# Subdiffusion and many-body quantum chaos with kinetic constraints

Hansveer Singh  Department of Physics, University of Massachusetts, Amherst,
Massachusetts 01003, USA  Brayden A. Ware  Department of Physics, University
of Massachusetts, Amherst, Massachusetts 01003, USA  Romain Vasseur
Department of Physics, University of Massachusetts, Amherst, Massachusetts
01003, USA  Aaron J. Friedman  Department of Physics and Center for Theory of
Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA

###### Abstract

We investigate the spectral and transport properties of many-body quantum
systems with conserved charges and kinetic constraints. Using random unitary
circuits, we compute ensemble-averaged spectral form factors and linear-
response correlation functions, and find that their characteristic time scales
are given by the inverse gap of an effective Hamiltonian—or equivalently, a
transfer matrix describing a classical Markov process. Our approach allows us
to connect directly the Thouless time, $t^{\vphantom{\dagger}}_{\rm Th}$,
determined by the spectral form factor, to transport properties and linear
response correlators. Using tensor network methods, we determine the dynamical
exponent, $z$, for a number of constrained, conserving models. We find
universality classes with diffusive, subdiffusive, quasilocalized, and
localized dynamics, depending on the severity of the constraints. In
particular, we show that quantum systems with “Fredkin” constraints exhibit
anomalous transport with dynamical exponent $z\simeq 8/3$.

_Introduction.—_ Recent years have seen substantial progress in understanding
how isolated quantum systems thermalize under their own dynamics. The
eigenstate thermalization hypothesis (ETH) Deutsch (1991); Srednicki (1994)
proposes that entanglement between subsystems allows for local equilibration:
Generic unitary evolution scrambles local quantum information into highly
nonlocal degrees of freedom, which are inaccessible to local observables.
Early tests of ETH Rigol  _et al._ (2007, 2008); Santos and Rigol (2010);
Borgonovi  _et al._ (2016); D’Alessio  _et al._ (2016) relied on small scale
numerics and extensions of integrable models, which are fine tuned;
understanding the universal aspects of quantum chaotic dynamics requires a
more general approach.

A hallmark of chaotic systems is that they dynamically forget as much
information about their past as symmetries allow. Hence, the salient features
of chaotic systems are well captured by replacing the microscopic model with a
random matrix with the same symmetries. Random unitary circuits (RUCs) invoke
the potency of random matrix theory (RMT) while also introducing spatial
locality, with the system evolved by a brickwork “circuit” of $\ell$-site
gates Roberts and Yoshida (2017); Kos  _et al._ (2018); Nahum  _et al._ (2017,
2018); von Keyserlingk  _et al._ (2018); Sünderhauf  _et al._ (2018); Chan
_et al._ (2018a); Kos  _et al._ (2021); Friedman  _et al._ (2019). RUCs are
fully generic, and their study has elucidated the universal dynamics of
chaotic quantum systems: Entanglement grows linearly until saturating to a
volume law, with fluctuations in the KPZ universality class Nahum  _et al._
(2017); operator fronts (and out-of-time-ordered correlation functions)
propagate ballistically and broaden diffusively Nahum  _et al._ (2018); von
Keyserlingk  _et al._ (2018); Chan  _et al._ (2018a), etc.

However, these RUCs are designed to be featureless; an interesting question is
how these properties change as one reintroduces other physical ingredients,
such as symmetries. With conserved charges, one can consider transport; for
typical $U(1)$ symmetry, one expects conserved charges to diffuse Rakovszky
_et al._ (2018); Khemani  _et al._ (2018); Gharibyan  _et al._ (2018);
Friedman  _et al._ (2019). Operators that overlap with conserved quantities
are expected to have slower dynamics, dominated by hydrodynamic modes. It is
also interesting to study dynamics with more complicated symmetries or
constraints Valado  _et al._ (2016); Chandran  _et al._ (2016); Chen  _et al._
(2018); Gopalakrishnan and Zakirov (2018); Lan  _et al._ (2018); Everest  _et
al._ (2016); Pancotti  _et al._ (2020); Scherg  _et al._ (2021); Rakovszky
_et al._ (2019); Zhou and Ludwig (2020); Huang (2020). Fractons, e.g., are
excitations in systems with charge and dipole conservation, which are
constrained to move in pairs only Vijay  _et al._ (2015, 2016). This higher-
order symmetry can also be viewed as a _constraint_ ; recent studies of
fractons in the contexts of RUCs and hydrodynamics have found evidence for
subdiffusion, with dynamical exponent $z=2(m+1)$, where $m$ is the highest
conserved moment of charge Iaconis  _et al._ (2021); Gromov  _et al._ (2020);
Feldmeier  _et al._ (2020); Morningstar  _et al._ ; Iaconis  _et al._ (2019);
Glorioso  _et al._ (2021); Moudgalya  _et al._ (2021).

In this Letter we analyze the general consequences of kinetic constraints on
charge-conserving many-body quantum dynamics in one dimension. Kinetic
constraints restrict the local rearrangements of charges and have been
intensely studied as models of classical systems with glassy dynamics Valado
_et al._ (2016); Pancotti  _et al._ (2020); Scherg  _et al._ (2021); Lan  _et
al._ (2018); Everest  _et al._ (2016); Garrahan  _et al._ (2010); Garrahan
(2018); Ritort and Sollich (2003); Lesanovsky and Garrahan (2013). Depending
on the locally forbidden rearrangements, adding constraints may anomalously
slow down or completely freeze the process of thermalization. Using variations
of RUCs, we probe how imposing constraints on generic quantum systems leads to
new universality classes with slow dynamics. Using Floquet random circuits, in
the limit of large on-site Hilbert space, we relate the scaling of the many-
body Thouless time—the time scale for a system to show RMT spectral
rigidity—with system size Chan  _et al._ (2018a, b); Bertini  _et al._ (2018);
Gharibyan  _et al._ (2018, 2018); Fyodorov and Mirlin (1997); Sünderhauf  _et
al._ (2018); Friedman  _et al._ (2019); Winer and Swingle (2020, 2021); Kos
_et al._ (2021) to the inverse gap of the transfer matrix of a stochastic
classical model; or equivalently, of an effective Hamiltonian, which lies at a
Rokhsar-Kivelson (RK) point Rokhsar and Kivelson (1988); Moudgalya  _et al._
(2021); Friedman  _et al._ (2019). We show that the same transfer matrix also
controls the dynamics of linear response correlators, providing a general
relation between the Thouless time and transport Gharibyan  _et al._ (2018).
Depending on the severity of the model’s constraints, we find diffusive,
subdiffusive, quasilocalized, and localized dynamics, and identify a new
universality class of constrained $z\simeq 8/3$ “Fredkin” systems Movassagh
and Shor (2016); Salberger and Korepin (2016); Salberger  _et al._ (2017);
Chen  _et al._ (2017a, b); Zhang and Klich (2017); Udagawa and Katsura (2017);
Langlett and Xu (2021); Chen  _et al._ (2020).

_Spectral rigidity and transport correlators.—_ A useful indicator of quantum
chaos is _level repulsion_ Deutsch (1991); Srednicki (1994); Gharibyan  _et
al._ (2018), characterized by an RMT distribution of the eigenvalues of the
evolution operator Gharibyan  _et al._ (2018, 2018); Kos  _et al._ (2018);
Winer and Swingle (2020); Chan  _et al._ (2018a, b); Friedman  _et al._
(2019); Kos  _et al._ (2021). Periodically driven (Floquet) RUCs Chan  _et
al._ (2018a, b); Sünderhauf  _et al._ (2018); Friedman  _et al._ (2019),
afford such a spectrum, as time evolution follows from the Floquet unitary,
$\mathcal{F}$, that evolves the system by one time step. For Hamiltonian or
Floquet systems, it is convenient to measure the ratio of consecutive energy
gaps (the “$r$ ratio”) Oganesyan and Huse (2007); another robust probe of
spectral rigidity is the two-point spectral form factor (SFF) Bertini  _et
al._ (2018); Fyodorov and Mirlin (1997); Chan  _et al._ (2018a, b); Friedman
_et al._ (2019); Gharibyan  _et al._ (2018, 2018); Winer and Swingle (2020,
2021); Kos  _et al._ (2021),

$\displaystyle
K(t)\equiv\sum\limits_{m,n=1}^{\mathcal{D}}\,\overline{\,e^{\mathbf{i}\left(\theta_{m}-\theta_{n}\right)t}}\,=\overline{\,\left|{\rm
tr}[\,\mathcal{F}^{t}\,]\right|^{2}\,}~{}~{},~{}~{}~{}$ (1)

where $\\{\theta^{\,}_{m}\\}$ are the eigenphases of $\mathcal{F}$,
$\mathcal{D}=q^{L}$ with $L$ the number of sites and $q$ states per site, and
the overline denotes averaging over an ensemble of statistically similar
systems. In the limit $q\to\infty$, RUCs reproduce the spectral properties of
nonlocal random matrix models Brouwer and Beenakker (1996); Chan  _et al._
(2018a): $K(t)=t$ for $0<t<t^{\vphantom{\dagger}}_{\rm Heis}=\mathcal{D}$, the
Heisenberg time, and $K(t)=\mathcal{D}$ for $t>t^{\vphantom{\dagger}}_{\rm
Heis}$. In this limit, thermalization—characterized by a linear ramp $K=t$—is
instantaneous.

Away from this limit, one expects an initial overshoot of the linear ramp
until interactions thermalize the system Chan  _et al._ (2018b); Winer and
Swingle (2020). A noninteracting Floquet RUC has $K=t^{L}$; one can imagine
divvying the system into weakly interacting blocks of size $\xi(t)$, so that
$K(t)\sim t^{L/\xi(t)}$, with $\xi(0)\sim 1$. Under time evolution,
interactions lead $\xi(t)$ to grow, saturating to $\xi(t)=L$ for $t\geq
t^{\vphantom{\dagger}}_{\rm Th}$, so that $K(t)=t$ Gharibyan  _et al._ (2018);
Chan  _et al._ (2018a); Kos  _et al._ (2018); Bertini  _et al._ (2018);
Friedman  _et al._ (2019). The Thouless time, $t^{\vphantom{\dagger}}_{\rm
Th}$—in analogy to single-particle disordered wires Thouless (1974, 1977)—is
the time it takes for a chaotic system to thermalize fully, signaled by a
linear ramp, $K(t)=t$.

One can also observe delayed thermalization even for $q\to\infty$ with
conserved charges Gharibyan  _et al._ (2018); Friedman  _et al._ (2019).
Symmetries (and constraints) lead to independent sectors of $\mathcal{F}$
whose eigenvalues do not repel; thus, a chaotic system with $\mathcal{N}$
independent sectors will have $K(t)=\mathcal{N}\,t$ after thermalizing
Friedman  _et al._ (2019). Ref. 16 provides a recipe for computing the SFF in
the presence of symmetries, mapping $K(t)$ to a classical Markov process,
itself equivalent to a quantum Hamiltonian at an RK point Rokhsar and Kivelson
(1988); Friedman  _et al._ (2019); Moudgalya  _et al._ (2021). Study of the
corresponding classical lattice gas reveals that diffusion of the $U(1)$
conserved charge delays thermalization, with $K(t)\to\mathcal{N}\,t$ for
$t\gtrsim t^{\vphantom{\dagger}}_{\rm Th}\sim L^{2}$. Slower, subdiffusive
scalings of $t^{\vphantom{\dagger}}_{\rm Th}$ have also been observed in
systems with dipole-moment conservation Moudgalya  _et al._ (2021).

In this work, we investigate the effect of constraints and symmetries on
thermalization by studying the SFF and linear response (connected) transport
correlators,

$\displaystyle\mathcal{C}(x,t)\,=\,\left\langle\,\mathfrak{q}(x,t)\,\mathfrak{q}(0,0)\,\right\rangle^{\,}_{c}~{}~{},~{}~{}$
(2)

with $\mathfrak{q}(x)$ the local charge density, $Q=\int dx\,\mathfrak{q}(x)$
the conserved $U(1)$ charge, and
$\left\langle\,\dots\,\right\rangle\,=\,\mathcal{D}^{-1}\,{\rm
tr}\left[\,\dots\,\right]$, the equilibrium average at infinite temperature
111Subtraction of the disconnected part is implied. Connected correlators in
the spin and particle language are identical up to a factor of four.. We
provide a recipe for computing the structure factor (2) for arbitrary $q$, and
the SFF (1) for $q\to\infty$ in generic, quantum chaotic models, using the
machinery of RUCs. We show that the important physics of _both_ quantities is
controlled by the low energy properties of the _same_ transfer matrix,
$\mathcal{T}$, which also describes a discrete-time Markov process with the
same conservation laws and constraints 222Such a direct connection between the
spectral form factor and two-point correlators was first suggested in Ref.
19.. We can also view $\mathcal{T}^{t}\approx e^{-t\,H^{\,}_{\rm RK}}$, where
$H^{\,}_{\rm RK}$ lies at an unfrustrated RK point Rokhsar and Kivelson
(1988); Friedman  _et al._ (2019); Moudgalya  _et al._ (2021). Within a fixed
charge sector, the gap of $\mathcal{T}$ (or $H^{\,}_{\rm RK}$) scales as
$\Delta\sim L^{-z}$; its inverse is the Thouless time,
$t^{\vphantom{\dagger}}_{\rm Th}\sim L^{z}$, the time required for information
to relax throughout the system. The same dynamical exponent controls transport
properties from (2), and we find a universal scaling form
$\mathcal{C}(x,t)\sim t^{-1/z}f(x/t^{1/z})$, with $z=2$ and $f(\cdot)$
Gaussian for diffusive systems, and $z>2$ for subdiffusive systems.

Figure 1:  Models and setup. Top: Cartoon depiction of the allowed dynamical
moves for the five models presented;
${\color[rgb]{0,0,0}\bullet}\mathllap{\circ}$ indicates a particle and $\circ$
denotes a hole. Bottom Left: Cartoon sketch of the spectral form factor for a
generic chaotic system (red) and the RMT prediction (blue); the linear ramp
regime ($K(t)=\mathcal{N}t$) sets in for $t\gtrsim t^{\,}_{\rm Th}$, the
Thouless time, which scales as $L^{z}$. Bottom Right: Heat map of the
structure factor (charge two-point function), shown here for Fredkin RUCs,
with the variance used to extract $z$, the dynamical critical exponent
($z\simeq 8/3$ for Fredkin constraints).

_Models.—_ We consider several constrained models acting on a chain of $L$
qubits ($q=2$) which may be occupied
(${\color[rgb]{0,0,0}\bullet}\mathllap{\circ}$) or empty ($\circ$), with a
$U(1)$ conserved charge corresponding to particle number. A cartoon of the
allowed dynamical moves is given in Fig. 1: Essentially, particles are allowed
to hop if the neighboring sites are appropriately occupied/unoccupied. Time
evolution is generated by a circuit of gates with the general form

$\displaystyle\mathcal{U}^{\,}_{r}\,=\,\sum\limits_{\alpha}\,P^{\,}_{r,\alpha}\,U^{\,}_{r,\alpha}\,P^{\,}_{r,\alpha}\,+\,\sum\limits_{\beta}\,P^{\,}_{r,\beta}~{}~{},~{}~{}$
(3)

where $\alpha$ labels constraint-satisfying configurations of cluster $r$ with
fixed $U(1)$ charge $Q^{\,}_{r}=\sum_{j\,\in\,r}\,\mathfrak{q}^{\,}_{j}$;
$\beta$ labels individual constraint-violating configurations on $r$ (with no
corresponding unitary dynamics); and $U^{\,}_{r,\alpha}$ is a
$n^{\,}_{\alpha}\times n^{\,}_{\alpha}$ Haar unitary that mixes the
$n^{\,}_{\alpha}$ states in block $\alpha$ with a fixed $U(1)$ charge
Rakovszky  _et al._ (2018); Khemani  _et al._ (2018); Friedman  _et al._
(2019).

The allowed moves for the models considered are depicted in Fig. 1. The
Fredkin model Movassagh and Shor (2016); Salberger and Korepin (2016);
Salberger  _et al._ (2017); Chen  _et al._ (2017a, b); Zhang and Klich (2017);
Udagawa and Katsura (2017); Langlett and Xu (2021); Chen  _et al._ (2020),
allows hopping between sites $j$ and $j+1$ if $j+2$ is occupied _or_ $j-1$ is
unoccupied, respectively implemented by gates $\mathcal{U}^{\,}_{r,R}$ (right)
and $\mathcal{U}^{\,}_{r,L}$ (left). The Gonçalves-Landim-Toninelli (GLT)
model Gonçalves  _et al._ (2009) allows hopping if _either_ neighboring site
is occupied; XNOR allows hopping if both neighboring sites are in the _same_
state Yang  _et al._ (2020); Zadnik and Fagotti (2021); Feldmeier  _et al._
(2020); Pozsgay  _et al._ (2021); Pozsgay (2021); Bastianello  _et al._
(2021); $U(1)$ East allows hopping only if the right (“East”) neighbor is
occupied; and PXYP 333The PXYP model is a $U(1)$ conserving version of the PXP
model describing Rydberg atom chains. allows hopping only if _both_
neighboring sites are occupied Lesanovsky and Garrahan (2013); Bernien  _et
al._ (2017); Turner  _et al._ (2018); Chen  _et al._ (2018); Lin and Motrunich
(2019); Bluvstein  _et al._ (2021). Each model is implemented via minimal
gates of the form given in Eq. 3. Each type of $\ell$-site gate,
$\mathcal{U}^{\,}_{r}$, requires $\ell$ layers per “time step”, and is always
block diagonal in the charge basis sup . Models with different constraints or
encodings thereof are also discussed in the Supplement sup .

_Spectral form factor.—_ Evaluating the SFF (1) requires the use of Floquet
circuits to guarantee a spectrum: The unitaries comprising the first time
step, $\mathcal{F}$, are drawn independently; evolution to time $t$ is
generated by $\mathcal{F}^{\,t\,}$. For arbitrary $t$, ensemble averaging Eq.
1 is generally intractable Roberts and Yoshida (2017); Chan  _et al._ (2018a);
Friedman  _et al._ (2019); Moudgalya  _et al._ (2021); to simplify Haar
averaging—and to wash out any features not related to particular symmetries
and constraints—we include an ancillary $d$-dimensional qudit on each site
444Note that including the ancillary qudit eliminates any $\beta$-type blocks
from Eq. 3, as all blocks act nontrivially on the qudits. so $q=2d$, and take
the limit $d\to\infty$ Friedman  _et al._ (2019); Moudgalya  _et al._ (2021).
The leading contribution to $K(t)$ can be evaluated diagrammatically Brouwer
and Beenakker (1996), and yields $t$ equivalent “Gaussian” diagrams Brouwer
and Beenakker (1996); Chan  _et al._ (2018a); Friedman  _et al._ (2019). This
procedure is fully generic Friedman  _et al._ (2019); Moudgalya  _et al._
(2021); sup : The Haar averaging contracts the indices of gates in the two
traces, eliminating one trace as well as the $d$-state variables, leaving only
a single trace over the physical qubits,

$\displaystyle K(t)\,=\,t\,{\rm
tr}\left[\,\mathcal{T}^{\,t\,}\,\right]~{}~{},~{}~{}$ (4)

where the transfer matrix, $\mathcal{T}$, encodes the contribution of
configurations of the physical qubits to $K(t)$.

The form of $\mathcal{T}$ for such models is simple Friedman  _et al._ (2019);
Moudgalya  _et al._ (2021); sup : $\mathcal{T}$ is a circuit with the same
geometry as $\mathcal{F}$, comprising Hermitian 555Hermitian gates are to be
expected after averaging over a unitary and its conjugate. gates,
$T^{\,}_{r}$, i.e.
$\mathcal{T}\,=\,\bigotimes_{\lambda}\,\bigotimes_{r\in\lambda}\,T^{\,}_{r}$,
where $\lambda$ labels layers of the circuit, and $T^{\,}_{r}$, has the same
block structure as the corresponding $\mathcal{U}^{\,}_{r}$; each block has
uniform entries $1/n$, with $n$ the block size Friedman  _et al._ (2019);
Moudgalya  _et al._ (2021); sup :

$\displaystyle
T^{\,}_{r}\,=\,\sum\limits_{\alpha}\,\frac{1}{n^{\,}_{\alpha}}\,\sum\limits_{m,m^{\prime}\,\in\,\alpha\,}\,{|m\rangle}{\langle
m^{\prime}|}~{}~{},~{}~{}$ (5)

where $m,m^{\prime}$ run over the $n^{\,}_{\alpha}$ configurations in block
$\alpha$. Note that $\mathcal{T}$ describes a discrete-time Markov process for
a classical lattice gas with _the same_ constraints and conservation laws as
the quantum circuit Schütz (2001); Friedman  _et al._ (2019); Moudgalya  _et
al._ (2021); sup . Relatedly, we can define local Hamiltonian terms,
$H^{\,}_{r}=\mathbb{1}^{\,}_{r}-T^{\,}_{r}$, so that at long wavelengths,
$\mathcal{T}^{\,t\,}\approx e^{-t\,H^{\,}_{\rm RK}}$, where $H^{\,}_{\rm
RK}=\sum_{r}H^{\,}_{r}$ always lies at an unfrustrated RK point Rokhsar and
Kivelson (1988); Friedman  _et al._ (2019); Moudgalya  _et al._ (2021). The
Thouless time, $t^{\vphantom{\dagger}}_{\rm Th}$, marks the start of the
linear ramp regime, $K(t)=\mathcal{N}\,t$. Each of the $\mathcal{N}$ sectors
has largest eigenvalue unity; the linear ramp sets in after subleading
contributions have decayed: i.e., ${\rm
tr}\left[\mathcal{T}^{\,t\,}\right]\approx{\rm tr}\left[e^{-t\,H^{\,}_{\rm
RK}}\right]\,\to\,\mathcal{N}\left(1+e^{-t\,\Delta}\right)$ at late times,
where $\Delta=L^{-z}$ is the gap of $H^{\,}_{\rm RK}$ (or $\mathcal{T}$). We
extract the Thouless time from $K(t)$ as Gharibyan  _et al._ (2018); Friedman
_et al._ (2019); Moudgalya  _et al._ (2021)

$\displaystyle
K(t)\sim\,t\,\left(\mathcal{N}+e^{-t\,\Delta}+\dots\right)~{}\implies~{}t^{\vphantom{\dagger}}_{\rm
Th}=\frac{1}{\Delta}=L^{z}~{},~{}~{}~{}~{}$ (6)

with $z$ the dynamical exponent. Thus, $t^{\vphantom{\dagger}}_{\rm Th}$ gives
the time scale over which $K(t)\to\mathcal{N}\,t$, and lower bounds the time
required for generic models with the same symmetries and constraints to
thermalize Winer and Swingle (2020); Chan  _et al._ (2018a); Friedman  _et
al._ (2019); Moudgalya  _et al._ (2021); sup . For some of the models
considered, the low-energy properties of $H^{\,}_{\rm RK}$ have been reported:
The Fredkin Hamiltonian, e.g., has a gap $\Delta\sim L^{-z}$ with $z>2$ Chen
_et al._ (2017a, b). Our results imply that the same dynamical exponent, $z$,
also controls thermalization and transport properties (see below), for generic
many-body quantum systems in this class Gharibyan  _et al._ (2018).

Figure 2: Numerical results from the transfer matrix, $\mathcal{T}$. Left:
Variance of the spin profile in each of the five models. The variance
saturates in the PXYP model, indicating localization, while charges eventually
spread across the system in the other models. Middle: Apparent dynamical
exponent, $z$, versus time, $t$. For GLT, XNOR and Fredkin, $z(t)$ saturates
to $z=2$ (diffusion), $z=4$ and $z=8/3$ (subdiffusion), respectively. In the
$U(1)$ East model, $z(t)$ appears to grow without bound, $z(t)\sim\log t$,
indicating quasilocalized dynamics with spread slower than any power law —
another way to extract $z(t)$ using the return probability
$\mathcal{C}(x=0,t)\sim t^{-1/z}$ leads to a different and even larger
estimate $z\approx 7$ for the time scales we can access sup . Right: Collapse
of charge profiles for Fredkin when rescaled by the dynamical exponent
$z=8/3$. Inset: Collapse in momentum space showing ${\cal C}(k,t)\sim{\rm
e}^{-C|k|^{z}t}$ at small $k$. TEBD data use maximum bond dimension
$\chi^{\,}_{\rm max}=1024$ for Fredkin and $\chi^{\,}_{\rm max}=512$ for the
other models to ensure convergence.

_Two-point correlations.—_ To compute Haar-averaged two-point functions, we
dispense with the ancillary qudit and Floquet structure: The five models
considered act on $L$ qubits ($q=2$) with Haar unitaries independently drawn
at each time step. Correlators (2) in RUCs can generically be written in terms
of a transfer matrix,

$\displaystyle\mathcal{C}^{\,}_{i,j}=\left\langle\,\overline{\,\mathcal{O}^{\,}_{i}(t)\,\mathcal{O}^{\,}_{j}(0)\,}\,\right\rangle=\left(\mathcal{O}^{\,}_{i}\middle|\,\mathcal{T}^{\,t\,}\,\middle|\mathcal{O}^{\,}_{j}\right)~{}~{},~{}~{}$
(7)

where $\left|\mathcal{O}\right)$ is an element of the $q^{2L}$-dimensional
operator space, and $\mathcal{T}$ acts therein, is implicitly Haar averaged
666Each layer of the transfer matrix can be ensemble averaged independently,
as unitary gates are independently drawn at each time step., and has the same
circuit structure as $\mathcal{F}$ (and the SFF transfer matrix). For models
with Hilbert space dimension $q$ and unitaries given by Eq. 3, the gates of
$\mathcal{T}$ take the form sup
$\left(\sigma^{\mu}_{r}\middle|T^{\,}_{r}\middle|\sigma^{\nu}_{r}\right)\,=\,q^{-\ell}\,{\rm
tr}\left[\,\sigma^{\mu}_{r}\,\overline{\,\mathcal{U}^{\dagger}_{r}\,\sigma^{\nu}_{r}\mathcal{U}^{\vphantom{\dagger}}_{r}\,}\,\right]$,
where
$\left(\sigma^{\mu}_{r}\middle|\sigma^{\nu}_{r}\right)=\delta^{\,}_{\mu,\nu}$
are orthonormal basis operators (e.g. Pauli strings for $q=2$). Haar averaging
gives

$\displaystyle
T^{\,}_{r}\,=\,\sum\limits_{\alpha}\,\frac{1}{n^{\,}_{\alpha}}\,\sum\limits_{m,m^{\prime}\,\in\,\alpha\,}\,|\,\pi^{\,}_{m})(\pi^{\,}_{m^{\prime}}|~{}~{},$
(8)

for diagonal (i.e., charge conserving) operators, where
$\left|\pi^{\,}_{m}\right)\,=\,\sqrt{q}\,{|m\rangle}{\langle m|}$ is a
projector onto state $m$ in block $\alpha$, and
$\left(\pi^{\,}_{m}\middle|\pi^{\,}_{n}\right)=\delta^{\,}_{m,n}$ form an
orthonormal basis for the $q$ diagonal operators on each site sup .

Crucially, we note that $\mathcal{T}$ (8) is identical to the SFF transfer
matrix (5), with the $q$ states per site replaced by $q$ charge-conserving
operators. Thus, the universal features of both spectral and physical
correlations are controlled by the low energy spectrum of $\mathcal{T}$,
generically relating the Thouless time (related to spectral rigidity) to
transport properties, as proposed in Ref. 19. As an aside, we note that
nondiagonal (charge-changing) operators do not mix with diagonal operators
under $\mathcal{T}$, but evolve under a different transfer matrix if at all
sup . However, we need only consider correlators of diagonal (charge)
operators to extract universal transport properties.

_Numerics.—_ We efficiently simulate the dynamics generated by $\mathcal{T}$
using time evolving block decimation (TEBD) applied to matrix-product
operators (MPOs) Vidal (2003); Verstraete  _et al._ (2004); Vidal (2004);
Schollwöck (2011), exploiting slow entanglement growth compared to the
underlying unitary dynamics. We simulate infinite-temperature correlation
functions $\mathcal{C}(x,t)=\,\mathcal{D}^{-1}\,{\rm
tr}\left[\,\overline{\,\mathfrak{q}(x,t)\,\mathfrak{q}(0,0)\,}\,\right]$,
where $\mathfrak{q}(x,t)$ is the occupation of site $x$ at time $t$. We also
use the spatial variance of the correlator,
$\sigma^{2}(t)=\sum_{x}\,x^{2}\,\mathcal{C}(x,t)-(\sum_{x}\,x\,\mathcal{C}(x,t))^{2}\sim
t^{2/z}$, to extract the dynamical exponent, $z$—characterizing the transport
of charge—via $2/z(t)\equiv d\log\sigma^{2}/d\log t$ (shown in the center
panel of Fig. 2). For GLT, $z(t)\to 2$, indicating diffusive transport and
consistent with classical results Gonçalves  _et al._ (2009); for XNOR and
Fredkin, $z=4$ and $z\simeq 8/3$, respectively, indicating subdiffusion; for
PXYP, $\sigma^{2}(t)$ itself saturates, indicating localization; and for
$U(1)$ East, $z(t)$ grows slowly without saturation, indicating
quasilocalization.

The PXYP and $U(1)$ East cases can be understood in terms of Hilbert space
fragmentation Pai  _et al._ (2019); Khemani  _et al._ (2020); Sala  _et al._
(2020); Moudgalya  _et al._ (2019); Rakovszky  _et al._ (2020); Kohlert  _et
al._ (2021); Rakovszky  _et al._ (2020): The number of sectors, $\mathcal{N}$,
for both models scales exponentially in system size sup . In the terminology
of Ref. 87, PXYP is “strongly fragmented” and does not thermalize (i.e. there
is no transport; charges are localized), while $U(1)$ East is “weakly
fragmented” and thermalizes very slowly ($\sigma^{2}(t)$ grows more slowly
with $t$ than any power law). XNOR also shows weak fragmentation, and its
dynamical exponent, $z=4$, can be derived analytically from the unusual spin-
wave spectrum, $E(k)\sim k^{2}/L^{2}$ of the underlying effective RK
Hamiltonian sup . This subdiffusive transport with $z=4$ can also be
understood in terms of the “screening” of the effective charge carried by the
diffusive magnon excitations in this model Gopalakrishnan and Vasseur (2019);
sup .

We remark that $z(t)$ appears to approach $8/3$ for Fredkin constraints—a
similar numerical estimate, $z\approx 2.69$, was reported in Ref. 54 in the
context of low-temperature physics of the Fredkin Hamiltonian. While our
results derive from RUCs, we expect that this $z=8/3$ characterizes a new
dynamical universality class of _generic_ many-body quantum or classical
systems (Floquet, Hamiltonian, or noisy) with Fredkin constraints. The Fredkin
correlator satisfies the universal scaling form $\mathcal{C}(x,t)\sim
1/t^{1/z}f(x/t^{1/z})$, with $f(\cdot)$ a non-Gaussian function (see third
panel of Fig. 2 and Ref. 76). Remarkably, we find numerically that the Fredkin
RK Hamiltonian has a low-energy spectrum $E(k)\sim k^{4/3}/L^{4/3}$,
reminiscent of the XNOR model, suggesting the possibility of a similar
mechanism for subdiffusion in both models sup .

_Discussion.—_ We studied spectral and transport properties of many-body
quantum systems with conserved charges and kinetic constraints using random
unitary circuits. We computed ensemble-averaged spectral form factors and
linear-response correlation functions for various classes of constraints, and
showed that both relate to the same transfer matrix, $\mathcal{T}$, describing
a classical Markov process. This mapping holds for _any_ choice of symmetries
and constraints, and in any dimension; however, numerical simulation of
$\mathcal{T}$ becomes intractable for $d>1$. Our results establish a general
correspondence between the Thouless time and transport properties in
conserving systems, and we unearth a broad range of possible transport
properties depending on the constraint. The Fredkin universality class is
especially interesting: Further characterizing its dynamical exponent of
$z\simeq 8/3$ presents a clear direction for future work.

_Acknowledgments.—_ We thank U. Agrawal, J. T. Chalker, A. De Luca, and A.C.
Potter for useful discussions and collaborations on related work; we thank J.
P. Garrahan, S. Gopalakrishnan, A. Lucas, R. Nandkishore, B. Pozsgay, T.
Rakovszky, P. Sala, and W. Witczak-Krempa for the same and for their feedback
on this manuscript. We acknowledge support from the Air Force Office of
Scientific Research under Grant No. FA9550-21-1-0123 (RV and BAW) and the
Alfred P. Sloan Foundation through a Sloan Research Fellowship (RV).

Note Added.— While completing this manuscript, Ref. 92 appeared on the arXiv,
and reports subdiffusive hydrodynamics for the “Motzkin” Hamiltonian; Motzkin
constraints are very similar to Fredkin constraints, and appear to lie in the
same universality class with dynamical exponent $z\simeq 8/3$ sup .

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#  Resonant excitation of quasinormal modes of black holes

Hayato Motohashi  0000-0002-4330-7024 Division of Liberal Arts, Kogakuin
University, 2665-1 Nakano-machi, Hachioji, Tokyo, 192-0015, Japan

###### Abstract

We elucidate that a distinctive resonant excitation between quasinormal modes
(QNMs) of black holes emerges as a universal phenomenon at avoided crossing
near exceptional point through high-precision numerical analysis and theory of
QNMs based on the framework of non-Hermitian physics. This resonant phenomenon
not only allows us to decipher a long-standing mystery concerning the peculiar
behaviors of QNMs but also stands as a novel beacon for characterizing black
hole spacetime geometry. Our findings pave the way for rigorous examinations
of black holes and the exploration of new physics in gravity.

Introduction. The last decade has witnessed the advent of gravitational wave
(GW) astronomy, opening a new window to the universe, especially in the
dynamical and strong-field regime near black holes Abbott  _et al._ (2016);
Aasi  _et al._ (2015); Acernese  _et al._ (2015); Akutsu  _et al._ (2021).
General relativity predicts that black holes are the simplest astrophysical
objects in the universe since the only rotating astrophysical black hole
solution that satisfies the Einstein equation is the Kerr spacetime Israel
(1967); Carter (1971); Hawking (1972), characterized solely by mass and spin.
Remnant black holes after binary black hole mergers are expected to settle
into the state of a Kerr black hole, emitting ringdown GWs. Linear
perturbation theory predicts that the ringdown GWs are described by a linear
combination of intrinsic exponentially damped sinusoids known as quasinormal
modes (QNMs) Vishveshwara (1970); Press (1971); Teukolsky (1973);
Chandrasekhar and Detweiler (1975). Their timescales and amplitudes are
governed by complex frequencies and excitation factors Leaver (1985, 1986a,
1986b); Sun and Price (1988); Nollert and Schmidt (1992); Nollert and Price
(1999); Andersson (1995, 1997); Glampedakis and Andersson (2001, 2003), also
determined solely by mass and spin, playing a central role in the black hole
spectroscopy program Detweiler (1980); Echeverria (1989); Finn (1992); Dreyer
_et al._ (2004); Berti  _et al._ (2006). Despite extensive efforts to better
understand them, the peculiar behaviors of QNMs reflecting changes in black
hole parameters have remained elusive Kokkotas and Schmidt (1999); Nollert
(1999); Berti  _et al._ (2009); Konoplya and Zhidenko (2011).

In this Letter, we reveal the emergence of the resonant excitation phenomenon
associated with an avoided crossing of QNMs near the exceptional point (EP).
The avoided crossing in quantum mechanics, also known as level repulsion, is a
phenomenon where two energy eigenvalues cannot coincide unless a certain
condition is satisfied Hund (1927); von Neuman and Wigner (1929); Landau and
Lifshitz (1977); Arnold (1978). It plays a pivotal role in experiments and
observations across vast fields of physical science Ashcroft and Mermin
(1976); Landau (1932); Zener (1932); Majorana (1932); Stückelberg (1932);
Ivakhnenko  _et al._ (2023); Purcell (1946); Herzberg (1991); Smirnov (2003);
Wurm (2017); Giganti  _et al._ (2018). The EP Kato (1995); Özdemir  _et al._
(2019); Wiersig (2020); Parto  _et al._ (2021); Ding  _et al._ (2022), a
branch point singularity in the complex eigenvalue plane, is crucial in non-
Hermitian physics El-Ganainy  _et al._ (2018); Bergholtz  _et al._ (2021);
Ashida  _et al._ (2021), where avoided crossing occurs in its vicinity Heiss
(2000, 2012). With these concepts, we formulate the resonance of QNMs and
provide novel perspectives for exploring gravity. We employ geometric units
with $G=c=1$ throughout.

Kerr QNM frequencies and excitation factors. We first reveal the emergence of
the resonance of Kerr QNMs at avoided crossing and identify their unique
features. Linear perturbation theory in general relativity predicts that the
ringdown GW strain $h=h_{+}-ih_{\times}$ at far distance from perturbed Kerr
black hole is given by a superposition of QNM damped sinusoids Vishveshwara
(1970); Press (1971); Teukolsky (1973); Chandrasekhar and Detweiler (1975):

$\displaystyle h=\sum_{\ell,m,n}\frac{I_{\ell mn}B_{\ell mn}S_{\ell
mn}e^{im\varphi}}{\omega_{\ell mn}^{2}}e^{-i\omega_{\ell mn}t},$ (1)

in a decomposition in terms of spin-weighted spheroidal harmonics $S_{\ell
mn}(\theta,a\omega_{\ell mn})$ Teukolsky (1972, 1973); Press and Teukolsky
(1973). While a complex factor $I_{\ell mn}$ involves an integral depending on
the initial condition, recent studies suggest that for each $(\ell,m)$
multipole $I_{\ell mn}$ does not strongly depend on the overtone index $n$ and
black hole spin Oshita (2021); Cheung  _et al._ (2024). Therefore, the
ringdown GWs are mainly governed by complex QNM frequencies $\omega_{\ell mn}$
and excitation factors $B_{\ell mn}$ defined by Leaver (1986b)

$\displaystyle B_{\ell mn}=\left.\frac{A_{\rm
out}}{2\omega}\left(\frac{\mathop{}\\!\mathrm{d}A_{\rm
in}}{\mathop{}\\!\mathrm{d}\omega}\right)^{-1}\right|_{\omega=\omega_{\ell
mn}},$ (2)

with asymptotic amplitudes $A_{\rm out}$ and $A_{\rm in}$ for the outgoing and
ingoing waves, respectively. $\omega_{\ell mn}$ and $B_{\ell mn}$ depend only
on mass $M$ and spin $a$ of the black hole. Since the mass dependency is a
simple scaling, the dependency on a dimensionless spin parameter $a/M$ is of
primary interest.

Figure 1: Kerr QNM frequencies $\omega_{\ell mn}$ (left) of $(\ell,m)=(2,2)$
from $n=0$ (bottom red) to $7$ (top magenta) for $0\leq a/M\leq 1-10^{-6}$
with arrows indicating the direction of increasing spin. Excitation factors
$B_{\ell mn}$ with $n=5,6$ (middle) and other overtones (right), the latter of
which is normalized. The range $a/M=0.875$ (circle) – $0.915$ (triangle) is
highlighted with small dots with the spacing of $0.005$.

Figure 2: Kerr QNM frequencies $\omega_{\ell mn}$ (top) and excitation factors
$B_{\ell mn}$ (bottom) of $(\ell,m)=(3,1)$ from $n=4$ to $7$ for $0.95\leq
a/M\leq 1-10^{-6}$. Highlighted are the ranges $a/M=0.952$ (diamond) – $0.96$
(square), $0.9715$ (circle) – $0.973$ (triangle), and $0.978$ (plus) – $0.983$
(cross) with small dots with the spacing of $5\times 10^{-4}$, $5\times
10^{-5}$, and $5\times 10^{-4}$, respectively. Black dashed curves represent
the hyperbola (top) and lemniscate (bottom).

We revisit numerical calculation of the QNM frequencies and excitation factors
of Kerr black holes. We closely follow the strategy adopted in Zhang  _et al._
(2013) and implement some technical improvements to achieve high-precision
calculation up to higher overtones which will be presented elsewhere 111In
addition, we also correct some errors found in an intermediate step of Zhang
_et al._ (2013). Consequently, our results of excitation factors and the ones
in Zhang  _et al._ (2013) differ for a factor of
$e^{-i\omega(1-\sqrt{1-(a/M)^{2}})}$.. The dataset is publicly available
Motohashi (2024a).

For $(\ell,m)=(2,2)$ multipole of GWs, most of the Kerr QNM frequencies
uniformly migrate towards the accumulation point at $2M\omega_{\ell mn}=m$
Detweiler (1980); Glampedakis and Andersson (2001); Cardoso (2004); Hod
(2008); Yang  _et al._ (2013a, b). However, as depicted in Fig. 1, only the
fifth overtone suddenly turns over at $a/M\simeq 0.9$ with making a small
knot-shape loop and heads towards an isolated point far from the accumulation
point. This behavior was first pointed out by Onozawa Onozawa (1997) about
three decades ago and has been confirmed by other work Yang  _et al._ (2013b);
Cook and Zalutskiy (2014), but the physical reason has been veiled. While it
has been implicitly assumed that the fifth overtone solely behaves in a
strange manner, here we point out that, while less manifest, the trajectory of
the sixth overtone QNM frequency is slightly distorted when it passes near the
fifth overtone at $a/M\simeq 0.9$.

On the other hand, it was recently found that the absolute values of $B_{225}$
and $B_{226}$ become large for high-spin Kerr black holes Giesler  _et al._
(2019); Oshita (2021), despite the fact that the excitation factors tend to
zero at the extremal spin Ferrari and Mashhoon (1984); Berti and Cardoso
(2006); Zhang  _et al._ (2013) and asymptotically scale as $n^{-1}$ for higher
overtones Andersson (1997); Berti and Cardoso (2006). Again, the physical
reason has been unclear. However, when viewed on the complex plane in Fig. 1,
we notice that the excitation factors actually exhibit a suggestive behavior.
While other overtones follow spiral trajectories, it turns out that $B_{225}$
and $B_{226}$ are against the spirals and are enhanced in almost opposite
directions at $a/M\simeq 0.9$.

The implications of these observations are two-fold: First, this phenomenon
should be viewed as occurring in a pair of two modes rather than a single
mode. Second, there should be an underlying interplay between the
turning/distortion of the QNM frequencies and the strong excitations.

Actually, a similar phenomenon occurs more manifestly for higher multipoles.
In Fig. 2, we see that $(\ell,m)=(3,1)$ modes exhibit the distinctive
behaviors successively between multiple pairs. Among them the pair of $n=4$
and $5$ is the most manifest case, exhibiting a sharp repulsion between the
two QNM frequencies 222Such repulsion was also observed in the QNMs of scalar
field in Kerr Yang  _et al._ (2013b), (anti-)de Sitter black holes Jansen
(2017); Davey  _et al._ (2022); Kinoshita  _et al._ (2024), and charged black
holes Dias  _et al._ (2022a, b); Davey  _et al._ (2024).. In Fig. 2, we also
note that the mild repulsion between $n=6$ and $7$ overtones seems to be a
trigger of the looping trajectory of the sixth overtone found in Onozawa
(1997); Cook and Zalutskiy (2014). Further, during these peculiar behaviors of
QNM frequencies, the corresponding pairs of excitation factors exhibit
symmetric amplifications. The sharper the repulsion between two QNM
frequencies are, the more strongly and clearly point symmetrically the
corresponding pair of excitation factors are amplified.

Interestingly, for the most manifest case in Fig. 2, we find that the QNM
frequencies and excitation factors almost follow the hyperbola and the
lemniscate of Bernoulli, respectively. When written down in the polar
coordinates with $z=re^{i\theta}$, the hyperbola $r^{-2}=\cos 2\theta$ and
lemniscate $r^{2}=\cos 2\theta$ are inverses of each other. Albeit
approximate, it is remarkable that such intimately related curves are inherent
in QNM spectrum of Kerr GWs.

Figure 3: Absolute values of the difference between the excitation factors
(blue solid) and the inverse of the difference between QNM frequencies (orange
dashed), the latter of which is normalized to match the maxima, with a fit by
a quarter-power Lorentzian (black dotted).

We show in Fig. 3 that the inverse proportionality between the difference of
excitation factors $\Delta B$ and the difference of QNM frequencies
$\Delta\omega$ is actually a common feature of the pair excitations, providing
insight that they belong to the same underlying phenomenon. As expected, the
inverse proportionality is more manifest for the case of sharper repulsion and
stronger excitation. Further, we find that the peaks in Fig. 3, especially the
sharp ones, demonstrate an excellent agreement with a quarter-power Lorentzian
$f(a/M)$, where $f(x)=f_{0}[\gamma^{2}/\\{(x-x_{0})^{2}+\gamma^{2}\\}]^{1/4}$
with positive parameters $f_{0},\gamma,x_{0}$, rather than the Lorentzian
itself. Not only the absolute value, we also confirm that a product $\Delta
B\Delta\omega$ remains almost constant as a complex number during the resonant
excitation.

We find that this phenomenon ubiquitously occurs for Kerr QNMs, i.e., GWs with
other multipoles, and scalar and electromagnetic fields. Further, even in QNMs
away from Kerr, we find that the resonances with the same features show up
(see Supplemental Material). These are tantalizing hints to a universal nature
of QNMs.

Theory of quasinormal modes. Below we provide a theoretical explanation for
this phenomenon by developing a theory of QNMs. Let us consider the QNMs of a
general master equation for perturbations $\psi$ around a black hole after
Fourier and harmonic expansions,

$\displaystyle\left(\frac{\mathop{}\\!\mathrm{d}^{2}}{\mathop{}\\!\mathrm{d}x^{2}}+U\right)\psi=0,$
(3)

where $U$ is an effective potential and $x$ denotes the tortoise coordinate
with $x\to\pm\infty$ corresponding to the spatial infinity and event horizon
of the black hole, respectively. For nonrotating black holes,
$U=\omega^{2}-V(x)$ Regge and Wheeler (1957); Zerilli (1970) and hence (3)
reads $H\psi=\omega^{2}\psi$ with an effective Hamiltonian
$H=-\frac{\mathop{}\\!\mathrm{d}^{2}}{\mathop{}\\!\mathrm{d}x^{2}}+V(x)$,
which is analogous to the Schrödinger equation in quantum mechanics. For
radial perturbation equation of rotating black holes, $U$ depends on
$x,\omega$ and other parameters Teukolsky (1972, 1973), so (3) defines a
nonlinear eigenvalue problem. $U$ changes as the spin parameter varies,
causing the migration of QNM frequencies.

A crucial difference of QNMs from the bound state in quantum mechanics
originates from the boundary condition. Instead of decaying at infinity, the
QNM wave function is required to satisfy the in/outgoing wave condition
$\psi\to e^{\pm i\omega x}$ as $x\to\pm\infty$. This Siegert boundary
condition discretizes the QNM eigenvalues as complex number with negative
imaginary part. Hence, the QNM wave function diverges at $x\to\pm\infty$ (when
viewed at constant time hypersurface), and the standard inner product for
bound states is not well-defined for QNMs. Several kinds of definitions of the
inner product for QNMs of black holes have been considered Ching  _et al._
(1995); Leung  _et al._ (1997, 1998); Yang  _et al._ (2015); Zimmerman  _et
al._ (2015); Jaramillo  _et al._ (2021); Gasperin and Jaramillo (2022); Yang
and Zhang (2023); Green  _et al._ (2023); London (2023); London and Gurevich
(2023); Ma and Yang (2024).

In quantum mechanics, this kind of system is known as the Schrödinger problem
of open systems and has been utilized for studies of quantum resonances, for
which the Hamiltonian is not Hermitian Gamow (1928); Landau and Lifshitz
(1977); Kukulin  _et al._ (1989); Moiseyev (2011). More recently, the non-
Hermitian physics has been extensively studied both from theoretical and
experimental point of view El-Ganainy  _et al._ (2018); Bergholtz  _et al._
(2021); Ashida  _et al._ (2021). Here, we define a regular inner product, or
more precisely biorthogonal product, for QNMs inspired by the case of the
quantum resonance states Kukulin  _et al._ (1989); More (1971a, b); Sternheim
and Walker (1972). It corresponds to a generalization of Ching  _et al._
(1995); Leung  _et al._ (1997, 1998) to the case with (3) including rotating
black holes. In this formulation, QNM wave function $\psi_{n}$ with eigenvalue
$\omega_{n}$ corresponds to the resonance state, for which we introduce so-
called anti-resonance (or adjoint) state
$\tilde{\psi}_{n}\coloneqq\psi_{n}^{*}$ with eigenvalue
$\tilde{\omega}_{n}\coloneqq-\omega_{n}^{*}$. We then define the biorthogonal
product for QNMs along a contour $\mathcal{C}$ as

$\displaystyle(\tilde{\psi}_{n}|\psi_{k})\coloneqq\int_{\mathcal{C}}\mathop{}\\!\mathrm{d}{x}\frac{U_{n}-U_{k}}{\omega_{n}^{2}-\omega_{k}^{2}}\psi_{n}\psi_{k}-\frac{[\psi_{n}\psi^{\prime}_{k}-\psi^{\prime}_{n}\psi_{k}]_{\partial\mathcal{C}}}{\omega_{n}^{2}-\omega_{k}^{2}},$
(4)

where $U_{n}=U|_{\omega=\omega_{n}}$. From (3), it is straightforward to show
that the biorthogonality exactly holds for any $\mathcal{C}$, i.e.,
$(\tilde{\psi}_{n}|\psi_{k})=0$ when $n\neq k$. Formally taking the limit
$\omega_{k}\to\omega_{n}$, we define the “norm squared” for QNMs as

$\displaystyle\mathcal{A}_{n}^{2}\coloneqq\int_{\mathcal{C}}\mathop{}\\!\mathrm{d}{x}\frac{\partial
U_{n}}{\partial\omega_{n}^{2}}\psi_{n}^{2}-\left.\frac{\partial}{\partial\omega_{k}^{2}}[\psi^{\prime}_{n}\psi_{k}-\psi_{n}\psi^{\prime}_{k}]_{\partial\mathcal{C}}\right|_{\omega_{k}=\omega_{n}},$
(5)

which is in general complex 333As a special case, if we choose a contour
$\mathcal{C}$ along the real axis, (5) reduces to the Zel’dovich
regularization Zel’dovich (1961) (see also earlier work Kapur  _et al._
(1938)).. We can analytically show that $\mathcal{A}_{n}^{2}$ satisfies a
useful formula Motohashi (2024b)

$\displaystyle B_{n}=\frac{iA_{\rm out}^{2}}{2\omega_{n}\mathcal{A}_{n}^{2}},$
(6)

which allows for an alternative way to calculate the excitation factor $B_{n}$
defined by (2). The equivalence between (6) and (2) is also numerically
crosschecked for various specific cases including Schwarzschild and Kerr black
holes. The formula (6) tells us that the excitation factor $B_{n}$ becomes
large when the norm squared $\mathcal{A}_{n}^{2}$ becomes small, which
precisely occurs at the avoided crossing near the EP.

Resonant excitation at avoided crossing. To investigate how QNM frequencies
and excitation factors change when two QNM frequencies get close due to the
variation of $U$, let us regard the continuous changes as an accumulation of
infinitesimal changes $\delta U$. If a mode $\omega_{n}$ is far from
degeneracy, it is straightforward to generalize the Rayleigh-Schrödinger-like
perturbation theory for resonance state Zel’dovich (1961); Perelomov and
Zel’dovich (1998) to QNMs and obtain Motohashi (2024b)

$\displaystyle\delta\omega_{n}=-\frac{(\tilde{\psi}_{n}|\delta
U_{n}|\psi_{n})}{2\omega_{n}(\tilde{\psi}_{n}|\psi_{n})}=iB_{n}\frac{(\tilde{\psi}_{n}|\delta
U_{n}|\psi_{n})}{A_{\rm out}^{2}},$ (7)

which reveals another role of the excitation factor $B_{n}$, i.e., defining
the sensitivity of $\delta\omega_{n}$ to $\delta U$.

The avoided crossing, where two (or more) eigenvalues are close to degenerate,
requires a special care 444Note that the similarity between the avoided
crossing and repulsion of QNM eigenvalues was pointed out in Dias  _et al._
(2022a, b); Davey  _et al._ (2024) in the studies of charged black holes. Here
we examine QNM frequencies and also excitation factors and derive the resonant
excitation.. We employ a simple analysis of bifurcation theory of eigenvalues
about EP Heiss (2000); Seyranian and Mailybaev (2003) to capture the essential
features of the resonance. We focus on the nonrotating black hole case with
$U=\omega^{2}-V(x)$ 555The rotating black hole case may be investigated by
applying a prescription of auxiliary eigenvalue recently employed in Isobe
_et al._ (2024) in the context of the bulk-edge correspondence. and assume
that two QNM overtones are close $\omega_{1}\approx\omega_{2}$ and they are
isolated from other QNMs so that we can approximately treat them as a two-
level system. Suppose a small change $V\to V+\delta V$ as a part of continuous
changes, causing a change of the QNM frequencies $\omega_{i}\to\omega_{\pm}$,
which can be obtained as eigenvalues of the Hamiltonian
$H_{ij}=(\tilde{\psi}_{i}|H|\psi_{j})/(\mathcal{A}_{i}\mathcal{A}_{j})$:

$\displaystyle\omega^{2}_{\pm}=\mathcal{E}_{c}\pm\sqrt{\mathcal{E}_{d}^{2}+\Delta^{2}},$
(8)

where $\mathcal{E}_{c,d}\coloneqq(\mathcal{E}_{1}\pm\mathcal{E}_{2})/2$,
$\mathcal{E}_{i}\coloneqq\omega_{i}^{2}+\delta\omega_{i}^{2}$ with $i=1,2$,
and $\Delta\coloneqq(\tilde{\psi}_{1}|\delta
V|\psi_{2})/(\mathcal{A}_{1}\mathcal{A}_{2})$. When $\Delta$ is negligible,
(8) recovers $\mathcal{E}_{i}$, the result of perturbation theory. On the
other hand, $\omega_{\pm}$ are degenerate if
$\mathcal{E}_{d}^{2}+\Delta^{2}=0$ is satisfied, which defines EP. Since this
condition involves complex variables, it is in general not possible to be
satisfied by tuning a single real parameter, resulting the avoided crossing.

Let us consider the case where continuous variation of $V$ is driven by a
single parameter. At each point except the avoided crossing the perturbation
theory is valid, and hence $\omega_{i}^{2}$ follow $\mathcal{E}_{i}$. Suppose
that the perturbation theory predicts the two modes approach on the same line,
for which, denoting $\mathcal{E}_{d}=pe^{i\alpha}$ and
$\mathcal{E}_{c}=\omega_{\rm EP}^{2}$, $p$ is a real parameter and $\alpha$
and $\omega_{\rm EP}^{2}$ are constant. We also neglect the variation of
$\Delta=qe^{i\beta}$. The single-parameter degree of freedom of $V$ is thus
represented by $p$. Denoting $\omega_{\pm}^{2}-\omega_{\rm EP}^{2}=x+iy$, it
is straightforward to derive $X^{2}-Y^{2}=q^{2}\sin 2(\alpha-\beta)$, where
$X+iY$ is defined by rotation of $x+iy$ by $\pi/4-\alpha$. Unless
$\alpha=\beta$, which requires a tuning of an additional parameter,
$\omega_{\pm}^{2}-\omega_{\rm EP}^{2}$ follows a hyperbolic avoided crossing,
and so does $\omega_{\pm}-\omega_{\rm EP}$ itself, since this occurs at a
small region near EP. A simple calculation shows
$|\omega_{\pm}^{2}-\omega_{\rm EP}^{2}|\approx({\rm const}+p^{2})^{1/4}$ at
the peak when $p\ll 1$, which means $|\omega_{+}-\omega_{-}|^{-1}$ obeys the
quarter-power Lorentzian.

On the other hand, QNM eigenstates $\psi_{\pm}$ associated to $\omega_{\pm}$
are given by $\psi_{+}=\psi_{1}+T\psi_{2}$ and $\psi_{-}=-T\psi_{1}+\psi_{2}$,
where $T=1/(\delta+\sqrt{\delta^{2}-1})$ is governed by a dimensionless
complex parameter $\delta=\sqrt{\mathcal{E}_{d}^{2}/\Delta^{2}+1}$. If the two
modes are close and then $\delta\approx 0$ and
$\mathcal{A}_{1}^{2}\approx\mathcal{A}_{2}^{2}$ are satisfied, the eigenstates
are almost degenerate $\psi_{-}\approx i\psi_{+}$, and the norm squared nearly
vanish as $\mathcal{A}^{2}_{\pm}\propto\pm\delta$. These behaviors are nothing
but what is expected at EP Özdemir  _et al._ (2019); Wiersig (2020); Parto
_et al._ (2021); Ding  _et al._ (2022). With (6) and (8), we obtain
$B_{\pm}\propto\mathcal{A}^{-2}_{\pm}\propto(\omega_{\pm}^{2}-\omega_{\rm
EP}^{2})^{-1}$. Hence, the excitation factors are amplified following the
inverse of the hyperbola, i.e., the lemniscate of Bernoulli, and the resonance
peak obeys the quarter-power Lorentzian.

Conclusions and Discussion. When two QNM frequencies approach on the complex
plane, they are anomalously excited. We clarified this resonance phenomenon by
developing high-precision numerical analysis and theoretical framework of QNMs
in the language of non-Hermitian physics. The resonance of Kerr QNMs not only
allows us to decipher a long-standing mystery in black hole physics but also
stands as a novel beacon for characterizing black hole spacetime geometry.
Deviations from Kerr, whether due to astrophysical effects or correction to
general relativity, will manifest in the resonances in QNMs. Our findings thus
pave the way for rigorous examinations of black holes and the exploration of
new physics in gravity.

Acknowledgments. This work was supported by Japan Society for the Promotion of
Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) Grant No.
JP22K03639.

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## Supplemental material

In the main text we focus on the resonance at avoided crossing in the vicinity
of EP of Kerr QNMs. To argue that this phenomenon is a universal feature of
QNMs, it is helpful to study a simple toy model away from Kerr QNMs. Indeed,
the simplest rectangular barrier potential played an important role in a
pioneering work Chandrasekhar and Detweiler (1975) in the early days of QNM
research. Along the same line, here we consider a simple toy model

$\left(\frac{d^{2}}{dx^{2}}+\omega^{2}-V(x)\right)\psi=0,$ (9)

with double rectangular barriers:

$V(x)=\begin{cases}V_{0},&0\leq x\leq d,\\\ V_{1},&b\leq x\leq b+d,\\\ 0,&{\rm
otherwise},\end{cases}$ (10)

with $0\leq d\leq b$ and $V_{0}\geq 0$ and $V_{1}\geq 0$. We define the QNM
frequencies by requiring the purely right-/left-going boundary condition,
i.e., $\psi\to e^{\pm i\omega x}$, at $x>b+d$ and $x<0$, respectively. We can
then calculate the excitation factor by (2) or (6). This toy model allows us
to examine the intrinsic behavior of QNMs and excitation factors in a simple
and transparent manner.

Here, we introduce a hierarchy $V_{1}\ll V_{0}$ and investigate the behavior
of QNM frequencies and excitation factors with respect to the variation of
$b$. For small values of $b$, the QNM spectrum is almost the same as the
unperturbed one with $V_{1}=0$, but as we increase $b$, the QNM frequencies
migrate on the complex frequency plane.

The QNMs for a similar setup was investigated in Barausse  _et al._ (2014);
Cheung  _et al._ (2022); Kyutoku  _et al._ (2023) as the simplest variation of
the “flea on the elephant” effect Jona-Lasinio  _et al._ (1981); Graffi  _et
al._ (1984); Simon (1985); Santarsiero and Gori (2019), which has been
explored recently in the context of the spectral instability Nollert (1996);
Barausse  _et al._ (2014); Daghigh  _et al._ (2020); Jaramillo  _et al._
(2021); Qian  _et al._ (2021); Cheung  _et al._ (2022); Jaramillo  _et al._
(2022); Berti  _et al._ (2022); Kyutoku  _et al._ (2023); Cardoso  _et al._
(2024). It has been known that small deformation of potential at far distance
causes a drastic change of QNM frequencies while changes in the time-domain
ringdown signal shows up only at late time. However, the behavior of the
excitation factors has not been clarified.

Let us classify the QNM frequencies for the double rectangular barriers (10)
into two categories: The first sequence $\omega^{(1)}_{n}$ $(n=0,1,2,\cdots)$
denotes modes that are smoothly connected to the unperturbed modes with
$V_{1}=0$, whereas the second sequence $\omega^{(2)}_{k}$ $(k=0,1,2,\cdots)$
denotes modes that show up when $V_{1}>0$ and correspond to the resonance
frequencies for the effective cavity between two barriers. For small $b$, the
first sequence modes follow the logarithmic spiral around the unperturbed
modes which is consistent with Leung  _et al._ (1997). By using the
perturbation theory formula (7), we obtain

$\omega^{(1)}_{n}\simeq\bar{\omega}_{n}-\frac{i\bar{B}_{n}V_{1}\sin\bar{\omega}_{n}d}{\bar{\omega}_{n}}e^{i\bar{\omega}_{n}(2b+d)},$
(11)

where $\bar{\omega}_{n}$ and $\bar{B}_{n}$ are unperturbed QNM frequencies and
excitation factors, respectively. On the other hand, for the second sequence,
we can derive a simple analytical expression under $V_{1}\ll V_{0}$ as

$\omega^{(2)}_{k}\simeq\frac{1}{2b}\left[(2k+1)\pi-i\ln\frac{V_{0}}{V_{1}}\right].$
(12)

Thus, as we increase $b$, the second sequence modes accumulate to the origin
of the complex frequency plane, during which the second sequence modes (or
more generally, migrating modes including modes that were originally belonging
to the first sequence) approach to the first sequence modes, which then begin
to migrate together with the second sequence modes.

We find that when two QNM frequencies approach each other during the
migration, they exhibit the same behavior observed for the Kerr QNMs; The two
QNM frequencies repel each other, during which the excitation factors are
point-symmetrically amplified. In the left and middle columns of Fig. 4, we
depict the trajectories of QNM frequencies and excitation factors with respect
to the variation of $b$. In addition to $b$, we control the second parameter
$V_{1}$ to see how the avoided crossing and resonance change. The cases with
$V_{1}d^{2}=4.30\times 10^{-3}$ and $4.31\times 10^{-3}$ demonstrate sharp
resonances, where the trajectories follow the hyperbola and lemniscate. The
two cases switch about 90 degrees on the complex plane from each other, which
is predicted at the sign flip of the RHS of $X^{2}-Y^{2}=q^{2}\sin
2(\alpha-\beta)$ in the main text. The cases with $V_{1}d^{2}=3.50\times
10^{-3}$ and $5.50\times 10^{-3}$ show mild resonances. The trajectories of
the QNM frequencies and excitation factors can be regarded as distorted
versions of the hyperbola and lemniscate, respectively. These sharp and mild
resonances reproduce what we observed in the Kerr QNM frequencies and
excitation factors in the main text.

We also examine the characteristic resonant peak appearing in the difference
of the excitation factors and the inverse of the difference of the QNM
frequencies. As shown in the right column of Fig. 4, the peaks are well fitted
by the quarter-power Lorentzian, same as the Kerr case. For the sharp
resonances, the relative difference between any pair of $\Delta B$,
$1/\Delta\omega$ (normalized by multiplying a numerical constant), and the
quarter-power Lorentzian remains $\mathcal{O}(1)$% in the plotted region. The
mild resonances are still better fitted by the quarter-power Lorentzian over
the Lorentzian itself.

Note also that the resonance phenomenon occurs between the fundamental mode
and third overtone of the first sequence. It implies that, while the
resonances for Kerr QNMs studied in the main text occur for higher overtones,
the resonances involving the fundamental mode and/or lower overtones are in
general not prohibited. Indeed, we analyze the double rectangular barriers
with various parameter sets and confirm the resonances for various pairs of
two modes approaching each other. It suggests that the resonance is a
universal feature of QNMs.

Figure 4: Left and middle columns: QNM frequencies (left column) and
excitation factors (middle column) for the double rectangular barriers (10).
The fundamental mode $\omega^{(1)}_{0}$ (red) and third overtone
$\omega^{(1)}_{3}$ (green) of the first sequence are highlighted, while
overtones up to the seventh in the first (colored) and second (black)
sequences are depicted. We set parameters $V_{0}d^{2}=16$ and
$V_{1}d^{2}=(3.50,4.30,4.31,5.50)\times 10^{-3}$ (from first to fourth row)
and depict trajectories with respect to the variation of $b/d$ in the range
$1\leq b/d\leq 15$ with arrows indicating the direction of increasing $b/d$.
The logarithmic spiral (black dotted), hyperbola (black dashed), and
lemniscate of Bernoulli (black dashed) approximate the trajectories well.
Right column: $|B^{(1)}_{0}-B^{(1)}_{3}|$ (blue solid) and
$1/|\omega^{(1)}_{0}-\omega^{(1)}_{3}|$ (orange dashed), the latter of which
is multiplied by a numerical constant to match the maxima. The peak can be
well fitted by a quarter power of Lorentzian (black dotted) rather than the
Lorentzian itself (gray dotted).