually set by
the radius of the circle centered at the origin passing through the two
crosses.

Figure 4: The location of ${\mathfrak{q}}^{2}_{c}$ in the complex plane for
various values of $\Delta$ within the range $2\leq\Delta\leq 8$. Every dot
(cross), corresponding to a particular value of $\Delta$, lies on a circle
whose center is at the origin; the region enclosed by the circle is then the
domain of convergence of the derivative expansion about the first (second) QNM
associated with that value of $\Delta$. The dashed circle shows a transition
between two regimes: for values of $\Delta<\Delta_{\text{t}}$, the dot does
not lie on the circle passing through the crosses. In addition, dots are
located in the $(\text{Im}\,{\mathfrak{q}}^{2})$-axis. But for
$\Delta>\Delta_{\text{t}}$, dots and crosses lie on the same circle.

At $\Delta=\Delta_{\text{t}}$, the dot and the two crosses are all located at
the same distance to the origin. One is to conclude that at this special value
of $\Delta$, the radii of convergence about the lowest-lying and the second
QNM are the same but are determined by distinct points in the complex
${\mathfrak{q}}^{2}$-plane. For $\Delta>\Delta_{\text{t}}$, however, in each
side of the $(\text{Im}\,{\mathfrak{q}}^{2})$-axis, there is a dot and a cross
coinciding with each other. It simply shows that beyond the transition value
of $\Delta$, the radius of convergence of the derivative expansion about the
lowest-lying and the second QNMs is set by the same value and is identified
with the same critical point. See appendix B for a detailed discussion, and
figures 10 and 11 for illustrations of the collision points on the complex-
valued dispersion relations.

Figure 5:  Lowest-level-degeneracy. Poles of the retarded two-point function
of the scalar operator, in the complex ${\mathfrak{w}}-$plane, at various
values of the complexified momentum
${\mathfrak{q}}^{2}=|{\mathfrak{q}}^{2}|e^{i\theta}$. Blue dots correspond to
the location of poles for real ${\mathfrak{q}}^{2}$, namely for $\theta=0$. As
$\theta$ increases from $0$ to $2\pi$, each pole moves counter-clockwise
following the trajectory whose color changes continuously from blue to red.
The two top panels show the situation slightly before and after the collision
point marked by black dots. At the critical value
$|{\mathfrak{q}}_{c}^{2}|\approx 1.25$, the trajectories of the lowest-lying
modes collide. After the collision, the orbits of these modes are no longer
closed: the two of them exchange their positions as the phase $\theta$
increases from $0$ to $2\pi$. The second collision occurs at a higher
momentum. The two bottom panels show the situation slightly before and after
the second collision, marked by black crosses. At the critical value
$|{\mathfrak{q}}_{c}^{2}|\approx 1.54$, the trajectories of the second QNMs
collide with the common trajectory of the lowest-lying modes. After the
collision, the two lowest-lying modes together with the second QNMs exchange
their positions cyclically as the phase $\theta$ increase from $0$ to $2\pi$.

The existence of a transition value for $\Delta$ has an important outcome for
the complex life of the lowest-lying QNMs. Indeed, it distinguishes two
distinct types of scattering between complexified QNMs depending on whether
$\Delta<\Delta_{\text{t}}$ or $\Delta>\Delta_{\text{t}}$. Each type of
scattering point determines the critical point
$({\mathfrak{q}}_{c}^{(1)},{\mathfrak{w}}_{c}^{(1)})$ within the respective
regime of $\Delta$. To clarify this point, we demonstrate the collision points
for two values of $\Delta$ in figures 5, 7 and 6.

Figure 6:  Level-crossing.  Poles of the retarded two-point function of the
scalar operator, in the complex ${\mathfrak{w}}-$plane, at various values of
the complexified momentum
${\mathfrak{q}}^{2}=|{\mathfrak{q}}^{2}|e^{i\theta}$. Blue dots correspond to
the location of poles for real ${\mathfrak{q}}^{2}$, namely for $\theta=0$. As
$\theta$ increases from $0$ to $2\pi$, each pole moves counter-clockwise
following the trajectory whose color changes continuously from blue to red.
The two panels show the situation slightly before and after the collision
point marked by brown dot-crosses. Before the collision the orbits of all QNMs
are closed. At the critical value $|{\mathfrak{q}}_{c}^{2}|\approx 2.18$, the
trajectories of the lowest-lying modes collide with those of the second QNMs.
After the collision, orbits of mentioned modes are no longer closed: in both
left and right sides of the plane, the lowest-lying and the second QNMs
exchange their positions as the phase $\theta$ increases from $0$ to $2\pi$.

Let us first consider figure 5 which corresponds to
$\Delta=4<\Delta_{\text{t}}$. As was discussed earlier, in this range, radius
of convergence for the lowest-lying modes differs from that of the second
QNMs. For this reason, in the figure, we have separated the two cases. In the
two top panels we show the positions of complexified modes slightly before
(left plot) and after (right plot) the first collision. The collision occurs
at $|{\mathfrak{q}}^{2}_{c}|\approx 1.25$ setting the radius of convergence
about the lowest-lying QNM to be
$|{\mathfrak{q}}_{c}^{(1)}|\approx(1.25)^{1/2}$. 777This value agrees with the
result given in the tensor sector of metric fluctuations given in
Grozdanov:2019uhi , associated with the energy-momentum tensor which has
$\Delta=4$.  This collision point is actually the black dot already introduced
in figure 4 and as one expects, corresponds to the point with $\Delta=4$ on
the left red branch of figure 3 as well. Since at the collision point the two
lowest-lying modes coincide with each other, we refer to it as the lowest-
level-degeneracy.

By increasing $|{\mathfrak{q}}^{2}|$, the second QNMs (with mode number $n=2$)
collides with the first QNM. In the two bottom panels of figure 4, we have
shown the situation slightly before and after such a collision. The collision
occurs at $|{\mathfrak{q}}^{2}_{c}|\approx 1.54$ at points marked by two black
crosses in the figure. So $|{\mathfrak{q}}_{c}^{(2)}|\approx(1.54)^{1/2}$ sets
the radius of convergence for the derivative expansions about the second QNMs.
The collision points are actually the black crosses already introduced in
figure 4 and as one expects, correspond to the point with $\Delta=4$ on the
green branch of figure 3.

As an example for $\Delta>\Delta_{\text{t}}$, we show the case $\Delta=6$ in
figure 6. As was discussed earlier, in this range, the radius of convergence
for the lowest-lying modes is equal to that of the second QNMs. So one expects
one single type of collision between the trajectories of complexified modes to
solely determine the radius of convergence. In figure 6, we indicate the
positions of complexified modes slightly before and after such a collision,
marked by brown dot-crosses. The collision occurs at
$|{\mathfrak{q}}^{2}_{c}|\approx 2.18$ setting the radius of convergence about
all four of the lowest QNMs to be
$|{\mathfrak{q}}_{c}^{(1)}|=|{\mathfrak{q}}_{c}^{(2)}|\approx(2.18)^{1/2}$.
The two collision points are actually the brown dot-crosses already introduced
in figure 4 and as one expects, correspond to the point with $\Delta=6$ on the
plot of figure 3. Since at each collision point one of the lowest-lying QNMs
collides with the respective second QNM, we refer to this collision as level-
crossing Grozdanov:2019kge .

Figure 7:  Simultaneous level-crossing and lowest-level-degeneracy at
transition.  This figure shows the same modes as figures 5 and 6 but now at
the intermediate value of the conformal operator dimension $\Delta=5.23$. That
is approximately the transition dimension $\Delta_{\text{t}}$ which separates
the two types of collision regimes, level-crossing versus lowest-level-
degeneracy, from each other. It is apparent that now both types of collisions
appear simultaneously. Before the collision, in the left panel, the two
lowest-lying QNMs ($n=1$) have two separate trajectories, and the two second
QNMs ($n=2$) have two separate trajectories. These four trajectories then
collide with each other simultaneously at the three critical points indicated
by the magenta dot and the magenta crosses. These collision points are also
displayed in magenta in the complex momentum plane in figure 4.

Finally, at the transition point $\Delta_{\text{t}}$, shown in figure 7,at
$|{\mathfrak{q}}_{c}^{2}|$, three collisions occur. So a level-crossing occurs
simultaneously with a lowest-level-degeneracy. By “simultaneously occur”, we
mean that they correspond to the same $|{\mathfrak{q}}_{c}^{2}|$, although not
at a common $\theta$. From the field theory point of view, this constellation
of poles may be viewed as the defining feature, fixing the value of
$\Delta_{\text{t}}=5.23$. An analytic understanding and physical
interpretation of this value remains to be understood. This same constellation
of poles at $\Delta_{\text{t}}$ is illustrated in the complex momentum plane
in figure 4. The magenta dot and crosses there indicate the same poles as the
magenta dot and crosses in figure 7.

In summary, we find that the transition first observed in figure 3 is
translated to a transition from the lowest-level-degeneracy to the level-
crossing behavior of the QNMs at complex momenta. As shown by the magenta dots
and crosses in figure 4, just at the transition value of $\Delta$, the lowest-
level-degeneracy and the level-crossing occur simultaneously. Thus they
jointly determine the radius of convergence about the lowest-lying QNM.

##  4 Bound on the radius of convergence from pole-skipping

In section 2 we saw how the pole-skipping points constrain the dispersion
relation of QNMs at imaginary momenta. In this section we will show that the
pole-skipping points also constrain the radius of convergence of the
derivative expansion about the lowest-lying QNMs.

In figure 8 we have compared the above mentioned convergence radius, namely
$|{\mathfrak{q}}_{c}^{(1)}|$, with the absolute value of the momenta
associated with the pole-skipping points at frequencies:
${\mathfrak{w}}=-i,-2i,-3i$. More precisely, at each of these frequencies, we
consider the pole-skipping point with the smallest absolute value of its
momentum. In the language of (2.17), the red, green, blue, purple and cyan
curves in figure 8 correspond to $|{\mathfrak{q}}^{*}_{1,1}|$,
$|{\mathfrak{q}}^{*}_{2,1}|$, $|{\mathfrak{q}}^{*}_{3,1}|$,
$|{\mathfrak{q}}^{*}_{4,1}|$ and $|{\mathfrak{q}}^{*}_{5,1}|$, respectively.

We observe that the black curve in figure 8, indicating the radius of
convergence, $|{\mathfrak{q}}_{c}^{(1)}|$, is actually the envelope of the
pole-skipping curves. In other words, the radius of convergence of the
derivative expansion about the lowest-lying mode is bounded from above. The
upper-bound is given by the distance between the origin and that pole-skipping
point which is closest to the origin. One may write this as

$|{\mathfrak{q}}_{c}^{(1)}|\leq\text{min}\\{|{\mathfrak{q}}^{*}_{1,1}|,|{\mathfrak{q}}^{*}_{2,1}|,|{\mathfrak{q}}^{*}_{3,1}|,\cdots\\}\,.$
(4.25)

Figure 8: Comparison between the radius of convergence of the derivative
expansion about the lowest-lying gapped QNM, determined by
$|{\mathfrak{q}}_{c}^{(1)}|$, and the smallest absolute value of the momentum
at pole-skipping points. The red, green and blue curves can be continued to
larger values of $\Delta$, but here we focus on the lower values of
$|{\mathfrak{q}}|$. Higher QNMs with $n=4,\,5$ have been considered and do not
intersect or coincide with the black line of
$|{\mathfrak{q}}_{c}^{(1)}|$-values in the range $2\leq\Delta\leq 8$.

At specific values of $\Delta$ this bound is saturated. We find three such
saturation points, namely those at which the red, green and blue curves
coincide with the black curve in figure 8. One of the saturation values,
interestingly, is the transition value $\Delta_{\text{t}}$ that we already
discussed in section 3. Recall that the black curve has a kink at exactly this
transition value $\Delta_{\text{t}}$. One might guess that the transition from
lowest-level-degeneracy to the level-crossing behavior of QNMs is the
consequence of the fact that $|{\mathfrak{q}}_{c}^{(1)}|$ cannot exceed the
bound (4.25). One may further infer that the kink in the black curve is also
related to this bound. Analytic solutions would help to reveal these potential
relations.

##  5 Vector and tensor operators

Our discussion so far has been limited to a scalar gapped operator. Similar
arguments can be made for gapped operators of higher spins. In this section we
do so in the case of vector and tensor operators in the boundary theory.

Let us start with studying a vector operator $\mathcal{V}$; this operator is
dual to a massive vector field $A^{\mu}$ in the bulk. The corresponding
equation of motion is Aharony:1999ti ; Kabat:2012hp

$\nabla_{\mu}F^{\mu}_{\nu}-m^{2}A_{\nu}=\,0\,.$ (5.26)

We consider transverse perturbations of this field around the AdS5 black brane
solution (2.3):

$A=A_{\mu}\,dx^{\mu}=\int d^{4}k\,A_{x}(u;\omega,k)\,e^{-i\omega
t+ikz}\,dx\,.$ (5.27)

Defining $E_{x}=-i{\mathfrak{w}}A_{x}$, we arrive at

$E_{x}^{\prime\prime}+\frac{f^{\prime}}{f}\,E_{x}^{\prime}+\frac{4u({\mathfrak{w}}^{2}-{\mathfrak{q}}^{2}f)-m^{2}f}{4u^{2}f^{2}}\,E_{x}=\,0\,.$
(5.28)

Then the spectrum of QNMs associated with the field $E_{x}$ in the bulk
corresponds to non-hydrodynamic gapped modes of the operator $\mathcal{V}$,
excited by $E_{x}$ on the boundary. The conformal weight of $\mathcal{V}$ is
given in terms of $m$, namely the mass of $A^{\mu}$ in AdSd+1, as follows

$\Delta=\frac{d}{2}+\sqrt{\frac{(d-2)^{2}}{4}+m^{2}}\,\,\,\,\,\,\xrightarrow[]{d=4}\,\,\,\,\,\Delta=2+\sqrt{1+m^{2}}\,.$
(5.29)

Let us recall the unitarity bound Minwalla:1997ka  on operators with the spin
$\ell$,

$\Delta\geq d+\ell-2\,.$ (5.30)

So vector operators ($\ell=1$) in $d=(3+1)$ dimensions are bounded by
$\Delta\geq 4+1-2=3$.

It turns out that the spectral curve of $E_{x}$ has similar properties to that
of the scalar field in the bulk. We have checked that the dispersion relations
of its QNMs at imaginary momenta behave qualitatively just like those shown in
figure 1. Thus the statement that QNMs are constrained by pole-skipping points
continues to hold for $\mathcal{V}$ in perfect analogy to the scalar operator
case. The transition between the lowest-level-degeneracy and the level-
crossing (see figure 3) is observed in this case as well. We find the
transition value of $\Delta$ for vector operators to be

$\Delta_{\text{t}}\approx 4.35\,.$ (5.31)

In order to compare the mentioned transition with that of the scalar operator,
in figure 9, we have considered operators of dimension
$1/2<\Delta/\Delta_{\text{t}}<3/2$ with integer spins. The black curve in
figure 8 is identical to the grey curve shown in figure 9.

For this vector operator we have confirmed that the pole-skipping points and
the radius of convergence satisfy the same relations displayed for the scalar
operator in figure 8. The figure for the vector is qualitatively the same and
hence we do not show it here. However, the transition conformal dimension for
the vector operator, given in eq. (5.31), differs from that of the scalar
operator, given in eq. (3.24).

In figure 9, we have also shown the results concerning a gapped tensor
operator, $\mathcal{T}$, in the boundary theory. This operator is dual to the
transverse component of a massive spin two field in the bulk. Let us refer to
the massive spin two field as $\varphi_{\mu\nu}$. The corresponding equations
of motion are given by Benini:2010pr :

$\begin{split}0=&\,(\Box-m^{2})\varphi_{\mu\nu}+2R_{\mu\lambda\nu\rho}\varphi^{\lambda\rho}\,,\\\
0=&\,D^{\mu}\varphi_{\mu\nu}\,,\\\ 0=&\,\varphi^{\mu}_{\mu}\,.\end{split}$
(5.32)

The fluctuation of the $(xy)$-component decouples from the rest of the
fluctuations. This $\varphi_{xy}$ sources the fluctuations of $\mathcal{T}$ on
the boundary. In a $d$-dimensional boundary theory, the conformal dimension of
the operator $\mathcal{T}$ is given by

$\Delta=\frac{1}{2}(d+\sqrt{d^{2}+4m^{2}})\,\,\,\,\,\,\xrightarrow[]{d=4}\,\,\,\,\,\Delta=2+\sqrt{4+m^{2}}\,.$
(5.33)

Note that the unitarity bound (5.30) forces the traceless tensor operator
dimension to obey $\Delta\geq 4$. Taking

$\varphi_{xy}=\int d^{4}k\,\frac{\psi(u;\omega,k)}{u}\,e^{-i\omega t+ikz},$
(5.34)

it turns out that $\psi(u)\equiv\psi(u;\omega,k)$ obeys exactly the same
equation that a massive scalar fields does obey in the bulk, namely (2.6).
Hence, the horizon expansion and location of pole-skipping points for the
tensor sector are identical to those presented in the scalar sector above.

Figure 9: The radius of convergence of the derivative expansion about the
lowest-lying QNMs versus the scaling dimension of the boundary operator,
within the range $1/2\leq\Delta/\Delta_{\text{t}}\leq 3/2$. The black dot,
which corresponds to transverse perturbations of the metric in the bulk was
found in Grozdanov:2019uhi . The dashed parts of the magenta and cyan curves
show the values of $\Delta$ excluded by the unitarity bound for the vector and
tensor, respectively. The black curve for the scalar operator coincides over
the whole range with the cyan curve for the tensor operator. However, for the
scalar operator all values of $\Delta$ displayed here are allowed by the
scalar’s unitarity bound, $\Delta\geq 2$.

Thus, as expected, we find the QNMs of $\mathcal{T}$ to feature exactly the
same behavior as the scalar operator’s QNMs. This can be seen in figure 9. The
gray and cyan curves coincide with each other. The black dot shown on the cyan
curve corresponds to $\mathcal{T}$ with $\Delta=4$ (or equivalently with
$\Delta/\Delta_{\text{t}}=0.76$). An example for such an operator is the shear
component of the energy-momentum tensor, namely $T_{xy}$. For this $\Delta=4$
we find $|{\mathfrak{q}}_{c}|=1.11942$ which is in agreement with the critical
point momentum computed from the spin-2 tensor perturbations of the metric,
$h_{xy}$, provided in equation (4.38a) of Grozdanov:2019uhi  as well as with
figure 13 in that same paper.

Let us note another feature displayed in figure 9 which is related to the
unitarity bound (5.30). For vector and tensor operators the minimum allowed
value of $\Delta$ is identical to the $\Delta$ of the conserved vector and
tensor operators, namely the $U(1)$ current with $\Delta=3$ and the energy-
momentum tensor with $\Delta=4$, respectively. It is remarkable that the upper
bound (4.25) is obeyed by the vector operator for all values in agreement with
the unitarity bound $\Delta\geq 3$ which we have checked. However, if the
unitarity bound is violated, also the bound (4.25) is violated.

##  6 Comments on the BTZ case ($AdS_{3}$)

Our analysis can be simply extended to the $AdS_{d+1}$ case with $d>2$. In
three bulk dimensions, however, the situation is different. In this case, one
studies the QNMs around the BTZ black hole.

From a non-extremal BTZ black hole, the retarded Green’s function of an
operator with the conformal dimension $\Delta$ in the boundary CFT can be
computed analytically. For a non-integer $\Delta$ Son:2002sd :

$\begin{split}G^{R}({\mathfrak{w}},{\mathfrak{q}})=&\,C_{\Delta}\,\Gamma\left(\frac{\Delta}{2}+\frac{i({\mathfrak{w}}-{\mathfrak{q}})}{2}\right)\,\Gamma\left(\frac{\Delta}{2}-\frac{i({\mathfrak{w}}-{\mathfrak{q}})}{2}\right)\,\Gamma\left(\frac{\Delta}{2}+\frac{i({\mathfrak{w}}+{\mathfrak{q}})}{2}\right)\\\
&\times\Gamma\left(\frac{\Delta}{2}-\frac{i({\mathfrak{w}}+{\mathfrak{q}})}{2}\right)\bigg{[}\cosh(\pi{\mathfrak{q}})-\cos(\pi\Delta)\cosh(\pi{\mathfrak{w}})+i\sin(\pi\Delta)\sin(\pi{\mathfrak{w}})\bigg{]}\,,\end{split}$
(6.35)

where $C_{\Delta}$ is a normalization constant and we have assumed
$T_{R}=T_{L}=T$. The poles of the response function are given by

${\mathfrak{w}}_{n}({\mathfrak{q}})=\,\pm{\mathfrak{q}}-i(2n+\Delta),\,\,\,\,\,\,\,n=0,1,2,\,\dots\,.$
(6.36)

The pole-skipping points and critical points of the spectral curve associated
with the above response function have been found in Grozdanov:2019uhi . The
nearest critical points to the mode $n=0$ in (6.36) is found to be
Grozdanov:2019uhi :

${\mathfrak{q}}_{c}=\pm i,\,\,\,\,w_{c}=\,-i(1+\Delta).$ (6.37)

On the other hand the radius of convergence of
${\mathfrak{w}}({\mathfrak{q}})$ expanded around any value of ${\mathfrak{q}}$
is determined by the closest singularity of ${\mathfrak{w}}({\mathfrak{q}})$
to that point in the complex ${\mathfrak{q}}$-plane Heller:2020hnq . Since the
dispersion relation (6.36) is an entire function, it cannot be singular at the
critical points (6.37). More precisely, the dispersion relation (6.37) does
not have any singular point. Thus the radius of convergence of the derivative
expansion about the QNMs in the BTZ case is infinite.888We thank Alexandre
Serantes for pointing this out. In summary, we do not expect any relation
between pole-skipping and the radius of convergence in the BTZ case. In
contrast to BTZ, in $AdS_{d+1}$ with $d>2$, the dispersion relations have
square-root singularities leading to finite radius of convergence about the
QNMs, as discussed in last sections.

##  7 Discussion

In this paper, we have computed the pole-skipping points and critical points
of scalar (spin-0), vector (spin-1), and tensor (spin-2) perturbations with
different masses, see sections 2 and 3, respectively. First, this is
holographically dual to pole-skipping points in the correlation functions of
scalar, vector, and tensor operators with different operator dimensions.
Second, it yields the radius of convergence of a low-energy expansion in small
momenta of gapped, non-hydrodynamic modes. The latter expansion is analogous
to the momentum space expansion of gapless, hydrodynamic modes. Our analysis
yields four main results:

  1. 1.

First, the number of pole-skipping points lying on the dispersion relation of
the $n^{th}$ QNM decreases with the mode level $n$, as indicated in figure 1.
The pole-skipping points, i.e. the dots in figure 1, restrict the dispersion
relation of each QNM with a given color (red, green and purple, …) to run
through the set of pole-skipping points of the corresponding color. From
figure 1 it is also obvious that the $n^{th}$ QNM runs through the $n^{th}$
pole-skipping point closest to the imaginary momentum axis. That is the pole-
skipping point with $n^{th}$ smallest imaginary momentum,
$\text{Im}\,\mathfrak{q}$ at a given value of $\text{Im}\,{\mathfrak{w}}$. For
example, the lowest-lying mode with $n=1$ indicated by the red trajectory,
runs through those pole-skipping points which at each value of
$\text{Im}\,\mathfrak{q}$ are the ($1^{st}$) closest to the
$\text{Im}\,\mathfrak{q}$-axis in figure 1.

  2. 2.

Second, the pole-skipping points provide an upper bound on the values of the
critical points, see eq. (4.25). This bound implies that the magnitude of the
critical momentum, $|\mathfrak{q}_{c}^{(1)}|$, of the derivative expansion,
$\mathfrak{w}(\mathfrak{q})$, around the lowest gapped quasinormal frequency
is bounded from above by the values of the pole-skipping momenta, as
illustrated in figure 8. This establishes a relation between pole-skipping
points and critical points which could not be observed before at fixed values
of operator dimensions $\Delta=3,\,4$ Grozdanov:2019uhi .

  3. 3.

Our third result is that for particular values of the operator dimension,
$\Delta\approx 3.1$, $4.3$, and $\Delta=\Delta_{\text{t}}$, the convergence
radius is equal to the momentum of the pole-skipping point, as seen from
figure 8 999It should be emphasized that the analytic structure of the
spectral curve in holography is not known in general. From the known examples,
we think that there should not be any non-analyticity in the spectral curve of
holographic models. Following this logic and by use of the implicit function
theorem, in order to find the radius of convergence, we focused on critical
points of the spectral curve instead of explicitly finding the singular points
of dispersion relations. However, we saw that in the BTZ case this
simplification failed to work. This is due to the fact that in that case the
dispersion relations are exact and non-singular. Thus the main question which
remains to be answered is that of the structure of spectral curves in
holography Heller:2020hnq .. This indicates that at particular values of
$\Delta$ the critical points may be computed by the relatively simple
computation for pole-skipping points. This creates a possibility for finding
analytic solutions (as opposed to numerical results) for critical points.

  4. 4.

There exists a transition between two distinct types of critical points at a
particular conformal dimension $\Delta=\Delta_{\text{t}}$. This classification
results from the two distinct collision behaviors of the relevant QNMs
determining a given critical point. We refer to the first type of QNM behavior
at $\Delta<\Delta_{\text{t}}$ as lowest-level-degeneracy behavior, illustrated
in figure 5. The behavior of QNMs at $\Delta>\Delta_{\text{t}}$ is coined
level-crossing, as illustrated in figure 6. This behavior is further detailed
in appendix B.

All four results are valid for all operator dimensions we considered, i.e. in
the regimes of level-crossing as well as the regime of lowest-level-
degeneracy. These results are also valid for fields with distinct spins under
spatial rotations, i.e. for massive scalar, massive vector, and massive tensor
fluctuations, as discussed in section 5.

Figure 4 reveals two more features. Critical points above $\Delta_{\text{t}}$
have complex-valued squared critical momentum, $\mathfrak{q}_{c}^{2}$, while
for $\Delta<\Delta_{\text{t}}$ the squared critical momentum is purely
imaginary. Figure 4 clearly shows that at $\Delta_{\text{t}}$ a discontinuity
occurs: the critical points with purely imaginary momentum instantly jump to
critical points with nonzero momentum when $\Delta$ is increased
infinitesimally above $\Delta_{\text{t}}$. At $\Delta=\Delta_{\text{t}}$, the
bound (4.25) from result “2.” is saturated and there is a kink in the curve
for, $|\mathfrak{q}_{c}^{(1)}|$ (the radius of convergence about the lowest
QNM) as a function of $\Delta$. It seems plausible that the kink in this curve
is associated with the discontinuity seen in figure 4 which we just discussed.

All the results of this paper are related to gapped QNMs. One naturally may
think if analogous results would exist in the case of gapless excitations. In
the case of gapless hydrodynamic excitations, the corresponding pole-skipping
points have been discussed in detail in Blake:2019otz . For the shear and
diffusion channels, the pole-skipping points lie on the dispersion relations
in the Im ${\mathfrak{w}}$ \- Re ${\mathfrak{q}}$ plane where Im
${\mathfrak{w}}=-i\,n$. For the sound channel, however, there is not any pole-
skipping point at Im ${\mathfrak{w}}=-i\,n$. Thus, let us focus on the shear
and diffusion channels, corresponding to the spin-1 hydrodynamic excitations,
namely momentum diffusion and charge diffusion modes. In contrast to the
gapped excitations discussed in this work, the pole-skipping in the spin-1
hydrodynamic excitations occurs at real momenta, while the critical points of
the dispersion relations have complex momentum Grozdanov:2019kge . It is clear
that the latter two sets of points do not lie in the same plane. Then it would
be natural not to find any relation analogous to that of gapped excitations
(figure 8) for these cases.

Here we have focused on the Schwarzschild background dual to uncharged thermal
states. However, we expect our results to hold for other theories and other
background metrics as well. As an extension of our analysis one may consider
the QNMs of the Reissner-Nordström Abbasi:2020ykq ; Jansen:2020hfd , the
rotating Garbiso:2020puw  or charged magnetic Ammon:2017ded ; Ammon:2020rvg ;
Ammon:2016fru ; Grieninger:2017jxz  black branes. It would be interesting to
repeat the above computation in theories with gapped half integer spin
perturbations corresponding to introducing fermionic operators Liu:2009dm ;
Ammon:2010pg ; Ceplak:2019ymw .

It is urgently desirable to find analytic solutions for critical points. One
way may be to use simpler pole-skipping equations of motion at the saturation
values of $\Delta$ where pole-skipping points coincide with critical points,
as we had indicated before. One may also consider different systems, i.e.
different gravitational actions, different black brane solutions, possibly in
the extremal limit. The study of $AdS_{4}$ in Grozdanov:2020koi  has shown
that the analysis of pole-skipping points is significantly simplified in the
lower dimension compared to $AdS_{5}$. In the same vain, it would be
interesting to further investigate the saturation values of $\Delta$, perhaps
through studying some models which are analytically under control, such as the
SYK model Choi:2020tdj .

Through the connection of the trajectories of hydrodynamic and non-
hydrodynamic modes, it is clear that these two types of modes are related in
the complexified momentum space. Lessons for gapped modes have been learned
recently from gapless modes Withers:2018srf . Turning this logic around, the
relations we derived for gapped modes in this present work may help to
understand gapless modes better.

Pole-skipping points in gapless spin-0 (sound) QNMs have been related to the
quantum Lyapunov exponent, $\lambda$, and butterfly velocity, $v_{B}$,
describing the spreading quantum chaos across a system Maldacena:2015waa .
While we have derived relations for gapped modes, our results may have
implications for quantum chaos and bounds on $\lambda$ and $v_{B}$. In
Grozdanov:2019uhi  it was shown that in the case of a Schwarzschild black
brane in Einstein gravity the chaos values $\lambda$ and $v_{B}$ are also
encoded in the critical points of the gapless spin-2 excitations of the
metric. This may provide a connection between our results and quantum chaos.
As a second approach, one may consider an infinitesimal change of the operator
dimension of the metric. Then one may repeat our analysis for the spin-0
(sound) QNMs including the now infinitesimally gapped (formerly hydrodynamic
sound) modes.

In Blake:2019otz ; Grozdanov:2019uhi , the constraints imposed on hydrodynamic
modes from pole-skipping points at real momenta have been discussed. It would
be interesting to investigate whether the pole-skipping points at imaginary
momenta constrain hydrodynamic-dispersion relations, too.

Finally, it is highly desirable to rigorously prove all of the relations which
we have stated in this work based on numerical observations. Analytic proofs
will allow us to apply our results in practice. As one example, once one has
an analytic proof for the bound discussion in section 4, then it may be
possible to find analytic constraints on the response functions. In the case
of a scalar field theory in the Schwarzschild background, this may not be
feasible. But for solvable models, like that of reference Choi:2020tdj , there
would be a better chance for finding relations between pole-skipping and
critical points. We aim to check all the observations of the manuscript in
that model in the future.

###### Acknowledgements.

We would like to thank Ali Davody for helpful discussions, and Sašo Grozdanov
for comments on a draft of this paper. We would like to thank Szabolcs Horvát
for providing his excellent MaTeX package. This work was supported, in part,
by the U.S. Department of Energy grant DE-SC-0012447, and by grant number
561119208 “Double First Class” start-up funding of Lanzhou University, China.

##  Appendix A Near-horizon expansions & vector pole-skipping points

###  A.1 Scalar perturbations

The coefficients $M_{rs}$ corresponding to the first four near horizon
equations are given in table 1. These coefficients are introduced in section
2.

$M_{11}$ | $m^{2}+4{\mathfrak{q}}^{2}+6i{\mathfrak{w}}$
---|---
$M_{21}$ | $3m^{2}+4({\mathfrak{q}}^{2}+3i{\mathfrak{w}})$
$M_{22}$ | $\pi T(-20+m^{2}+4{\mathfrak{q}}^{2}+18i{\mathfrak{w}})$
$M_{31}$ | $3(m^{2}+2i{\mathfrak{w}})$
$M_{32}$ | $\pi T(-30+3m^{2}+4{\mathfrak{q}}^{2}+24i{\mathfrak{w}})$
$M_{33}$ | $(\pi T)^{2}(-60+m^{2}+4{\mathfrak{q}}^{2}+30i{\mathfrak{w}})$
$M_{41}$ | $m^{2}$
$M_{42}$ | $\pi T(-20+3m^{2}+10i{\mathfrak{w}})$
$M_{43}$ | $(\pi T)^{2}(3m^{2}+4(-20+{\mathfrak{q}}^{2}+9i{\mathfrak{w}}))$
$M_{44}$ | $(\pi T)^{3}(-120+m^{2}+4{\mathfrak{q}}^{2}+42i{\mathfrak{w}})$
Table 1:  Examples of coefficients for the equations of motion for the scalar
bulk field expanded near the horizon.

###  A.2 Vector perturbations

In analogy to the calculation of scalar perturbation pole-skipping points in
section 2, here we sketch the near-horizon calculation for the vector
perturbations discussed in section 5. The transverse vector field is expanded
as follows

$A_{x}(r)=\sum_{n=0}^{\infty}a_{n}(r-r_{h})^{n}=\,a_{0}+(r-r_{h})a_{1}+\cdots\,,$
(A.38)

and

$\displaystyle 0$ $\displaystyle=$ $\displaystyle N_{11}a_{0}+4(\pi
T)\,(i{\mathfrak{w}}-1)a_{1}\,,$ (A.39) $\displaystyle 0$ $\displaystyle=$
$\displaystyle N_{21}a_{0}+N_{22}a_{1}+8(\pi
T)^{2}\,(i{\mathfrak{w}}-2)a_{2}\,,$ (A.40) $\displaystyle 0$ $\displaystyle=$
$\displaystyle N_{31}a_{0}+N_{32}a_{1}+N_{33}a_{2}+12(\pi
T)^{3}\,(i{\mathfrak{w}}-3)a_{3}\,,$ (A.41) $\displaystyle 0$ $\displaystyle=$
$\displaystyle N_{41}a_{0}+N_{42}a_{1}+N_{43}a_{2}+N_{44}a_{3}+\,16(\pi
T)^{4}\,(i{\mathfrak{w}}-4)a_{4}\,,$ (A.42)

while the coefficients are given in table 2.

$N_{11}$ | $m^{2}+4{\mathfrak{q}}^{2}+2i{\mathfrak{w}}$
---|---
$N_{21}$ | $2(m^{2}+i{\mathfrak{w}})$
$N_{22}$ | $\pi T(-8+m^{2}+4{\mathfrak{q}}^{2}+10i{\mathfrak{w}})$
$N_{31}$ | $m^{2}$
$N_{32}$ | $2\pi T(-5+m^{2}+3i{\mathfrak{w}})$
$N_{33}$ | $(\pi T)^{2}(-28+m^{2}+4{\mathfrak{q}}^{2}+18i{\mathfrak{w}})$
$N_{41}$ | $0$
$N_{42}$ | $\pi T(-2+m^{2})$
$N_{43}$ | $2(\pi T)^{2}(-14+m^{2}+5i{\mathfrak{w}})$
$N_{44}$ | $(\pi T)^{3}(-60+m^{2}+4{\mathfrak{q}}^{2}+26i{\mathfrak{w}})$
Table 2:  Examples of coefficients for the equations of motion for the vector
bulk field expanded near the horizon.

The pole-skipping points vor the vector operator are then given by

$\begin{split}{\mathfrak{w}}=-i:\,\,\,\,\,\,\,{\mathfrak{q}}^{*}_{1,1}=&\frac{1}{2}\,i\,\sqrt{\Delta^{2}-4\Delta+5}\,,\\\
{\mathfrak{w}}=-2i:\,\,\,\,\,\,\,{\mathfrak{q}}^{*}_{j,2}=&\frac{1}{2}\,i\,\sqrt{\Delta^{2}-4\Delta+11+\,(-1)^{j}2\sqrt{2}\sqrt{\Delta^{2}-4\Delta+7}},\,\,\,\,j=1,2\,.\end{split}$
(A.43)

##  Appendix B Two types of critical points

In this section we explicitly illustrate the positioning of the critical
points along the dispersion relations for two cases: for
$\Delta=4<\Delta_{\text{t}}$ and then for $\Delta=6>\Delta_{\text{t}}$.

In figure 10, we depict the lowest-level-degeneracy which is the indication
for the critical point of the spectral curve when $\Delta<\Delta_{\text{t}}$.
We are considering the case $\Delta=4$ in this figure.

Figure 10: Left panel: dispersion relations of the lowest four QNMs at
$\Delta=4$ and ${\mathfrak{q}}^{2}=|{\mathfrak{q}}^{2}|e^{i\theta}$ with
$\theta=\pi$. Right panel: spectrum of the lowest four QNMs at $\Delta=4$ and
${\mathfrak{q}}=0$. The black dot in the left panel is identical to the black
dot located on the negative part of the Re$-{\mathfrak{q}}^{2}$ axis in figure
4.

As discussed in the main text, the lowest-level-degeneracy corresponds to the
collision of the lowest level QNMs at
${\mathfrak{q}}^{2}=-|{\mathfrak{q}}^{2}|$ on the Im ${\mathfrak{w}}$ axis.
For this reason and by considering
${\mathfrak{w}}\equiv{\mathfrak{w}}({\mathfrak{q}}^{2})=\,{\mathfrak{w}}(|{\mathfrak{q}}^{2}|e^{i\theta})$,
we have plotted Im ${\mathfrak{w}}$ as a function of $|{\mathfrak{q}}|$ at the
constant phase $\theta=\pi$. We find that the point at which the red curve
splits into two dashed red curves, namely the black dot, is exactly the
collision point we had found before (see black dot in figure 4):

${\mathfrak{q}}_{c}^{2}=-1.25309,\,\,\,\,\,\,\,|{\mathfrak{q}}_{c}|=\,1.11942\,.$
(B.44)

In figure 11, we have depicted the level-crossing which is the indication for
the critical point of the spectral curve when $\Delta>\Delta_{\text{t}}$. We
consider the case $\Delta=6$ in this figure.

Figure 11: Left panel: dispersion relations of the lowest three QNMs at
$\Delta=6$ and ${\mathfrak{q}}^{2}=|{\mathfrak{q}}^{2}|e^{i\theta}$ with
$\theta=1.04\pi$. Right panel: spectrum of the lowest three QNMs at $\Delta=6$
and ${\mathfrak{q}}=0$. The brown cross-dot in the left panel shows exactly
the brown cross-dot marked in figure 4. Thick curves in the left panel
correspond to the big stars on the right side of the right panel. Dashed
curves in the left panel show the dispersion of the modes after the collision.

Let us recall that the level-crossing discussed in the text corresponds to the
collision of a lowest-lying QNM with the second QNM having Re ${\mathfrak{w}}$
with the same sign. As shown in figure 4, at $\Delta=6$, this collision occurs
at

${\mathfrak{q}}_{c}^{2}=2.17972\,e^{i(1.04)\pi},\,\,\,\,\,|{\mathfrak{q}}_{c}|=1.47638\,.$
(B.45)

Thus we take
${\mathfrak{w}}\equiv{\mathfrak{w}}({\mathfrak{q}}^{2})=\,{\mathfrak{w}}\big{(}|{\mathfrak{q}}^{2}|e^{i(1.04)\pi}\big{)}$
and plot Im ${\mathfrak{w}}-|{\mathfrak{q}}|$. As can be seen in figure 11,
starting from $|{\mathfrak{q}}|=0$ and keeping $\theta$ constant at $1.04\pi$,
the degeneracy between the two lowest-lying modes is split. The splitting
becomes wider when $|{\mathfrak{q}}|$ increases. It means that in the right
panel, by taking $\theta=1.04\pi$, the two red stars have the same
$\text{Im}\,{\mathfrak{w}}$ only at $|{\mathfrak{q}}|=0$. For a non-vanishing
$|{\mathfrak{q}}|$ at $\theta=1.04\pi$, the red star on the right side has a
more negative Im ${\mathfrak{w}}$ than the one in the left side. The former
then follows the thick red curve in the left panel.

Also in figure 11 the same behavior is observed for the next higher QNMs with
mode number $n=2$. It is indicated by the big green star on the right side of
the right panel. Again, keeping fixed $\theta=1.04\pi$, when
$|{\mathfrak{q}}|$ increases, the green star on the right side follows the
thick green curve in the left panel. At some point, marked by a brown cross-
dot, the red and green thick curves collide with each other. This point is
exactly (B.45). The collision between two curves with different colors is the
manifestation of level-crossing. There is also another level-crossing at the
same value of $|{\mathfrak{q}}|$ which corresponds to the collision of small
red and and small green stars on the left side of the right panel. Since this
collision occurs at $\theta=2\pi-1.04\pi=0.96\pi$, it cannot be observed in
the left panel.

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