# Kibble-Zurek Mechanism and Beyond: Lessons from a Holographic Superfluid
Disk

Chuan-Yin Xia  Center for Gravitation and Cosmology, College of Physical
Science and Technology, Yangzhou University, Yangzhou 225009, China  Hua-Bi
Zeng<EMAIL_ADDRESS>Center for Theoretical Physics, Hainan
University, Haikou 570228, China  Center for Gravitation and Cosmology,
College of Physical Science and Technology, Yangzhou University, Yangzhou
225009, China  András Grabarits Department of Physics and Materials Science,
University of Luxembourg, L-1511 Luxembourg, Luxembourg  Adolfo del Campo
<EMAIL_ADDRESS>Department of Physics and Materials Science, University
of Luxembourg, L-1511 Luxembourg, Luxembourg  Donostia International Physics
Center, E-20018 San Sebastián, Spain

###### Abstract

The superfluid phase transition dynamics and associated spontaneous vortex
formation with the crossing of the critical temperature in a disk geometry is
studied in the framework of the $AdS/CFT$ correspondence by solving the
Einstein-Abelian-Higgs model in an $AdS_{4}$ black hole. For a slow quench,
the vortex density admits a universal scaling law with the cooling rate as
predicted by the Kibble-Zurek mechanism (KZM), while for fast quenches, the
density shows a universal scaling behavior as a function of the final
temperature, that lies beyond the KZM prediction. The vortex number
distribution in both the power-law and saturation regimes can be approximated
by a normal distribution. However, the study of the universal scaling of the
cumulants reveals non-normal features and indicates that vortex statistics in
the newborn superfluid is best described by the Poisson binomial distribution,
previously predicted in the KZM regime [Phys. Rev. Lett. 124, 240602 (2020)].
This is confirmed by studying the cumulant scalings as a function of the
quench time and the quench depth. Our work supports the existence of a
universal defect number distribution that accommodates the KZM scaling, its
breakdown at fast quenches, and the additional universal scaling laws as a
function of the final value of the control parameter.

##  I Motivation

The description of strong coupling problems in quantum many-body systems is
challenging in equilibrium and far from it. In this context, the use of gauge-
gravity duality, AdS/CFT duality, or holography offers an exciting prospect by
relating a quantum field theory without gravity to a classical gravity with
one more dimension [1, 2, 3]. Strongly coupled systems can be calculated non-
perturbatively and accurately within the framework of weak-coupling gravity
theory, and the use of holographic techniques has already proved useful in
condensed matter theory (AdS/CMT)[4, 5, 6, 7, 8].

An important application of the holographic setup is the description of
strongly-coupled systems with broken continuous symmetries such as a
superconductor [9, 10, 11]. Such formulation not only sheds insights into the
equilibrium state of superconductors and superfluids but also in the study of
vortex dynamics [12, 13, 14, 15, 16, 17, 18, 19], and more generally, in the
tenets of nonequilibrium statistical mechanics, such as the Kibble-Zurek
mechanism (KZM) [20, 21, 22, 23, 24, 25, 26, 27, 28]. The KZM was initially
proposed in cosmology and then applied in other fields, ranging from condensed
matter physics to quantum computing [29, 30, 31, 32, 33, 34, 35]. It links the
dynamics of continuous phase transitions to the equilibrium critical exponent,
explains the generation of topological defects, and predicts that their
density follows a universal power law that experiments and simulations can
test. According to KZM, when a system is quenched by modulating a control
parameter $T$ across a critical point $T_{c}$ in a finite quench time scale
$\tau_{Q}$, topological defects form as a result of the breaking of
adiabaticity. The latter is unavoidable when crossing a continuous phase
transition in the thermodynamics limit due to the divergence of relaxation
time, which is responsible for the critical slowing down. Specifically, KZM
exploits the scaling relations for the relaxation time $\tau$ and the
correlation length $\xi$, encoded in the power laws
$\tau=\tau_{0}|\epsilon|^{-z\nu}$ and $\xi=\xi_{0}|\epsilon|^{-\nu}$,
respectively. Here, $\nu$ is the correlation-length critical exponent, and $z$
is known as the dynamic critical exponent. The parameters $\tau_{0}$ and
$\xi_{0}$ are microscopic constants, and $\epsilon$ determines the proximity
to the critical point, being defined as

$\epsilon(T)=\frac{(T_{c}-T)}{T_{c}}.$ (1)

Consider the modulation in time of the control parameter $T(t)$, and thus, of
$\epsilon(t)=\epsilon[T(t)]$. According to the adiabatic-impulse
approximation, in the high-symmetry phase, the system adapts adiabatically to
the modulation of the control parameter $\epsilon(T)$ until the time left
until reaching the critical point is comparable with the instantaneous
relaxation time. In the subsequent stage of the quench, macroscopic degrees of
freedom are then effectively frozen until, once in the broken-symmetry phase,
the order parameter starts to grow in a time scale set by the freeze-out time.
The latter is identified by matching the instantaneous relaxation time is
equal to the so-called freeze-out time. For a linear quench near the critical
point,

$\displaystyle T(t)=\begin{cases}T_{c}(1-t/\tau_{Q})&0\leq t<t_{f}\\\
T_{f}&t\geq t_{f}\end{cases}.$ (2)

The time $t_{f}$ is the time when the quench is completed, satisfying
$T(t_{f})=T_{f}$ that is, $t_{f}=\tau_{Q}(1-T_{f}/T_{c})$. The freeze-out
timescale, in this case, reads

$\hat{t}=(\tau_{0}\tau_{Q}^{z\nu})^{\frac{1}{1+z\nu}}.$ (3)

KZM estimates the average domain size after the finite-time crossing of the
phase transition by the value of the equilibrium correlation length
$\hat{\xi}$ at the freeze-out time $\hat{t}$, i.e.,
$\hat{\xi}\propto\xi(T(\hat{t}))=\xi_{0}(\tau_{Q}/\tau_{0})^{\frac{\nu}{1+z\nu}}$.
Topological defects are formed with a certain probability at the junction
between adjacent domains. As a result, the average defect number $n$ inherits
a power-law dependence on the quenching time $\tau_{Q}$

$n\propto\frac{1}{\hat{\xi}^{d}}\propto\tau_{Q}^{-\frac{d\nu}{1+z\nu}},$ (4)

where $d$ is the spatial dimension. This is the central prediction of the KZM.
Its verification has motivated continuous research efforts over decades. Early
experiments focused on probing the spontaneous formation of defects in several
systems, such as liquid crystals [36, 37, 38], and superfluid helium [39, 40,
41, 42]. Subsequent efforts focused on verifying the power-law scaling, using
Josephson junctions [43, 44, 45, 46], thin film superconductors [47, 48],
linear optical quantum simulators [49], trapped ions [50, 51, 52], quantum
annealers [53, 54] and ultra-cold gases [55, 56, 57, 58, 59, 60].

This work explores the universal character of the critical dynamics beyond the
KZM using a holographic superfluid phase transition as a test bed. It
establishes that the mean density, considered by the KZM, and the complete
defect number distribution are universal. Such universality is not restricted
to slow quenches with low defect density where the KZM applies but extends to
arbitrary values of the quench time. In particular, it holds even for sudden
quenches, for which the defect density saturates at a plateau value, leading
to a universal breakdown of the KZM. Our work specifically demonstrates that
the defect number distribution is described by a Poisson binomial distribution
and that its cumulants exhibit power-law scaling laws as a function of the
quench time and the depth of the quench. These predictions are experimentally
testable by probing the spontaneous vortex formation scenarios in newborn
superfluids.

The following section details the relevant background on physics beyond KZM,
deviations from KZM, and the role of boundary conditions. A comparison of our
findings with previous literature is provided in Sec. III. Section IV contains
the holographic setting, which can be skipped by readers solely interested in
the universal aspects of critical dynamics. The universal dynamics for slow
and fast quenches is established in Sec. V. The discussion VII establishes the
Poisson binomial distribution as the universal distribution that describes
defect statistics as a function of the quench time and depth in and out of the
KZM regime. We close with a summary and outlook in Sec. VIII.

##  II Background

###  II.1 Beyond KZM physics

Beyond the KZM predictions, the fluctuations of the number of topological
defects have been predicted to be universal. In one-dimensional systems, the
point-like defect number distribution is well described by can usually be
described by a binomial or Poisson-binomial distribution [61, 62]. As a
result, the universal scaling with the quench time extends beyond the KZM
prediction to the variance and any other higher-order cumulant [61, 62], as
confirmed by numerical simulations [61, 62, 63, 64, 65, 27, 28, 24] and
experiments [66, 67, 68, 54]. KZM and beyond-KZM physics have also been
established by strongly-coupled systems described by holography [20, 16, 22,
69, 23, 70, 71, 24, 25, 26, 27, 28, 21], showcasing their universality.
Specifically, numerical simulations have been reported for one-dimensional
[21, 23, 24, 27, 28] and two-dimensional [20, 25, 26] holographic
superconductors and superfluids. But for some features [20, 21], all results
are rather similar to those of the mean-field theory and verify the validity
of KZM in holographic systems.

###  II.2 Deviations from KZM physics and fast-quench universality

Naturally, the validity of the KZM scaling laws is limited by the onset of
adiabaticity that occurs when $\hat{\xi}$ is comparable to the system size.
There are also reports on the impact of finite size on KZM [72, 73, 23]. When
the system size is close to the correlation length, the number of topological
defects will deviate from the prediction of KZM. Numerical studies of the
finite-size effects [23] show deviations of KZM universality, explaining
experimental findings in Ref. [73].

By contrast, when the quenching time is much shorter than the time scale in
which the order parameter grows, the density of topological defects no longer
satisfies the universal power law predicted by the KZM. Instead, it saturates
at a constant value called the plateau. This phenomenon was found in
experiments [59, 60, 68, 74] after being observed in numerical simulations
[72, 75, 21, 76, 23, 70, 69, 71, 70, 22, 25, 26]. This universal breakdown of
the KZM scaling at fast quenches is to be expected, as its derivation is
restricted to slow quenches near the adiabatic regime [33, 34, 77, 35]. The
nature of the plateau, the value of the plateau density, and its onset have
remained a matter of debate. Recently [69], its universal character was
established, along with the fact that the plateau density is independent of
the quench rate, being governed by the equilibrium relaxation time associated
with the final value of the control parameters. These findings have been
further verified in a holographic setting involving a structural phase
transition [71].

For rapid quenches, the end time $t_{f}$ is smaller than the relaxation time,
that is,

$t_{f}\leq\tau(t_{f})=\tau(T_{f}),$ (5)

so the freezing will not occur when $0<t<t_{f}$. As the time of evolution goes
by and $t$ increases, it eventually matches the minimum relaxation time
$\tau(T_{f})$. Thus, the freeze-out time in a rapid quench scales universally
as a function of $\epsilon_{f}=(T_{c}-T_{f})/T_{c}$

$\hat{t}=\tau(T_{f})\propto\epsilon_{f}^{-\nu z},$ (6)

at variance with the KZM prediction (3). In addition, the correlation length
is also determined by the final coupling value $T_{f}$ at the freeze-out time,
leading to the average number of defects

$n\propto\frac{1}{\xi^{d}(T_{f})}\propto\epsilon_{f}^{d\nu}.$ (7)

To sum up, quench protocols that satisfy the constraints of Eq. (5) can be
called rapid. They lead to a non-equilibrium configuration of the system that
is solely determined by the final value of the coupling constant $T_{f}$ and
the equilibrium critical exponents. As a result, the defect density Eq. (7)
forms a plateau in the rapid quenching region, independent of the quench time.
Furthermore, by matching the quench end time $t_{f}$ and the relaxation time
$\tau(T_{f})$, the crossover quenching time $\tau_{Q}^{c1}$ can be obtained,
as shown in Fig. 2 below. It is called the first critical quench rate and
satisfies the following condition

$t_{f}=\tau_{Q}^{c1}(1-T_{f}/T_{c})=\tau(T_{f})=\tau_{0}(T_{c}-T_{f})^{-\nu
z},$ (8)

which yields

$\tau_{Q}^{c1}\propto\epsilon_{f}^{-(\nu z+1)}.$ (9)

These results establish the universality of critical dynamics in rapid
quenches in a holographic framework.

###  II.3 Role of boundary conditions

In addition to these advances, the possible breakdown of KZM and beyond-KZM
predictions has been explored in various scenarios. This includes the effect
of boundary conditions on KZM. While open boundary conditions are natural in
most experiments, numerical simulations are often facilitated by imposing
periodic boundaries. It is further known that the role of boundary conditions
cannot be ignored, regardless of the system size. For instance, the Poincaré-
Hopf theorem dictates that the total topological charge of spontaneously
formed defects needs to vanish identically in manifolds with zero Euler
characteristic [78]. As a result, breaking of $\mathbb{Z}_{2}$ symmetry
restricts defect formation to kink-antikink pairs on a ring topology [79].
Similarly, the thermal quench of a superconducting or superfluid ring leads to
a vanishing winding number [32, 80, 81, 21, 82, 23]. In the same manner,
breaking of $U(1)$ associated with vortex formation is conditioned to
vanishing total vorticity in samples with a torus topology [64, 83]. Results
obtained in a lattice model indicate that even under extremely slow quenching,
periodic, anti-periodic, and free boundary conditions will not suppress the
universal power law scaling for the mean density but may affect high-order
cumulants and the defect number distribution [79]. In studies of KZM physics,
experimental setups often involved open boundaries [39, 40, 41, 42, 47, 48,
55, 56, 57, 58, 59, 60]. However, two-dimensional systems with boundaries have
not been numerically simulated in a holographic setting. In this article, we
use the holographic superfluid model on an open disk geometry [6] as a test
bed for vortex formation and the validity of both KZM and beyond-KZM physics.
As we use Neumann boundary conditions, the Poincaré-Hopf theorem does not
restrict the total vorticity. This geometry involves a boundary at the maximum
radius, making it possible to test whether the finite-size boundary affects
the universal finite-quench scaling behavior.

##  III Findings and comparison to previous works

In this work, we demonstrate the far-from-equilibrium universality in the
defect number distribution both as a function of the quench time and the
quench depth, that is, the final value of the control parameter. Our work
largely transcends the scope of the celebrated KZM, focused on the mean
density of topological defects as a function of the quench time [35].
Likewise, our findings extend the notion of the universality of critical
dynamics with respect to previous works on beyond-KZM physics. Specifically,
previous works reporting the universality of the defect number statistics in
classical and quantum systems are restricted to the power-law scaling regime
associated with slow quenches in which KZM holds, in theoretical [61, 62, 63,
64, 65, 27, 28, 24] and experimental studies [66, 67, 68, 54]. Our manuscript
established the universality of the defect number distribution at arbitrarily
quench rates for the first time, from the slow to the sudden regimes.
Specifically, we establish that the Poisson binomial distribution describes
the universal statistics of spontaneously formed point-like topological
defects in all regimes. It is a curious accident that previous studies have
evaded the universal signatures of the defect number statistics in the power-
law scaling regime for decades, possibly due to the lack of a theoretical
framework, despite the signal being in the noise often measured in the
experiments. Once noticed, such a universal distribution is compatible with
the observation that the KZM regime is restricted to slow quenches with very
low defect densities. Our results are remarkable in that they establish the
universality of the defect number distribution even at the highest density
values achievable by spontaneous defect formation through sudden quenches that
are associated with the saturation plateau and the breakdown of the KZM.

##  IV Model of holography superfluid disk and numerical scheme for phase
transition dynamics

A simple action for a holographic superfluid consists of a complex scalar
field $\Psi$ with mass $m$, minimally coupled to a $U(1)$ gauge field
$A_{\mu}$ [10, 9, 11],

$S=\int
d^{4}x\sqrt{-g}\Big{[}-\frac{1}{4}F^{2}-|D\Psi|^{2}-m^{2}|\Psi|^{2}\Big{]},$
(10)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$,
$D_{\mu}=\partial_{\mu}-iqA_{\mu}$ with charge $q$. The theory can be defined
in a $AdS_{4}$ black hole background with Eddington-Finkelstein coordinates,

$ds^{2}=\frac{\ell^{2}}{u^{2}}\left(-f(u)dt^{2}-2dtdu+dr^{2}+r^{2}d\theta^{2}\right),$
(11)

in which $\ell$ is the AdS radius, $u$ is the AdS radial coordinate of the
bulk and $f(u)=1-(u/u_{h})^{3}$. Thus, $u=0$ is the AdS boundary, while
$u=u_{h}$ is the horizon; $r$ and $\theta$ are, respectively, the radial and
angular coordinates of the dual $2+1$ dimensional boundary, which is a
holographic superfluid disk as in Ref. [12, 13]. The Hawking temperature is
$T=3/(4\pi u_{h})$. From dimensional analysis, the ratio $T/\mu$ between the
Hawking temperature $T$ and the chemical potential $\mu=A_{t}|_{u=0}$ is
dimensionless. The temperature on the boundary can be expressed as

$T=T_{c}\frac{\mu_{c}}{\mu}.$ (12)

Thus, increasing the charge density is equivalent to decreasing the
temperature. For simplicity, the probe limit is adopted in this work by
assuming that the matter fields do not affect the gravitational fields. After
rescaling $\ell=u_{h}=1$, the equations of motions (EoMs) can be written as

$(-D^{2}+m^{2})\Psi=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \partial_{\mu}F^{\mu\nu}=J^{\nu},$ (13)

where $J^{\mu}=i(\Psi^{*}D^{\mu}\Psi-\Psi D^{\mu}\Psi^{*})$ is the bulk
current. Furthermore, the axial gauge $A_{u}=0$ is adopted as in [11]. Without
loss of generality, we set $m^{2}$ to $-2$.

The fully expanded equation of motion can be written as

$\begin{split}-\left(\partial_{u}\left(f\partial_{u}\Psi\right)+i\left(\partial_{u}A_{t}\right)\Psi+2iA_{t}\partial_{u}\Psi\right)\\\
+\left(-\partial_{r}^{2}\Psi+i\left(\partial_{r}A_{r}\right)\Psi+2iA_{r}\partial_{r}\Psi-\frac{1}{r}\partial_{r}\Psi\right)\\\
+\frac{1}{r^{2}}\left(-\partial_{\theta}^{2}\Psi+i\left(\partial_{\theta}A_{\theta}\right)\Psi+2iA_{\theta}\partial_{\theta}\Psi\right)\\\
+\left(A_{r}^{2}+\frac{iA_{r}}{r}+\frac{A_{\theta}^{2}}{r^{2}}+u\right)\Psi+2\partial_{t}\partial_{u}\Psi=0,\end{split}$
(14)

$\begin{split}\partial_{u}^{2}A_{t}-\partial_{u}\partial_{r}A_{r}-\frac{1}{r}\partial_{u}A_{r}-\frac{1}{r^{2}}\partial_{u}\partial_{\theta}A_{\theta}\\\
-i\left(\Psi^{*}\partial_{u}\Psi-\Psi\partial_{u}\Psi^{*}\right)=0,\end{split}$
(15)

$\begin{split}-\frac{1}{r}\partial_{u}A_{r}-\partial_{t}\partial_{u}A_{t}-\partial_{t}\partial_{r}A_{r}-\frac{1}{r^{2}}\partial_{t}\partial_{\theta}A_{\theta}+\frac{f}{r}\partial_{u}A_{r}\\\
+f\partial_{u}\partial_{r}A_{r}+\frac{f}{r^{2}}\partial_{u}\partial_{\theta}A_{\theta}+\partial_{r}^{2}A_{t}+\frac{1}{r}\partial_{r}A_{t}+\frac{1}{r^{2}}\partial_{\theta}^{2}A_{t}\\\
-i\left(\Psi^{*}\partial_{t}\Psi-\Psi\partial_{t}\Psi^{*}\right)-2A_{t}\Psi^{*}\Psi\\\
+if\left(\Psi^{*}\partial_{u}\Psi-\Psi\partial_{u}\Psi^{*}\right)=0,\end{split}$
(16)

$\begin{split}2\partial_{t}\partial_{u}A_{r}-\partial_{u}\partial_{r}A_{t}-\partial_{u}\left(f\partial_{u}A_{r}\right)+\frac{1}{r^{2}}\left(\partial_{r}\partial_{\theta}A_{\theta}-\partial_{\theta}^{2}A_{r}\right)\\\
+i\left(\Psi^{*}\partial_{r}\Psi-\Psi\partial_{r}\Psi^{*}\right)+2A_{r}\Psi^{*}\Psi=0,\end{split}$
(17)

$\begin{split}2\partial_{t}\partial_{u}A_{\theta}-\partial_{u}\left(f\partial_{u}A_{\theta}\right)-\partial_{u}\partial_{\theta}A_{t}-\partial_{r}^{2}A_{\theta}+\frac{1}{r}\partial_{r}A_{\theta}\\\
+\partial_{r}\partial_{\theta}A_{r}-\frac{1}{r}\partial_{\theta}A_{r}+i\left(\Psi^{*}\partial_{\theta}\Psi-\Psi\partial_{\theta}\Psi^{*}\right)+2A_{\theta}\Psi^{*}\Psi=0.\end{split}$
(18)

The five partial differential equations Eq. (14)-(18) are not independent, so
we can choose any four of them. In this work, we choose Eq. (14)-(17) as in
Ref. [12]. In addition to the above settings, proper boundary conditions
should be imposed to solve them. We impose the regular boundary conditions of
all physical solutions at the horizon of the black hole. Explicitly, we set
$A_{t}=0$ on the horizon as in [16, 12, 13]. Boundary conditions for other
fields at the horizon are automatically satisfied because they are implicit in
their respective equations rather than being set on an ad hoc basis.

Near the boundary $u=0$, the general solutions take the asymptotic form as,

$\displaystyle A_{\nu}(t,u,r,\theta)$ $\displaystyle=$ $\displaystyle
a_{\nu}(t,r,\theta)+b_{\nu}(t,r,\theta)u+\mathcal{O}(u^{2}),$ (19)
$\displaystyle\Psi(t,u,r,\theta)$ $\displaystyle=$
$\displaystyle\Psi_{1}(t,r,\theta)u+\Psi_{2}(t,r,\theta)u^{2}+\mathcal{O}(u^{3}).$
(20)

From the AdS/CFT dictionary, the coefficients $a_{r,\theta}$ can be treated as
the superfluid velocity along the coordinates $r,\theta$, while $b_{r,\theta}$
are the conjugate currents [84]. The coefficients $a_{t}$ and $b_{t}$ are
respectively interpreted as the chemical potential and charge density in the
boundary field theory; $\Psi_{1}$ is the source term, while $\Psi_{2}$ is the
vacuum expectation value $\langle O\rangle$ of the dual scalar operator. On
the boundary $u=0$, we always set $\Psi_{1}\equiv 0$ in order to have
spontaneously broken symmetry of the order parameter. We also fix the chemical
potential, that is, $a_{t}=\mu$. From Eq. (12), when $\mu>\mu_{c}$, the
temperature $T$ below $T_{c}$, we can get the superfluid phase. The critical
chemical potential in our setting takes the value $\mu_{c}\sim 4.0637$. The
critical exponents of the phase transition of this model are consistent with
those of the mean-field theory [85, 86].

Figure 1:  Vortices and anti-vortices formed in a holographic superfluid disk
after thermal quench: (a) Growth of the average order parameter as a function
of time, where $\tau_{Q}=10$, $T_{f}=0.94T_{c}$ and $R=50$. Panels (b) and (c)
correspond to snapshots of the spatial dependence of absolute value and phase
of the local order parameter at the quasi-equilibrium stage ($t=275$) of the
phase transition marked by red squares in panel (a). A red rectangular dashed
line surrounds a positive vortex and an antivortex, while a red circular
dashed line surrounds a vortex pair. The snapshot has 8 positive vortices and
7 anti-vortices, among which the vortex pairs are annihilating.

There are two spatial coordinates $\theta,r$ whose boundary conditions need to
be specified. For a disk, all fields along the $\theta$ have periodic boundary
conditions. When a phase transition occurs, the condensed value of the order
parameter grows dynamically, so we cannot fix it. Along the radial coordinate
of the disk, we impose Neumann boundary conditions
$\partial_{r}A_{\nu}=\partial_{r}\Psi=0$ at $r=R$, rather than Dirichlet
boundary conditions. In our model, $R$ is the radius of the disk embedded in
the boundary of the bulk. Note that this Neumann boundary condition is imposed
on the whole range of $u$.

Chebyshev spectral methods are used in the $(u,r)$ direction for numerical
simulation. Fourier spectral methods are used in the $\theta$ direction. In
the time direction, we use the fourth-order Runge-Kutta method to simulate the
evolution of the system and use boundary conditions at each step of the
evolution to constrain the values of the boundaries in other directions. In
addition, all fields need to be given initial values. The specific scheme will
be introduced in detail in the next section.

Before driving the system by a thermal quench, the normal fluid should be
prepared above the critical temperature at equilibrium. Numerically, we first
give a set of normal state solutions at $\mu=\mu_{c}$ corresponding to the
critical temperature $T_{c}$. It is uniform in the $r$ and $\theta$ direction,
and written as

$\displaystyle A_{t}=\mu_{c}(1-u),$ $\displaystyle A_{x}=A_{y}=\Psi=0.$ (21)

Then, we added random noise to $\Psi$ in the hole bulk to simulate the thermal
fluctuation of the system, that is, $\Psi=S(t,r,\theta)$. In this way, the
normal fluid that grows at the boundary also inherits random noise
$s(t,r,\theta)$. The noise $s$ satisfies the normal distribution, with a
vanishing statistical average $\langle s(t,r,\theta)\rangle=0$, and two-point
correlations $\langle
s(t,r,\theta)s(t^{\prime},r^{\prime},\theta^{\prime})\rangle=h\delta(t-t^{\prime},r-r^{\prime},\theta-\theta^{\prime})$,
where the amplitude $h={10}^{-4}$. After a little heating, the fluid disk is
ready for quenches.

##  V Universal scaling in slow and fast quench region

Using Eq. (12), we can simulate the linear cooling process by varying the
chemical potential in the following way

$\displaystyle\mu(t)=\begin{cases}\mu_{c}(1-t/\tau_{Q})^{-1}&0\leq t<t_{f}\\\
\mu_{f}&t\geq t_{f}\end{cases}.$ (22)

The time $t_{f}$ at which the modulated coupling reaches its final value,
$\mu(t_{f})=\mu_{f}$, will play a key role in what follows and is given by

$t_{f}=\tau_{Q}(1-\mu_{c}/\mu_{f}).$ (23)

A single realization after the thermal quench is shown in Fig. 1 for the
choice of parameters $\tau_{Q}=10$, $T_{f}=0.94T_{c}$, and $R=50$. Like the
superfluid in the boundaryless tread geometry [20], the superfluid in the disk
geometry with open boundaries can also generate vortices through the KZM, and
the vortices will also annihilate. However, the difference is that the number
of positive and negative vortices need not be equal in every quenching
experiment, only on average. Positive and negative vortices adjacent to each
other will annihilate in pairs. This phenomenon is in stark contrast to what
occurs in a rotating superfluid under disk geometry [12, 13], when all
vortices have equal circulation. In addition, vortices may also be lost at the
disk boundary.

Figure 2: Universal scaling laws at fast and slow quenches in a log-log
representation: The mean number $n$ of kinks as a function of the quench time
$\tau_{Q}$. The behavior at slow quenches exhibits a universal power-law
scaling with the quench time, which is in agreement with KZM. As the quench
time is reduced, there is a crossover to a plateau region with a value of the
vortex average number that is independent of the quench time. The crossover
occurs for $\tau_{Q}$ shorter than $\tau_{Q}^{c1}$. The data in the KZM power
law region is fitted by the $n=(992\pm 71)\tau_{Q}^{-0.518\pm 0.018}$.

Fig. 2 reports universal scaling laws at fast and slow quenches in a log-log
representation. Each sampling point is an average of 2000 realizations. In
panel 2, from top to bottom, the curves for the average number of kinks are
shown for three different values of the end temperatures
$T_{f}=0.82T_{c},0.90T_{c},0.94T_{c}$, in a larger disk $R=50$. The right
sides of the three curves all meet on a common line with a negative slope,
indicative of universal nonequilibrium behavior, and fitted to $n=(992\pm
71)\tau_{Q}^{-0.518\pm 0.018}$. It is in reasonable agreement with the
universal KZM power law scaling in Eq. (4), with a power law exponent
$\nu/(1+z\nu)=1/4$ using the verified values of the equilibrium critical
exponents $\nu=1/2$ and $z=2$. The proximity of the numerical value of the
fitted power-law exponent to the KZM prediction supports the validity of the
mean-field description and the validity of the KZM in holography.
Additionally, the plateau and deflection on the left side of Fig. 2 hint at
the destruction of KZM. Fig. 3 further shows the breakdown of the KZM scaling
law Eq. (4) at fast quenches, On the left of the Fig. 3, the results are
fitted to $n=(416\pm 10)\epsilon_{f}^{1.03\pm 0.01}$, which exhibits a
universal power-law scaling of the topological defects and the quench depth
$\epsilon_{f}$, this is in agreement with the universal prediction (7). The
numerical points fit by $\hat{t}=(12.4\pm 0.2)\epsilon_{f}^{-0.957\pm 0.008}$
in the right of Fig. 3 also shows the corresponding freeze-out timescales
varying universally at fast quenches according to (6) as a function of the
quench depth.

Figure 3: Universal scaling laws at fast quenches in a log-log representation.
(a) The mean number $n$ of vortices and (b) the freeze-out time $\hat{t}$
scale universally as a function of the quench depth $\epsilon_{f}$. The
scaling laws are determined using about $10^{5}$ trajectories. The
corresponding data is fitted by the following power-laws $n=(416\pm
10)\epsilon_{f}^{1.03\pm 0.01}$, $\hat{t}=(12.4\pm 0.2)\epsilon_{f}^{-0.957\pm
0.008}$.

##  VI Universal statistics of vortices

The estimation of the low-order cumulants of the vortex number distribution is
based on the assumption that the generation of vortices at different locations
is governed by independent and equally distributed random events [62, 64, 63,
83]. In our model, when the normal fluid undergoes a phase transition into a
superfluid, it partitioned into “protodomains” of size set by the correlation
length $\hat{\xi}$, and there is a certain probability $p$ of vortex
generation at the interface between adjacent domains. The number of possible
domain locations for defect formation can be estimated as
$\mathcal{N}_{d}=A/(f\hat{\xi}^{2})$, for a newborn superfluid of area $A$,
where $f$ is a fudge factor. The probability of forming $N$ defects is given
by the binomial distribution

$\displaystyle P(N)$ $\displaystyle=$ $\displaystyle B(N,\mathcal{N}_{d},p)$
(24) $\displaystyle=$
$\displaystyle\binom{\mathcal{N}_{d}}{N}p^{N}(1-p)^{\mathcal{N}_{d}-N}.$ (25)

This distribution encodes the universal scaling of its cumulants $\kappa_{q}$
with the quench time $\tau_{Q}$ in the slow-quench limit

$\kappa_{q}\propto\left(\frac{\tau_{0}}{\tau_{Q}}\right)^{\frac{2\nu}{1+z\nu}},$
(26)

and with the quench depth $\epsilon_{f}$ for fast quenches

$\kappa_{q}\propto\epsilon_{f}^{\nu}.$ (27)

Figure 4 shows the low-order cumulants of the vortex number as a function of
the quench time $\tau_{Q}$. A transition from the fast-quench plateau to the
power-law scaling regime is observed. In the latter, the first three cumulants
$\kappa_{1}$, $\kappa_{2}$ and $\kappa_{3}$ follow the scaling law
$\kappa_{1}=(992\pm 71)\tau_{Q}^{-0.518\pm 0.018}$, $\kappa_{2}=(932\pm
101)\tau_{Q}^{-0.522\pm 0.016}$ and $\kappa_{3}=(670\pm
130)\tau_{Q}^{-0.522\pm 0.029}$. The universal scaling of the cumulants
generalizes the beyond-KZM predictions in [62] to two-spatial dimensions, with
fitted power-law exponents slightly above the expected value of $1/2$ in the
mean-field regime in each case ($q=1,2,3$). The situation is reminiscent of
that in recent related experiments, where the observed power-law exponents for
the mean slightly surpassed the theoretical prediction [58, 87, 88]. Beyond
the mean $\kappa_{1}$, cumulants are highly sensitive to the quality of the
sampling, and a total of half a million trajectories are used for the
convergence of the third cumulant. This indicates the difficulty of probing
the universal scaling of high-order cumulants. The nonzero value and scaling
of the third cumulant confirm the non-Gaussian features of the underlying
distribution.

Fig. 5 shows the low-order cumulants of the vortex number as a function of the
quench depth $\epsilon_{f}$ in the fats-quench limit. The first three
cumulants $\kappa_{1}$, $\kappa_{2}$ and $\kappa_{3}$ scale us
$\kappa_{1}=(416\pm 10)\epsilon_{f}^{1.03\pm 0.01}$, $\kappa_{2}=(378\pm
9)\epsilon_{f}^{1.034\pm 0.001}$ and $\kappa_{3}=(294\pm
9)\epsilon_{f}^{1.08\pm 0.08}$. These power-law numerical fits are in good
agreement with the beyond-KZM prediction in Eq. (26) and (27), with mean-field
critical exponents $\nu=1/2$ and $z=2$. The binomial distribution approaches a
normal distribution in the large $\mathcal{N}_{d}$ limit

$P(N)=\frac{1}{\sqrt{2\pi(1-p)\langle N\rangle}}\exp\left[-\frac{(N-\langle
N\rangle)^{2}}{2(1-p)\langle N\rangle}\right].$ (28)

Figure 6 shows the histograms for the vortex-number probability distribution
in slow quench region at different $\tau_{Q}$. Similarly, Fig. 7 shows the
histograms for the vortex-number probability distribution in the fast quench
region with different $\epsilon_{f}$. For both slow and fast quenches, all
histograms fit well to the normal distribution in Eq. (28), showing its
robustness and lending support to the beyond-KZM prediction. However, the
calculation of the trace norm distance between the numerical data and the
distribution indicates the limits of this approximation, as we discuss next.

Figure 4: Universal scaling of low-order cumulants of the vortex number
distribution versus the quench time $\tau_{Q}$. The first two cumulants
$\kappa_{1}$ and $\kappa_{2}$ are the mean value and the variance of the
vortex number, respectively. The third cumulant $\kappa_{3}$ is related to the
skewness of $n$. In the KZM power law region, the first cumulant $\kappa_{1}$
is fitted to $\kappa_{1}=(992\pm 71)\tau_{Q}^{-0.518\pm 0.018}$, the second
cumulant $\kappa_{2}$ is fitted to $\kappa_{2}=(932\pm 101)\tau_{Q}^{-0.522\pm
0.016}$ and the third cumulant $\kappa_{3}$ is fitted to $\kappa_{3}=(670\pm
130)\tau_{Q}^{-0.522\pm 0.029}$. For each $\tau_{Q}$, $5\times 10^{5}$ are
considered. Black triangles denote the theoretical values of a PBD of two
different Bernoulli weights. The numerically fitted Bernoulli probabilities
and the ratio of the average number of vortices
$p_{1}=0.13,\,p_{2}=0.1,\,r=0.38$ were obtained numerically. The inset shows
the relative square distance from the numerical results as a function of
$\tau_{Q}$. Figure 5: Universal scaling of low-order cumulants of the vortex
number distribution versus the quench depth $\epsilon_{f}$ at fast quenches.
The first cumulant $\kappa_{1}$ is fitted to $\kappa_{1}=(416\pm
10)\epsilon_{f}^{1.03\pm 0.01}$, the second cumulant $\kappa_{2}$ is fitted to
$\kappa_{2}=(378\pm 9)\epsilon_{f}^{1.034\pm 0.001}$ and the third cumulant
$\kappa_{2}$ is fitted to $\kappa_{3}=(294\pm 9)\epsilon_{f}^{1.08\pm 0.08}$.
Values for each $\epsilon_{f}$ involve $5\times 10^{5}$ trajectories. Black
triangles denote the cumulant values obtained by the best-fitting PBD with the
same parameters as in Fig. 4. The inset shows the relative square distance
from the numerical results as a function of $\tau_{Q}$. Figure 6: Histograms
for the vortex-number probability distribution for different values of the
quench time. Both even and odd outcomes of the vortex number are possible as a
result of the Neumann boundary conditions on the disk. The normal
approximation is shown in red. The PBD accurately fits the data in all
regimes, accounting also for the non-binomial nature of the first three
cumulants in Figs. 4 and 5. The solid line denotes the PBD fitting with the
same parameters as in Figs. 4 and 5 accounting for the additional skewness up
to good precision. The inset shows the trace distance norm between the PBD and
numerically computed histograms as a function of $\tau_{Q}$. The comparison
between the numerical data and the normal distribution reveals higher values
of the trace norm distance. Figure 7: Histogram for the vortex-number
probability distribution for fast quenches for different quench depths.
Histograms are fitted to the normal approximation and the PDB. The inset shows
the $\epsilon_{f}$ dependence of the corresponding trace distance norm,
indicating the accuracy of the PBD over the normal distribution.

##  VII Disscussion

To date, studies of the defect statistics beyond KZM have focused on defect
formation in one spatial dimension [62, 63, 65, 79], see as well [61, 66, 53,
67, 54, 89] in the quantum case. Studies in two spatial dimensions are scarce
and have focused on other aspects, addressing only the normal features of the
defect number distribution [83]. Only one study to date has characterized non-
normal features in a two-dimensional system consisting of a holographic
superconductor with periodic boundary conditions [79]. In such a setting, the
vortex number distribution is confined to even outcomes; as such, it deviates
from the binomial distribution and is best described as an even-Poissonian
model (a limiting case of the binomial distribution conditioned to even
outcomes). Thus, evidence of the universality of the defect number
distribution in two (and higher) spatial dimensions is lacking.

The number of defects in the superfluid disk with open boundary conditions is
unrestricted and involves both even and odd outcomes. The non-zero values of
the third cumulant and its universal scaling can be accommodated by the
binomial distribution; see Eqs. (24) and (26). Cumulant ratios depend solely
on the defect-formation probability $p$ in the binomial distribution. For the
first three cumulants, these ratios are fixed as $\kappa_{2}/\kappa_{1}=1-p$,
$\kappa_{3}/\kappa_{1}=(1-p)(1-2p)$. However, the fitted power-laws are
inconsistent with the binomial model, as they require values of $p$ outside
the physical range $[0,1]$.

The best-fitted distribution is instead a generalization of the binomial
distributions to independent, but not identically distributed, Bernoulli
trials. This is known as the Poisson binomial distribution (PBD) [90, 91] and
was predicted in Ref. [62] for the most general scenario of spontaneously
formed point-like defects. Incidentally, the same distribution describes
defect number distribution in quantum systems [61], as experimentally
demonstrated under unitary and open quantum dynamics [66, 53, 54].

The PBD generalizes the binomial model. To derive it in the context of beyond-
KZM physics, one can still invoke the notion of a random tesselation of the
superfluids by assuming that the order parameter grows independently at
different locations, forming protodomains. The spatial extension of such
protodomains is determined by the KZM correlation length $\hat{\xi}$. We gloss
over this pictorial description being handwaving, as such protodomains are
ill-defined in dimensions larger than one.

Topological defects, such as vortices, are then expected to be formed where
adjacent domains meet with a given probability, in the spirit of the geodesic
rule [29]. However, the number of domains that meet at a point varies, e.g.,
in a random tesselation. Two effects come into play in such a picture. First,
the frequency of occurrence of a boundary merging different domains varies
with the number of them. Second, numerical simulations and experiments done by
merging independent Bose-Einstein condensates show that the probability of
vortex formation depends on the number of defects that meet [92].

The total number of events for defect formation $\mathcal{N}_{d}$ can be thus
split as $\mathcal{N}_{d}=\sum_{m}\mathcal{N}_{m}$, where
$\mathcal{N}_{m}=\mathcal{N}_{d}r_{m}$ accounts for the number of events for
defect formation as a result of merging $m$-protodomains. Here, the
probability of finding $m$ protodomains merging at a point is $r_{m}\geq 1$,
and thus $\sum_{m}r_{m}=1$. We further consider the conditional probability
$p_{m}$ for a defect to be formed when $m$-protodomains meet.

The explicit form of the PBD [91]

$\displaystyle P(N)=\sum_{A\in F_{n}}\prod_{m\in A}p_{m}\prod_{n\in
A^{c}}(1-p_{n}),$ (29)

involves the set $F_{n}$ of all subsets of $n$ integers that can be selected
from ${1,\dots,\mathcal{N}}_{d}$, and $A^{c}$, the complement of $A$. This
distribution still predicts the universal scaling of all cumulants with the
quench time. The corresponding cumulant generating function is simply the sum
of the corresponding cumulant generating functions of the individual and
independent Bernoulli random variables coming as multiples of
$\mathcal{N}_{m}$ for the given $m=1,2,\dots,M$:

$\displaystyle\log\hat{P}(\varphi)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\,\kappa_{n}\frac{(i\varphi)^{n}}{n!}$ (30)
$\displaystyle=$
$\displaystyle\sum_{m=1}^{M}\,\mathcal{N}_{m}\log(1-p_{m}+p_{m}\,e^{i\varphi})$

Thus, cumulants of $P(N)$ obey

$\kappa_{n}=\sum_{m=1}^{M}\mathcal{N}_{m}\,\kappa^{(m)}_{n},$ (31)

where the type-$m$ cumulant is simply the cumulant of the corresponding
Bernoulli variable with defect-formation probability $p_{m}$ and is given by
$\kappa^{(m)}_{n}=(-i\partial_{\varphi})^{n}\log\hat{P}(\varphi)|_{\varphi=0}$.
Specifically, the first few cumulants of the Poisson binomial distribution
read

$\displaystyle\kappa_{1}=\mathcal{N}_{d}\sum_{m=1}^{M}r_{m}p_{m},$ (32)
$\displaystyle\kappa_{2}=\mathcal{N}_{d}\sum_{m=1}^{M}r_{m}p_{m}(1-p_{m}),$
(33)
$\displaystyle\kappa_{3}=\mathcal{N}_{d}\sum_{m=1}^{M}\,r_{m}p_{m}(1-p_{m})(1-2p_{m}),$
(34)

with the $n$-th cumulant being

$\displaystyle\kappa_{n}$ $\displaystyle=$
$\displaystyle\mathcal{N}_{d}\sum_{m=1}^{M}\,r_{m}p_{m}(1-p_{m})\partial_{p_{m}}\kappa^{(m)}_{n-1}$
(35) $\displaystyle\propto$
$\displaystyle\left(\frac{\tau_{0}}{\tau_{Q}}\right)^{\frac{2\nu}{1+z\nu}}.$
(36)

The resulting model (29) loses predictive power with respect to the simpler
binomial model due to the appearance of multiple parameters, i.e., the set
$\\{r_{m},p_{m}\\}$. However, two key features are preserved: all cumulants
exhibit a universal power-law scaling as a function of the quench time, and
cumulant ratios are set by constant values, given in terms of
$\\{r_{m},p_{m}\\}$. We further note that on idealized models of random
tesselations, it is possible to go beyond and make predictions about the
values of the probability $r_{m}$ for the merging of $m$-protodomains and the
probability $p_{m}$ for a defect to be formed at such location using only
geometric probability arguments.

Here, we focus on the best-fitting distribution that can be determined
numerically. The first three cumulants have three free parameters as the
$r_{m}$ values are assumed to be independent of both $\tau_{Q}$ and
$\epsilon_{f}$. Thus, the $m=2$ case with $p_{1},\,p_{2},\,r$ provides the
simplest model beyond the binomial case to unambiguously characterize the
vortex statistics, assuming two kinds of Bernoulli variables. The average
number of vortices does not determine the total number of possible vortex
formation points. Therefore, $r$ is used to parameterize the ratio between the
events of ”type”-$1$ for vortex formation and the total average number of
vortices, $p_{1}\mathcal{N}_{1}=r\langle N\rangle$. Correspondingly,
$p_{2}\mathcal{N}_{2}=(1-r)\langle N\rangle$. To find the best fitting values,
the cost function $\mathcal{C}$ of the average square distance of the three
cumulants for all given $\tau_{Q}$ and $\epsilon_{f}$ values is minimized,

$\mathcal{C}=\frac{\sum_{i}\left[\left(\kappa^{\text{(num.)}}_{1,i}-\kappa^{\text{(PBD)}}_{1,i}\right)^{2}+\left(\kappa^{\text{(num.)}}_{2,i}-\kappa^{\text{(PBD)}}_{2,i}\right)^{2}+\left(\kappa^{\text{(num.)}}_{3,i}-\kappa^{\text{(PBD)}}_{3,i}\right)^{2}\right]}{\sum_{i}\left[\left(\kappa^{\text{(num.)}}_{1,i}\right)^{2}+\left(\kappa^{\text{(num.)}}_{2,i}\right)^{2}+\left(\kappa^{\text{(num.)}}_{3,i}\right)^{2}\right]}\,,$
(37)

where $i$ runs over all the $\tau_{Q}$ and $\epsilon_{f}$ values. This yields
the estimated parameters $p_{1}=0.13,\,p_{2}=0.1,\,r=0.38$ with the relative
mean error of the cumulants $\approx 0.0011$ averaged over the $\tau_{Q}$ and
$\epsilon_{f}$ values in the power-law and saturation regimes, simultaneously.
In doing so, the fitted model posits that a universal distribution function
-the PBD- accommodates the universal KZM scaling, its breakdown as fast
quenches, and the additional universal scaling laws as a function of the depth
of the quench, i.e., the final value of the temperature.

In the plateau regime, in the range of values for $\epsilon_{f}$ explored, as
well as in the power-law regime, as a function of $\tau_{Q}$, the trace norm
distance between the data and the closest PBD does not exceed 0.02.

For the sake of illustration, we also performed the fitting separately for the
dataset in the saturation and power law regimes and compared the corresponding
errors, summarized in the following table:

Optimization Dataset: | $\tau_{Q}$ | $\epsilon_{f}$ | $\tau_{Q}\,{\rm and}\,\epsilon_{f}$
---|---|---|---
$D_{\mathrm{TN,\tau_{Q}}}$ | $0.0113$ | $0.0175$ | $0.0111$
$D_{\mathrm{TN,\epsilon_{f}}}$ | $0.0111$ | $0.0158$ | $0.0108$
$\Delta\kappa_{q}(\tau_{Q})$ | $0.0009$ | $0.0025$ | $0.0008$
$\Delta\kappa_{q}(\epsilon_{f})$ | $0.0034$ | $0.0007$ | $0.0023$

Errors are approximately below the same value $\sim 0.02$ with slightly
smaller values for the cumulants for the datasets according to which the
minimization itself is performed. Additionally, the best
$\mathcal{B}(p,\mathcal{N})$ binomial fit with $p\approx 0.08$ and
$\mathcal{N}=\lceil\langle N\rangle/p\rceil$ based on the minimization of the
trace norm results in $\sim 2.5$ times larger errors than the PBD fittings.
This error analysis supports the identification of the PBD as the universal
defect number distribution across all the parameter space up to numerical
precision.

##  VIII Summary and outlook

The KZM is an important paradigm in nonequilibrium statistical mechanics that
utilizes equilibrium properties to predict the dynamics across a quantum phase
transition. It predicts the spontaneous formation of topological defects with
a density that scales universally as a function of the quench time. Recent
progress has shown that additional universal features exist in the dynamics
across a phase transition that can be described in terms of equilibrium
properties, such as the statistics of topological defects. However, the
validity of such predictions is limited to slow quenches where the density of
defects is low. Faster quenches are no longer described by KZM and show a
saturation of the density of defects with the quench time.

The recent observation that the mean defect density is also universal for fast
quenches constitutes the starting point of our study. In this work, we have
established that defect statistics is universal for arbitrary values of the
quench time and quench depth. Specifically, we have shown that the defect
number distribution follows a Poisson binomial distribution. This is the case
both at slow quenches, for which KZM holds, and for fast quenches when KZM
breaks down. A testable prediction of our analysis involves the universal
scaling laws of the cumulants of the defect statistics as a function of the
quench time and the quench depth. Such tests could be implemented in
experiments exploring superfluid formation in which the defect number
statistics can be measured. Ultracold gases offer a relevant example, where
the crossing of the superfluid phase transition can be induced by quenches of
temperature, chemical potential (via a dimple trap), or interatomic
interactions. In such systems, direct imaging of the vortex patterns is
possible, and the vortex number distribution can be analyzed [56, 60, 68, 74,
93, 88]. Similar studies may be at reach in other systems, such as polaritonic
condensates and quantum fluids of light [94], colloidal monolayers [95], and
multiferroics [96, 97].

###### Acknowledgements.

It is a pleasure to acknowledge discussions with Matteo Massaro, Seong-Ho
Shinn, and Kasturi Ranjan Swain. This work is supported by the National
Natural Science Foundation of China (under Grants No. 11275233), the
Postgraduate Research & Practice Innovation Program Jiangsu Province
(KYCX22_3450), and the Luxembourg National Research Fund (FNR), grant
references 17132054 and 17132060. For the purpose of open access, the author
has applied a Creative Commons Attribution 4.0 International (CC BY 4.0)
license to any Author Accepted Manuscript version arising from this
submission.

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