††institutetext: 1Department of Physics, Yamagata University, Kojirakawa-machi
1-4-12, Yamagata, Yamagata 990-8560, Japan,
2Research and Education Center for Natural Sciences, Keio University, Hiyoshi
4-1-1, Yokohama, Kanagawa 223-8521, Japan
3Department of Physics, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa
223-8521, Japan

# Quantum nucleation of topological solitons

Minoru Eto1,2, Muneto Nitta2,3

###### Abstract

The chiral soliton lattice is an array of topological solitons realized as
ground states of QCD at finite density under strong magnetic fields or rapid
rotation, and chiral magnets with an easy-plane anisotropy. In such cases,
topological solitons have negative energy due to topological terms originating
from the chiral magnetic or vortical effect and the Dzyaloshinskii-Moriya
interaction, respectively. We study quantum nucleation of topological solitons
in the vacuum through quantum tunneling in $2+1$ and $3+1$ dimensions, by
using a complex $\phi^{4}$ (or the axion) model with a topological term
proportional to an external field, which is a simplification of low-energy
theories of the above systems. In $2+1$ dimensions, a pair of a vortex and an
anti-vortex is connected by a linear soliton, while in $3+1$ dimensions, a
vortex is string-like, a soliton is wall-like, and a disk of a soliton wall is
bounded by a string loop. Since the tension of solitons can be effectively
negative due to the topological term, such a composite configuration of a
finite size is created by quantum tunneling and subsequently grows rapidly. We
estimate the nucleation probability analytically in the thin-defect
approximation and fully calculate it numerically using the relaxation
(gradient flow) method. The nucleation probability is maximized when the
direction of the soliton is perpendicular to the external field. We find a
good agreement between the thin-defect approximation and direct numerical
simulation in $2+1$ dimensions if we read the vortex tension from the
numerics, while we find a difference between them at short distances
interpreted as a remnant energy in $3+1$ dimensions.

††preprint: YGHP-22-3

##  1 Introduction

Topological solitons are ubiquitous in nature, playing significant roles in
quantum field theories Rajaraman:1987 ; Rubakov:2002fi ; Manton:2004tk ;
Shnir:2005vvi ; Vachaspati:2006zz ; Dunajski:2010zz ; Weinberg:2012pjx ;
Shnir:2018yzp , supersymmetric gauge theories Tong:2005un ; Tong:2008qd ;
Eto:2006pg ; Shifman:2007ce ; Shifman:2009zz , QCD Eto:2013hoa , cosmology
Kibble:1976sj ; Kibble:1980mv ; Vilenkin:1984ib ; Hindmarsh:1994re ;
Vachaspati:2015cma ; Vilenkin:2000jqa , and various condensed matter systems
Mermin:1979zz  such as helium-3 superfluids Volovik:2003fe , superfluids
Svistunov:2015 , superconductors Blatter:1994zz ; Giamarchi:2002 , Josephson
junctions of superconductors Ustinov2015 , Bose-Einstein condensates (BECs) of
ultracold atomic gasses Kawaguchi:2012ii , nonlinear media Pismen ;
Bunkov:2000 , liquid crystals Pismen  and magnets Nagaosa2013 ; Gobel:2020mqd
. Among various salient features of topological solitons, formations of
topological solitons are one of the most important aspects. Topological
solitons are known to be created at phase transitions accompanied with a
spontaneous symmetry breaking (SSB). This is known as the Kibble-Zurek
mechanism Kibble:1976sj ; Kibble:1980mv ; Zurek:1985qw ; Zurek:1996sj  and was
experimentally confirmed in various condensed matter systems such as liquid
crystals Bowick:1992rz , helium-4 superfluids Hendry:1994 , helium-3
superfluids Baeuerle:1996zz ; Ruutu:1995qz , and ultracold atomic gasses
Sadler:2006 ; Weiler:2008 . In two spatial dimensions, a pair of a vortex and
an anti-vortex is created at finite temperature, which is known as
Berezinskii-Kosterlitz-Thouless transition (BKT) transitions Berezinsky:1970fr
; Berezinsky:1970-2 ; Kosterlitz:1972 ; Kosterlitz:1973xp  (see also Refs.
Kobayashi:2018ezm ; Kobayashi:2019sus  for recent studies). This was
experimentally confirmed in thin-film helium-4 superfluids PhysRevLett.40.1727
, superconductors PhysRevLett.42.1165 , and ultracold atomic gases
Hadzibabic:2006 . There are several other mechanisms for creating topological
solitons such as bubble collisions Hawking:1982ga ; Copeland:1996jz ;
Digal:1995vd , monopole pair creation under strong magnetic field (as a dual
to the Schwinger mechanism) Affleck:1981ag ; Affleck:1981bma , and brane
annihilations Sen:2004nf  (see Refs. Bradley:2008zza ; Takeuchi:2012ee ;
Nitta:2012hy ; Takeuchi:2012ap  for condensed matter analogues).

The purpose of this paper is to propose a yet another mechanism for a creation
of topological solitons, that is, quantum nucleation through quantum
tunneling. This mechanism works when the ground state is “solitonic”. When the
Lagrangian or Hamiltonian contains a certain type of a topological term with
its coefficient larger than a certain critical value, the energy of
topological solitons is negative and thus they are spontaneously created in
uniform states. However, one cannot place infinite number of solitons since
they repel each other, and thus the ground state is a lattice of topological
solitons. A typical example of solitonic ground states is given by chiral
soliton lattices (CSLs) which are periodic arrays of domain walls or solitons,
appearing in various condensed matter systems: cholesteric liquid crystals
DEGENNES1968163  and chiral magnets togawa2012chiral ; togawa2016symmetry ;
KISHINE20151 ; PhysRevB.97.184303 ; PhysRevB.65.064433 ; Ross:2020orc  with
the Dzyaloshinskii-Moriya (DM) interaction Dzyaloshinskii ; Moriya:1960zz .
The latter has an important nanotechnological application in information
processing such as magnetic memory storage devices and magnetic sensors
togawa2016symmetry . The $O(3)$ sigma model together with the DM term reduces
to the sine-Gordon model plus a topological term at low energy, and the CSL is
a sine-Gordon lattice. Another condensed matter example of solitonic ground
states is given by magnetic skyrmions Bogdanov:1989 ; Bogdanov:1995  in chiral
magnets, which typically constitute a triangular lattice in the ground state
in the parameter region in which the DM term is strong enough Rossler:2006 ;
Han:2010by ; Lin:2014ada ; Ross:2020hsw . Since they have been realized in
laboratory experiments doi:10.1126/science.1166767 ; doi:10.1038/nature09124 ,
there has been great interests such as an application to information carriers
in ultradense memory and logic devices with low energy consumption
doi:10.1038/nnano.2013.29 . The other examples are for instance
${\mathbb{C}}P^{N-1}$ skyrmion lattices in $SU(N)$ magnets Akagi:2021dpk ;
Akagi:2021lva ; Zhang:2022 ; Amari:2022boe  and 3D skyrmions Kawakami:2012zw
in spin-orbit coupled BECs with background gauge fields as generalizations of
the DM term.

Recently, it has been predicted that CSLs are also ground states of QCD at
finite density under strong magnetic field Son:2007ny ; Eto:2012qd ;
Brauner:2016pko ; Chen:2021vou ; Gronli:2022cri  or under rapid rotation
Huang:2017pqe ; Nishimura:2020odq ; Eto:2021gyy , due to a topological term
originated from the chiral magnetic effect (CME) Son:2004tq ; Son:2007ny
which is the vector current in the direction of the magnetic field, or chiral
vortical effect (CVE) Vilenkin:1979ui ; Vilenkin:1980zv ; Son:2009tf  which is
the axial vector current in the direction of the rotation axis, respectively.
They also appear with thermal fluctuation Brauner:2017uiu ; Brauner:2017mui ;
Brauner:2021sci  (see also Refs. Yamada:2021jhy ; Brauner:2019aid ;
Brauner:2019rjg ). In the CSLs, the number density of solitons is determined
by the strength of external fields such as a magnetic field or rotation (or
the DM term for chiral magnets). As external fields are larger above the
critical value, the soliton number density is larger. Thus, when one gradually
increases(decreases) the strength of the external field, the mean inter-
soliton distance decreases(increases) accordingly. One of natural questions is
how they are created from the vacuum (uniform state). When one instantaneously
changes the external field from the value below the critical value to the one
above the critical value, it is unnatural that a flat soliton (domain wall)
with infinite world-volume instantly appears. Instead, quantum nucleation can
occur in this case as we propose in this paper.

To explain our mechanism, it is worth to recall quantum decay of a metastable
false vacuum and bubble nucleation first formulated by Coleman Coleman:1977py
; Callan:1977pt ; Coleman:1987rm  (see Refs. Rubakov:2002fi ; Weinberg:2012pjx
; Coleman:1985rnk  as a review). Decay probabilities can be calculated by
evaluating the Euclidean action values for bounce solutions. In the thin-wall
approximation, one can evaluate the decay probability in terms of tensions of
domain walls. Preskill and Vilenkin studied quantum decays of metastable
topological defects Preskill:1992ck  (see Ref. Vilenkin:2000jqa  as a review,
and Refs. Monin:2008mp ; Monin:2008xx ; Monin:2009gi  for recent studies). One
of typical cases is given by an axion model, in which a domain wall (or
soliton) terminates on a string. Thus, a domain wall is metastable and can
decay by quantum tunneling with creating a hole bounded by a closed string.
Again in the thin-wall approximation, one can evaluate the decay probability
of the domain wall in terms of tensions of domain walls and strings. Some
examples are given by domain walls in two-Higgs doublet models Eto:2018hhg ;
Eto:2018tnk  and axial domain wall-vortex composites in QCD Eto:2013bxa .
Another case is a string (vortex) ending on a monopole. In this case, a string
is metastable and decays by cutting the string into two pieces whose endpoints
are attached by a monopole and an anti-monopole through quantum tunneling.
Examples can be found for instance for electroweak $Z$-strings in the standard
model Nambu:1977ag ; Achucarro:1999it ; Eto:2012kb  and non-Abelian strings in
dense QCD Vinci:2012mc ; Eto:2013hoa .

In this paper, we study quantum nucleation of topological solitons through
quantum tunneling. For definiteness, we discuss chiral solitons in a complex
$\phi^{4}$ model (an axion model with the domain wall number one) with a
topological term, which is a simplification of low-energy theories of chiral
magnets (with an easy-plane anisotropy) and QCD at finite density under strong
magnetic field or rapid rotation. The origin of the topological term is the DM
interaction for chiral magnats, while it is CME and CVE for QCD under strong
magnetic field or rapid rotation, respectively. If the external field $B$ is
larger than a certain critical value $B_{c}$, the soliton tension is
effectively negative, and therefore it can be created by quantum tunneling. We
estimate the nucleation probability analytically in the thin-defect
approximation in any dimension, and fully calculate it numerically in $2+1$
and $3+1$ dimensions by using the relaxation (gradient flow) method. In $2+1$
dimensions, a vortex is particle-like, a soliton is string-like, and a pair of
a vortex and an anti-vortex is connected by a linear soliton, while in $3+1$
dimensions, a vortex is string-like, a soliton is wall-like, and a disk of a
soliton wall is bounded by a string loop. Once such a composite configuration
of a finite size is created by quantum tunneling, it grows rapidly. The
nucleation probability is maximized when the direction of the soliton is
perpendicular to the external field. We also find that decay (nucleation) is
prohibited for $B>B_{c}$ ($B<B_{c}$). We find that the nucleation
probabilities calculated in the thin-defect approximation and in the direct
numerical simulations show a good agreement in $2+1$ dimensions once we read
the vortex tension from the numerics. On the other hand, in $3+1$ dimensions,
we find a difference between them at short distances at the subleading order
which we interpret as a remnant energy.

This paper is organized as follows. In Sec. 2, we give a brief review of
quantum decay of a soliton in the complex $\phi^{4}$ model (the axion model
with the domain wall number one) without a topological term. In Sec. 3, we
present our model (the complex $\phi^{4}$ model with a topological term) and
discuss quantum nucleation and decay probabilities of solitons in the thin-
defect approximation. In Sec. 4 we numerically calculate the creation
probabilities of solitons in $2+1$ and $3+1$ dimensions and compare those in
the thin-defect approximation. Section 5 is devoted to a summary and
discussion. In Appendix A, we present an asymptotic behavior of the scalar
field outside a pair of vortex and an anti-vortex connected by a soliton. In
Appendix B, we give a derivation of some formula used in the quantum
nucleation.

##  2 Quantum decay of solitons by nucleation of holes: a review

We start with giving a brief review of the quantum decay of solitons (domain
walls) by quantum nucleations of holes in a complex $\phi^{4}$ model (an axion
model with the domain wall number one). The minimal model in $3+1$ dimensions
is

$\displaystyle{\cal L}_{\rm
UV}=\left|\partial_{\mu}\phi\right|^{2}-\frac{\lambda}{4}\left(|\phi|^{2}-v^{2}\right)^{2}+vm^{2}(\phi+\phi^{*}),$
(2.1)

with $v$ and $m$ are parameters whose mass dimension is 1, and $\lambda$ is
dimensionless. If the third term in Eq. (2.1) is absent, the model is the
Goldstone model invariant under a global $U(1)$ transformation $\phi\to
e^{i\eta}\phi$ spontaneously broken in the homogeneous vacuum $|\phi|=v$.
There is a Nambu-Goldstone (NG) mode and a Higgs mode whose mass is
$m_{h}=v\sqrt{\lambda}$. When we turn on the third term, the $U(1)$ symmetry
is explicitly broken, leaving the unique vacuum where the NG mode becomes a
pseudo-NG mode with the mass $m$. The $U(1)$ symmetry is an approximate
symmetry when the mass of the pseudo-NG mode is sufficiently small

$\displaystyle m_{h}\gg m\quad\Leftrightarrow\quad m^{2}\ll\lambda v^{2}.$
(2.2)

There the vacuum expectation value can be approximated as
$\left<\phi\right>\simeq v\left(1+\frac{m^{2}}{m_{h}^{2}}\right).$

The model admits two kinds of solitonic objects, namely vortices and solitons
(or domain walls). The vortex is a global string with co-dimension two which
is a topological defect if the explicit $U(1)$ breaking term is absent.
Thickness of the string and the tension of the string for $m=0$, namely the
energy per unit length, are given by

$\displaystyle\delta_{\rm st}\sim m_{h}^{-1},\qquad\mu\big{|}_{m\to 0}\sim\pi
v^{2}\log(m_{h}L),$ (2.3)

where $L$ is a long distance cutoff. When the $U(1)$ breaking term is not
zero, the string is no longer topological and it is always accompanied by the
soliton which is a wall-like object with co-dimension one.

Probably the soliton can be most clearly seen in the limit of
$\lambda\to\infty$ where the wine bottle potential becomes infinitely steep
($m_{h}\to\infty$), so that the amplitude of $\phi$ freezes out as $|\phi|=v$.
Writing $\phi=ve^{i\theta}$ and plugging it into Eq. (2.1), we are lead to the
sine-Gordon model

$\displaystyle{\cal L}_{\rm
IR}=v^{2}\left[(\partial_{\mu}\theta)^{2}+2m^{2}(\cos\theta-1)\right],$ (2.4)

where we have subtracted the constant $2m^{2}v^{2}$ for the minimum of the
potential energy to be 0 for convenience. The ground state is homogeneous as
$\theta=0$ with the redundancy of $2\pi n$ $(n\in\mathbb{Z})$. The ground
state energy $E^{({\rm vac})}=0$. In addition, there is a sine-Gordon soliton
which we take perpendicular to the $z$-axis without loss of generality:

$\displaystyle\theta=4\tan^{-1}e^{mz}.$ (2.5)

This connects $\theta=0$ at $z\to-\infty$ and $\theta=2\pi$ at $z\to\infty$.
The thickness and the tension, namely the energy per unit area, of the soliton
are given by

$\displaystyle\delta_{\rm
dw}=m^{-1},\qquad\sigma\big{|}_{\lambda\to\infty}=16mv^{2}.$ (2.6)

Note that the soliton in the sine-Gordon limit is classically stable but it
could be quantum-mechanically unstable. This is because it can end on a sting
which is infinitely thin ($\delta_{\rm st}\to 0$) in the $\lambda\to\infty$
limit, so that holes surrounded by the strings can be created by quantum
tunneling effect.

The instability of the soliton for a finite $\lambda$ is two fold: 1)
classical instability and 2) quantum instability.

1) The soliton in the UV theory is a metastable non-topological soliton. This
is because a loop surrounding the $S^{1}$ vacuum manifold which slightly
slants by $vm^{2}(\phi+\phi^{*})\sim 2m^{2}v^{2}\cos\theta$ can pass slip the
potential barrier around $|\phi|=0$ and shrink to the unique vacuum. This is
the classical instability of the soliton in the UV theory with finite
$\lambda$. Roughly speaking, if the approximate $U(1)$ condition in Eq. (2.2)
is satisfied, the soliton remains classically meta-stable. As for the tension
of the string, it is quite different for the massive case $m\neq 0$ from the
massless case $m=0$ in which the tension is logarithmically divergent as in
Eq. (2.3).111 In Ref. Preskill:1992ck , a string is attached by a sine-Gordon
domain soliton as our case, but it is assumed that the energy of the vortex is
logarithmically divergent. The key point is that the amplitude of the scalar
field converges to the VEV exponentially fast as numerically confirmed in
Appendix A, in contrast to the massless case for which the amplitude
polynomially approaches to the VEV. Thus, the tension is finite

$\displaystyle\mu\big{|}_{m>0}=\text{const.}$ (2.7)

in contrast to the massless case in Eq. (2.3). This fact is crucial for the
nucleation of topological soliton. Note that the approximate $U(1)$ condition
in Eq. (2.2) implies $\delta_{\rm st}\ll\delta_{\rm dw}$.

2) Even when the soliton is classically metastable, it would quantum
mechanically be unstable because of the nucleation of a hole. Let us assume a
shape of a hole is circular. If the radius $R$ of the hole is much greater
than the soliton thickness $\delta_{\rm dw}$, we can use the thin-defect
approximation, providing the decay probability Preskill:1992ck

$\displaystyle P_{\text{decay}}=Ae^{-S},\quad
S=\frac{16\pi\mu^{3}}{3\sigma^{2}},\quad R=\frac{2\mu}{\sigma}$ (2.8)

where $\mu$ is the constant tension of the string and the soliton tension
$\sigma$ is well approximated by Eq. (2.6) if $\lambda\gg 1$. The thin-defect
approximation is justified for $R\sim\mu/\sigma\sim\mu/(mv^{2})\gg 1/m$,
namely $\mu/v^{2}\gg 1$. Thus, the value of the bounce action $S$ is justified
in the thin defect approximation if
$S\sim\mu^{3}/\sigma^{2}\sim\mu^{3}/(m^{2}v^{4})=(v^{2}/m^{2})(\mu/v^{2})^{3}\gg
v^{2}/m^{2}$. On the other hand, the classical stability requires
$v^{2}/m^{2}\gg 1/\lambda$. Hence, since the bounce action can be of order one
or larger depending on the parameters, the decay rate can be either large or
smaller, respectively.

##  3 Quantum nucleation and decay of solitons in external fields in the thin-
defect approximation

In this section, we give the models (sine-Gordon model and complex $\phi^{4}$
model in the external field) in Subsec. 3.1 and estimate nucleation
probability of solitons in any dimensions $d$ in terms of tensions of solitons
and strings (vortices) in the thin-defect approximation in Subsec. 3.2. We
also calculate decay probability of solitons in the external field in Subsec.
3.3.

###  3.1  The models with external fields

We consider the sine-Gordon model under a constant background field ${\bm{B}}$
in $3+1$ dimensions, given by

$\displaystyle{\cal L}_{\rm
IR}=v^{2}\left[(\partial_{\mu}\theta)^{2}+2m^{2}(\cos\theta-1)+c{\bm{B}}\cdot\nabla\theta\right],$
(3.1)

where the overall constant $v$ is dimensionful, $m$ is a mass parameter, and
the mass dimension of $c$ is $-1$. The last term is a total derivative and is
a topological term. This Lagrangian is a simplification of low-energy
effective Lagrangians for various interesting systems: chiral magnets with
easy-axis anisotropy in which the topological term is the DM term, or chiral
Lagrangian for pions under a strong magnetic field (rapid rotation) in which
the last term originates from the CME (CVE). Here, we use a notation of
${\bm{B}}$ for either a magnetic field, a rotation or DM term. The Hamiltonian
reads

$\displaystyle{\cal H}_{\rm
IR}=v^{2}\left[\dot{\theta}^{2}+(\nabla\theta)^{2}-2m^{2}(\cos\theta-1)-c{\bm{B}}\cdot\nabla\theta\right].$
(3.2)

Since the last term in Eq. (3.1) is of the first order in the derivative, it
does not affect the equation of motion (EOM). Indeed, the homogeneous
configuration $\theta=2\pi n$ remains the solution of EOM. The energy density
is also unchanged from zero. The soliton solution also remains to be the same

$\displaystyle\theta=4\tan^{-1}e^{m\hat{\bm{n}}\cdot\bm{x}},$ (3.3)

where we have introduced an arbitrary unit vector $\hat{\bm{n}}$ perpendicular
to the soliton. The solution itself is unchanged, however, the soliton tension
receives a correction from the background field. Let $\alpha$ be relative
angle between $\hat{\bm{n}}$ and the constant background field ${\bm{B}}$.
Then, the additional energy per unit area of the single soliton reads

$\displaystyle\delta\sigma=-2\pi v^{2}cB\cos\alpha,$ (3.4)

where $B=|{\bm{B}}|$, and we have used the fact that $\theta$ increases by
$2\pi$ when one traverses the soliton along $\hat{\bm{n}}$. The net tension of
the single soliton reads

$\displaystyle\sigma=16mv^{2}-2\pi v^{2}cB\cos\alpha.$ (3.5)

This is minimized when $\alpha=0$ ($\pi$) for $c>0$ ($<0$). Namely, the most
stable soliton is perpendicular to the external field ${\bm{B}}$. This implies
that the soliton is tensionless at the critical value

$\displaystyle B=\frac{8m}{\pi|c|}\equiv B_{\rm c}.$ (3.6)

Moreover, it is negative for $B>B_{\rm c}$. Therefore, the homogeneous
configuration $\theta=2\pi n$ is no longer ground state, but the soliton is
the true ground state when $B>B_{\rm c}$. The multiple solitons are created by
increasing $B$, and in general the ground state is a periodic lattice of
soliton which is called the CSL.

The CSLs have been recently studied in various fields. However, most of the
previous arguments are static and they do not address how the homogeneous
ground state is replaced by the soliton when $B$ increases from the value
below $B_{\rm c}$ to the one above $B_{\rm c}$. Is the infinitely large
soliton suddenly created at the moment of $B=B_{\rm c}$? This sounds quite
unphysical. In order to answer to this elementary question, we study quantum
nucleation of the soliton in this paper.

However, the sine-Gordon model in Eq. (3.1) is not the most suitable for that
purpose. This is because the soliton is topologically stable within the
framework of the sine-Gordon model, and so we cannot discuss neither decay nor
nucleation. Thus, we are naturally guided to a linear sigma model as a UV
completion222From the viewpoint of the microscopic fundamental theory like
QCD, the Lagrangian (3.7) should be regarded as a low energy effective theory.
by including a massive degree of freedom (the Higgs mode). We consider a
complex $\phi^{4}$ model (the axion model with the domain wall number one)
with a constant background field ${\bm{B}}$:

$\displaystyle{\cal L}_{\rm
UV}=\left|\partial_{\mu}\phi\right|^{2}-\frac{\lambda}{4}\left(|\phi|^{2}-v^{2}\right)^{2}+vm^{2}(\phi+\phi^{*})+c{\bm{j}}\cdot{\bm{B}}$
(3.7)

with

$\displaystyle
j^{\mu}=-\frac{i}{2}(\phi^{*}\partial^{\mu}\phi-\phi\partial^{\mu}\phi^{*})=|\phi|^{2}\partial^{\mu}\theta\,,\qquad\phi=|\phi|e^{i\theta}.$
(3.8)

This reduces back to the sine-Gordon model in Eq. (3.1) in the limit
$\lambda\to\infty$. Since the last term of Eq. (3.7) does not contribute to
EOM, all the solutions for $c=0$ remain to be solutions for $c\neq 0$.
Therefore, meta-stable solitons attached by strings should exist in the model
(3.7) with $c\neq 0$ if the approximate $U(1)$ condition in Eq. (2.2) is
satisfied.

###  3.2 Nucleation probability of a soliton in the thin-defect approximation

Now let us take $B$ which is ssmaller than $B_{c}$. The ground state should be
the homogeneous configuration. Then, we increase $B$ to any value above
$B_{c}$ instantaneously. The ground state should be solitonic in this case. To
estimate the probability of nucleation of a soliton, we reverse the arguments
about the soliton decay in Sec. 2 in which the nucleation probability of a
hole on the domain wall was calculated, see Fig. 1.

Figure 1: The schematic picture of the decay of soliton by nucleating a hole
(left) and the creation of soliton by nucleating a disk soliton (right). Grey
objects stands for the solitons and the red objects corresponds to the
strings.

Here, we consider a disk of a soliton bounded by a string loop in the
homogeneous vacuum. Let us consider the spatial dimension $d$, and later we
will set $d=2,3$. In the thin-defect approximation, the bounce action reads

$\displaystyle S(R)={\rm vol}(S^{d-1})R^{d-1}\mu+{\rm vol}(B^{d})R^{d}\sigma,$
(3.9)

where $R$ is the radius of the soliton, and ${\rm vol}(S^{d-1})R^{d-1}$ and
${\rm vol}(B^{d})$ are volumes of the unit hypersphere and hyperball, given by

$\displaystyle{\rm
vol}(S^{d-1})=\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}\right)},\qquad{\rm
vol}(B^{d})=\frac{\pi^{d/2}}{\Gamma\left(\frac{d+2}{2}\right)},$ (3.10)

respectively. Note that the string tension $\mu$ is not logarithmically
divergent but a finite constant in the presence of the $U(1)$ breaking term
(the third term in Eq. (3.7)). See Appendix A for some details.

Clearly, $\mu$ is always positive. In contrast, the soliton tension $\sigma$
given in Eq. (3.5) can be either positive or negative. In the absence of the
topological term ($c=0$), $\sigma$ is positive, and $S$ has no stationary
points except for $R=0$. Namely, quantum nucleation of the disk is prohibited.

However, the situation drastically changes in the presence of the topological
term, $c\neq 0$, because the soliton tension can be negative $\sigma<0$ for
$B>B_{\rm c}$. Then, a nontrivial stationary point exists at

$\displaystyle R=\frac{d-1}{d}\frac{{\rm vol}(S^{d-1})}{{\rm
vol}(B^{d})}\frac{\mu}{(-\sigma)}.$ (3.11)

and the nucleation probability can be calculated as

$\displaystyle P_{\text{nucleation}}=Ae^{-S},\quad
S=\frac{(d-1)^{d-1}}{d^{d}}\frac{\left[{\rm
vol}(S^{d-1})\right]^{d}}{\left[{\rm
vol}(B^{d})\right]^{d-1}}\frac{\mu^{d}}{(-\sigma)^{d-1}}.$ (3.12)

Since $-\sigma$ given in Eq. (3.5) is maximized at $\alpha=0$ ($\pi$) for
$c>0$ ($<0$), the bounce action is minimized there with the negative soliton
tension

$\displaystyle\sigma_{\perp}=16mv^{2}\left(1-\frac{B}{B_{\rm c}}\right)<0.$
(3.13)

Therefore, the nucleation probability is maximized for the soliton
perpendicular to ${\bm{B}}$. Once the disk perpendicular to ${\bm{B}}$ is
nucleated, it rapidly expands. The thin-defect approximation is justified for
$R\gg\delta_{\rm dw}$. This can be rewritten as $\mu/(-\sigma_{\perp})\gg
1/m$.

###  3.3 Decay probability of a soliton in external fields in the thin-defect
approximation

Here we consider quantum decay of a soliton in the external field. Consider an
infinitely large flat soliton perpendicular to the external field under the
external field larger than $B_{\rm c}$. The bounce action of a hole on a
soliton can be written in the thin-defect limit as

$\displaystyle S(R)={\rm vol}(S^{d-1})R^{d-1}\mu-{\rm
vol}(B^{d})R^{d}\sigma_{0}\left(1-\frac{B}{B_{\rm c}}\right),$ (3.14)

with $\sigma_{0}=16mv^{2}$. Since the second term is positive for $B>B_{\rm
c}$, the bounce action does not have stationary point, and therefore the decay
is prohibited. We should emphasized that the soliton which is metastable for
$B=0$ is completely stable for $B>B_{\rm c}$.

On the other hand, when we instantaneously decrease $B$ below $B_{\rm c}$, the
stationary point of the bounce action appears at

$\displaystyle R=\frac{B_{\rm c}}{B_{\rm c}-B}\frac{d-1}{d}\frac{{\rm
vol}(S^{d-1})}{{\rm vol}(B^{d})}\frac{\mu}{\sigma_{0}},$ (3.15)

and the value of the action reads

$\displaystyle S=\left(\frac{B_{\rm c}}{B_{\rm
c}-B}\right)^{d-1}\frac{(d-1)^{d-1}}{d^{d}}\frac{\left[{\rm
vol}(S^{d-1})\right]^{d}}{\left[{\rm
vol}(B^{d})\right]^{d-1}}\frac{\mu^{d}}{\sigma_{0}^{d-1}}.$ (3.16)

Comparing this with the bounce action without external field, we find

$\displaystyle S\big{|}_{B=0}<S\big{|}_{0<B<B_{\rm c}}.$ (3.17)

This implies that the decay rate of the soliton is suppressed by the external
field. As the external field $B$ increases toward $B_{\rm c}$ from below, the
action diverges and the quantum decay is strongly suppressed.

##  4 Numerical simulations for quantum nucleation of solitons

In this section, we numerically calculate nucleation probability of solitons.
In Subsec. 4.1 we rewrite the Lagrangian in terms of dimensionless variables.
In Subsecs. 4.2 and 4.3, we calculate nucleation probabilities of solitons by
numerically constructing bounce solutions in 2+1 and 3+1 dimensions,
respectively.

###  4.1 Preliminary

A great benefit of considering the linear sigma model in Eq. (3.7) is that we
can treat the soliton and strings as regular objects of finite sizes. With
them at hand we can go beyond the thin-defect limit.

We will numerically solve EOM of Lagrangian in Eq. (3.7). To this end, it is
useful to rewrite Eq. (3.7) in terms of the dimensionless variables

$\displaystyle\tilde{x}^{\mu}=mx,\quad\tilde{\phi}=v^{-1}\phi,\quad\tilde{\lambda}=\frac{m_{h}^{2}}{m^{2}},\quad\tilde{\bm{B}}=m^{-1}c{\bm{B}}.$
(4.1)

Then, we have the Lagrangian

$\displaystyle{\cal L}_{\rm
UV}=m^{2}v^{2}\left[|\tilde{\partial}_{\mu}\tilde{\phi}|^{2}-\frac{\tilde{\lambda}}{4}\left(|\tilde{\phi}|^{2}-1\right)^{2}+\tilde{\phi}+\tilde{\phi}^{*}+\tilde{\bm{j}}\cdot\tilde{\bm{B}}\right]$
(4.2)

where $\tilde{\lambda}$ is the unique parameter characterizing solutions. For
the (meta-)stable solitons and strings to exist, we need to assume
$\tilde{\lambda}\gg 1$ corresponding to the condition in Eq. (2.2). For
concreteness, we will assume that the soliton is perpendicular to the
$z$-axis. Therefore, the last term in the bracket can be written as

$\displaystyle\tilde{\bm{j}}\cdot\tilde{\bm{B}}=\tilde{B}\tilde{j}_{z}\cos\alpha.$
(4.3)

The Hamiltonian reads

$\displaystyle{\cal H}_{\rm UV}=m^{2}v^{2}\tilde{\cal H}_{\rm
UV},\quad\tilde{\cal H}_{\rm
UV}=\left|\tilde{\nabla}\tilde{\phi}_{z}\right|^{2}+\frac{\tilde{\lambda}}{4}\left(|\tilde{\phi}|^{2}-1\right)^{2}-(\tilde{\phi}+\tilde{\phi}^{*})-\tilde{j}_{z}\tilde{B}\cos\alpha.$
(4.4)

###  4.2 Quantum nucleation of a soliton in $2+1$ dimensions

Here, we investigate nucleation of solitons in $2+1$ dimensions, in which the
soliton is a linear object and the vortex is a particle object. The $2+1$
dimensional version of Eq. (3.9) with $d=2$ is

$\displaystyle S=2\pi R\mu+\pi R^{2}\sigma.$ (4.5)

Its extremum is given by Eq. (3.12) for $d=2$,

$\displaystyle R_{0}=\frac{\mu}{-\sigma},\qquad
S_{0}=\frac{\pi\mu^{2}}{-\sigma}.$ (4.6)

Note that we have $[v]=\frac{1}{2}$, $[\lambda]=1$, and $[m]=1$ in $2+1$
dimensions. We will compare this analytic formula in the thin-defect limit
with numerical simulations for the soliton with finite thickness.

Once we obtain a numerical solution for a soliton attached by a vortex and an
anti-vortex at its both ends, we can measure the dimensionless radius
$\tilde{R}$ (a half length) of the soliton and evaluate the dimensionless
total energy $\tilde{\cal E}$ by

$\displaystyle\tilde{\cal E}(\tilde{R})=\int d^{2}\tilde{x}\ \tilde{\cal
H}_{\rm UV}(\tilde{R}).$ (4.7)

From this, we can evaluate the bounce action through the following formula

$\displaystyle S=\int d^{3}x\,{\cal H}_{\rm UV}=\frac{v^{2}}{m}\int
d^{3}\tilde{x}\,\tilde{\cal H}_{\rm
UV}=\alpha_{1}\frac{v^{2}}{m}\int_{0}^{\tilde{R}}d\tilde{r}~{}\tilde{\cal
E}(\tilde{r}),$ (4.8)

with a constant $\alpha_{1}=\pi$ (see Appendix B for a derivation). To
understand the formula quickly, let us substitute the energy formula
$\tilde{\cal E}(\tilde{R})=2\tilde{\mu}+2\tilde{R}\tilde{\sigma}$ with
$\tilde{\mu}=\mu/v^{2}$ and $\tilde{\sigma}=\sigma/(mv^{2})$ in the thin-
defect limit (the soliton of the length $2\tilde{R}$ with two vortices). We
easily find that the bounce action in Eq. (4.5) is correctly reproduced.

Note that the formula in Eq. (4.8) is valid only for constant $\mu$. If $\mu$
depended on $R$ logarithmically as the usual global vortex, we cannot use Eq.
(4.8). In Appendix A, we show our numerical solution in which the profile of
the scalar field exponentially converges to the VEV in the asymptotic region,
in contrast to the usual global vortex without any domain walls for which the
profiles polynomially approaches to the VEV.

By differentiating $S$ by $R$, we have

$\displaystyle\frac{dS}{dR}=\alpha_{1}\frac{v^{2}}{m}\frac{d\tilde{R}}{dR}\tilde{\cal
E}(\tilde{R})=\alpha_{1}v^{2}\tilde{\cal E}(\tilde{R}).$ (4.9)

The extremum of $S$ is then identified with the zero of ${\cal E}$:

$\displaystyle\frac{dS}{dR}=0\quad\Leftrightarrow\quad\tilde{\cal
E}(\tilde{R})=0.$ (4.10)

The remaining task is constructing suitable numerical configurations with a
soliton bounded by a vortex and an anti-vortex. To this end, we use the
standard relaxation scheme. Our method consists of two steps. Firstly, we take
a product ansatz of a pair of a vortex and an anti-vortex separated at
distance $2\tilde{R}_{\rm ini}$ as an initial configuration of the relaxation.
At this stage, we fix the positions of the vortices. Then, the straight
soliton of the length $2\tilde{R}_{\rm ini}$ is generated and the
configuration converges quite soon. We use this convergent configuration as
the initial configuration for the second relaxation, in which we do not fix
the vortex positions. During the second relaxation process, the vortices
approach to each other due to the soliton tension, and eventually annihilate
each other. We repeatedly measure the distance $2\tilde{R}$ of the vortices
and compute $\tilde{\cal E}(\tilde{R})$.

Figure 2: The soliton attached by a vortex and anti-vortex in $2+1$
dimensions. The negative amplitude $(-|\tilde{\phi}|)$ is plotted.

To be concrete, we take $\tilde{\lambda}=100$ which is large enough for the
soliton and vortices to be classically metastable. The amplitudes
$(-|\tilde{\phi}|)$ for several different separations are shown in Fig. 2. The
two peaks correspond to the vortex and anti-vortex while the linear object
stretching between them is the soliton which is visible only for a large
$\tilde{R}$. We evaluate $\tilde{\cal E}$ with three different values of
$\tilde{B}\cos\alpha=\\{1,1/2,1/4\\}$ in Eq. (4.4).

Figure 3: The dimensionless total energy $\tilde{\cal E}$ is plotted as the
function of the dimensionless radius $\tilde{R}$. Three different choices of
$\tilde{B}\cos\alpha=\\{1,1/2,1/4\\}$ corresponding to $B\geq B_{c}$,
$B=B_{c}$ and $B\leq B_{c}$, respectively, are shown. The right panel is the
zoom-up around $\tilde{R}=0$ of the left one.

The results are shown in Fig. 3. The datas are well fitted by a linear
function in the large $\tilde{R}$ region

$\displaystyle\tilde{\cal E}=2\tilde{R}a+2b,$ (4.11)

which should be compared with the thin-defect limit $\tilde{\cal
E}=2\tilde{R}\tilde{\sigma}+2\tilde{\mu}$. The coefficient $a$ can be either
positive or negative because it is related to the soliton tension which
depends on $\tilde{B}\cos\alpha$. On the other hand, the constant $b$ should
be insensitive on $\tilde{B}\cos\alpha$ because it should be identified with
the vortex tension which is independent on the background field. Indeed, our
numerical solutions show that the three lines almost meet at $\tilde{R}=0$ in
Fig. 3. The coefficients read numerically are shown in Table.1. Thus we
numerically determine the tensions $\tilde{\mu}$ and $\tilde{\sigma}$.
Importantly, $\tilde{\mu}$ is a constant as we mentioned above, see also
Appendix A.

$\tilde{B}\cos\alpha~{}(d=2)$ | $a(\sim\tilde{\sigma})$ | $b(\sim\tilde{\mu})$
---|---|---
$1$ | $-15.6$ | $11.6$
$1/2$ | $0.07$ | $12.0$
$1/4$ | $7.85$ | $12.2$

$\tilde{B}\cos\alpha~{}(d=3)$ | $a(\sim\tilde{\sigma})$ | $b(\sim\tilde{\mu})$ | $c$
---|---|---|---
$1$ | $-15.3$ | $11.8$ | $45.6$
$1/2$ | $-0.15$ | $12.7$ | $28.4$
$1/4$ | $7.78$ | $12.3$ | $24.3$

Table 1: The numerically obtained coefficients in Eqs. (4.11) and (4.17).

Among the three different choices, only $\tilde{B}\cos\alpha=1$ leads to the
negative soliton tension, corresponding to the case of $B\geq B_{c}$. The
stationary point is $\tilde{R}_{0}\simeq 0.75$, and the value of the bounce
action is about $\tilde{S}\simeq b\tilde{R}\simeq 9.0$. Hence, the nucleation
probability can be estimated as

$\displaystyle P_{\rm nucleation}=A\exp\left(-\alpha_{1}\frac{v^{2}}{m}\times
9.0\right).$ (4.12)

Note that the numerically determined values
$(\tilde{R}_{0},\tilde{\sigma},\tilde{\mu})=(0.75,-15.6,11.6)$ is consistent
with the analytic formula $\tilde{R}_{0}=\tilde{\mu}/(-\tilde{\sigma})$ for
the thin-defect limit.

###  4.3 Quantum nucleation of a soliton in $3+1$ dimensions

Next, we numerically investigate quantum nucleation of a disk soliton bounded
by a string loop in $3+1$ dimensions. By putting $d=3$ in Eq. (3.9), the
bounce action of the thin-defect limit is given by

$\displaystyle S=\pi R^{2}\mu+\frac{4\pi}{3}R^{3}\sigma,$ (4.13)

and its extremum is given by Eq. (3.12) for $d=3$,

$\displaystyle R_{0}=\frac{2\mu}{-\sigma},\qquad
S_{0}=\frac{16\pi\mu^{3}}{3\sigma^{2}}.$ (4.14)

The numerical procedures we adopt in this subsection are the same as those in
the previous subsection except for differences due to the spacial dimensions.
We numerically evaluate the dimensionless mass $\tilde{\cal E}$ for a disk of
a soliton of the radius $R$

$\displaystyle\tilde{\cal E}(\tilde{R})=\int d^{3}\tilde{x}~{}\tilde{\cal
H}_{\rm UV},$ (4.15)

where the dimensionless Hamiltonian $\tilde{\cal H}_{\rm UV}$ is given in Eq.
(4.4). Then, we evaluate the bounce action $S$ by the following formula

$\displaystyle S(R)=\int d^{4}x\,{\cal H}_{\rm UV}=\frac{v^{2}}{m^{2}}\int
d^{4}\tilde{x}\,\tilde{\cal H}_{\rm
UV}=\alpha_{2}\frac{v^{2}}{m^{2}}\int_{0}^{\tilde{R}}d\tilde{r}~{}\tilde{\cal
E}(\tilde{r}).$ (4.16)

The constant factor is $\alpha_{2}=4$ whose derivation is given in Appendix B.
Again, we should note that this formula is valid for the constant $\mu$. As a
quick check of the formula, one can reproduce Eq. (4.13) by substituting
$\tilde{\cal E}=2\pi\tilde{R}\tilde{\mu}+\pi\tilde{R}^{2}\tilde{\sigma}$.

Figure 4: The three-dimensional disk soliton perpendicular to the $z$-axis
generated from the two-dimensional linear soliton. In the left panel the red
part shows the region for $|\tilde{\phi}|<0.9$, and the blurred-blue part
corresponds to the region for ${\rm Re}[\tilde{\phi}]<0.4$. The middle and
right panels show $-|\tilde{\phi}|$ on the $\tilde{x}=0$ and $\tilde{z}=0$
cross sections, respectively.

As before we take relatively large value $\tilde{\lambda}=100$ to assure the
classical stability of the solitons and strings. In order to prepare a disk
shape soliton, we recycle the numerical configuration of the linear soliton
attached by two vortices in $2+1$ dimensions. Set the two-dimensional linear
soliton along the $x$-axis on the $xz$-plane, and let $\phi_{\rm 2d}(x,z)$ be
the corresponding field configuration. Then, a three-dimensional disk soliton
perpendicular to the $z$-axis can be obtained by rotating it around the
$z$-axis, namely $\phi_{\rm 3d}(x,y,z)=\phi_{\rm
2d}(x\cos\theta+y\sin\theta,z)$ with $\tan\theta=y/x$, see Fig. 4.

Having this as an initial configuration, we evolve it by a standard relaxation
method. The disk soliton shrinks as the relaxation process proceeds. We
measure the radius and the mass, so that we determine the function
$\tilde{\cal E}(\tilde{R})$. Finally, we calculate the bounce action $S(R)$ by
plugging it to the formula in Eq. (4.16).

Figure 5: The disk soliton bounded by a ring string. The red part shows the
region for $|\tilde{\phi}|<0.9$, and the blurred-blue part corresponds to the
region for ${\rm Re}[\tilde{\phi}]<0.4$.

The time evolution of the disk soliton under the relaxation process is shown
in Fig. 5. The disk is initially large, and we can clearly observe a circular
closed string (red part: $|\tilde{\phi}|<0.9$) and a disk soliton (blurred-
blue part: ${\rm Re}[\tilde{\phi}]<0.4$). We determine the radius $\tilde{R}$
of the ring by seeking the point where $|\tilde{\phi}|=0$. Note that the
energy does not immediately vanishes when $\tilde{R}$ reaches zero. This is a
finite width effect which is missed in the thin-defect limit. Since the
soliton and string are regular objects with finite sizes, a remnant of energy
still exists and it gradually decays and finally disappears.

Figure 6: The dimensionless total energy $\tilde{\cal E}$ is plotted as the
function of the dimensionless radius $\tilde{R}$ for $d=3$. We set
$\tilde{B}\cos\alpha=\\{1,1/2,1/4\\}$. The right panel is the zoom-up near
$\tilde{R}=0$ of the left one.

The numerical results for $\tilde{B}\cos\alpha=\\{1,1/2,1/4\\}$ are shown in
Fig. 6. The fact that the energy of $\tilde{B}\cos\alpha=1$ is negative for
large $\tilde{R}$ indicates that the soliton tension is negative. The zero of
$\tilde{\cal E}$ is found as $\tilde{R}\simeq 2$ which should correspond to
the extremum point of the bounce action. The energy of
$\tilde{B}\cos\alpha=1/2$ is approximately a linear function of $\tilde{R}$
implying the soliton tension vanishes. That of $\tilde{B}\cos\alpha=1/4$ grows
faster than linear functions, implying that the soliton tension is positive.
These are well fitted by

$\displaystyle\tilde{\cal E}=\pi\tilde{R}^{2}a+2\pi\tilde{R}b+c.$ (4.17)

The values of these coefficients are shown in Table 1. The values of $a$
($\sim\tilde{\sigma}$) and $b$ ($\sim\tilde{\mu}$) are consistent between the
left ($d=2$) and right ($d=3$) tables in Table 1. The last term $c$
corresponds to the remnant energy at $\tilde{R}=0$. This is absent in the
thin-defect limit. Hence, the bounce action for the finite size soliton is
slightly larger than the one at the thin-defect limit.

By using the fit in Eq. (4.17), we can evaluate the action for
$\tilde{B}\cos\alpha=1$ as

$\displaystyle
S=\alpha_{2}\frac{v^{2}}{m^{2}}\int_{0}^{\tilde{R}_{0}=2}d\tilde{r}~{}\tilde{\cal
E}(\tilde{r})\simeq 111\alpha_{2}\frac{v^{2}}{m^{2}}.$ (4.18)

Thus, the nucleation probability can be obtained as

$\displaystyle P_{\rm
nucleation}=A\exp\left(-111\alpha_{2}\frac{v^{2}}{m^{2}}\right).$ (4.19)

Before closing this section, let us examine the relevance of the remnant
energy found above. If we fit the numerical data by ignoring the remnant
energy with forcing $c=0$, then we find $a=-16.7$ and $b=16.7$, see the red-
dashed curve in Fig. 6. The value of integration for the constrained fit is
$\int_{0}^{\tilde{R}_{0}=2}d\tilde{r}~{}\tilde{\cal E}\simeq 69$, so that the
nucleation probability is slightly increased. Here, $a=-16.7$ is still
consistent with the one obtained in the $d=2$ case, whereas $b=16.7$ shows a
relatively large discrepancy from $b$ in the $d=2$ case. Hence, the string
tension $b$ is not correctly captured by the constrained fit. Thus, we
conclude that the remnant energy $c$ included in the unconstrained fit is not
a sort of artifacts of the numerical simulation but it should be considered as
a real finite width effect.

In conclusion, we have succeeded in numerically evaluating the bounce action
for the soliton bounded by the string with finite thickness. The finite width
effect has been found and it slightly reduces the nucleation probability
compared with the thin-defect limit.

##  5 Summary and discussion

We have proposed quantum nucleation of topological solitons through quantum
tunneling, as a novel mechanism for formation of topological solitons. We have
discussed chiral solitons in a complex $\phi^{4}$ model (an axion model) with
a topological term, which is a low-energy theory of chiral magnets with an
easy-plane anisotropy and QCD at finite density under strong magnetic field or
rapid rotation. First, we have estimated the creation probability analytically
in terms of tensions of string (vortex) and soliton in the thin-defect
approximation in any dimension. Second, we have performed numerical
simulations in $2+1$ and $3+1$ dimensions by the relaxation (gradient flow)
method, and have obtained creation probabilities. We have found a good
agreement between the thin-defect approximation and direct numerical
simulation in $2+1$ dimensions and have found, in $3+1$ dimensions, a
difference between them at short distances at the subleading order, which can
be interpreted as the remnant energy.

We have considered the complex $\phi^{4}$ model as an UV theory for the sine-
Gordon model at IR limit appearing in various context. Sine-Gordon solitons
are almost insensitive to UV, but different UV theories give different
structures of strings (vortices). However, creation probabilities will be
insensitive to such details.

In this study, we have estimated the nucleation probability in the vacuum
where there are no solitons. In the case with the external field above the
critical value, the true ground state is a CSL. Formation of the CSL in the
homogeneous vacuum should occur in the following process. Let us turn on the
external field $B>B_{\rm c}$ in the homogeneous vacuum. Initially, disk
solitons of the critical radius $R_{0}$ in Eq. (4.14) are nucleated all over
with the creation rate in Eq. (4.19). They rapidly expand as in the right
panel of Fig. 1, growing up to infinitely large solitons. These solitons repel
each other and thus adjust intersoliton distances to minimize the total
energy, thereby eventually forming into a CSL as schematically shown in Fig.
7.

Figure 7: Nucleation of solitons and formation of a CSL.

Of course, this is a rough sketch and we need more detailed analysis of
dynamical process. We also need to calculate nucleation probabilities of
solitons not in the homogeneous vacuum but also in inhomogeneous soliton
backgrounds. For instance, once the CSL ground state is formed, nucleation
probabilities of solitons should be zero in such a background. If we
instantaneously increase (decrease) the external field in the CSL ground
state, the number density of solitons should be decreased (increased). We thus
need nucleation (decay) rates of solitons in the CSL background, which remain
as a future work.

In this paper, we have considered the Abelian sine-Gordon model for
simplicity. On the other hand, it was found in Ref. Eto:2021gyy  that in the
case of two-flavor baryonic matter under rotation, non-Abelian solitons with
non-Abelian ${\mathbb{C}}P^{N-1}$ moduli Nitta:2014rxa ; Eto:2015uqa  are also
present in the ground state of QCD in a certain parameter region. In this
case, a non-Abelian soliton is bounded by a non-Abelian global string
Nitta:2014rxa ; Eto:2021nle . In such a case, the creation probability may
depend on the dimension $N$ of the moduli as numerical factor.

In $2+1$ dimensions, the pseudo-NG mode is mapped to an electromagnetic field
under a duality, while vortices are mapped to charged particles. With nonzero
mass $m$, particles are confined by electric fluxes. In this duality, the
topological term will be mapped to a constant electric field. It remains as a
future problem to study nucleation probabilities in terms of the duality. In
$2+1$ dimensions, there is a BKT transition at finite temperature. It is
interesting to discuss whether there is any conflict between quantum
nucleation and the BKT transition.

Note added: While this paper is being completed, we were informed that the
authors of Ref. Nishimura  was preparing a draft which may have some overlap
with our work.

## Acknowledgements

This work is supported in part by JSPS Grant-in-Aid for Scientific Research
(KAKENHI Grant No. JP22H01221). The work of M. E. is supported in part by the
JSPS Grant-in-Aid for Scientific Research KAKENHI Grant No. JP19K03839 and the
MEXT KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas
“Discrete Geometric Analysis for Materials Design” No. JP17H06462 from the
MEXT of Japan.

##  Appendix A Asymptotic behavior of the scalar field of a vortex attached by
a soliton

Here, we show a numerical solution with some more details to demonstrate that
the amplitude of $\phi$ for a vortex attached by a soliton decays
exponentially fast, in contrast to the slow-tail with a negative power of
isolated global vortices without solitons. For this purpose, we use a
configuration of a vortex and an anti-vortex connected by a soliton discussed
in Sec. 4.2, see Fig. 2. When the vortex and anti-vortex are well separated
compared with the width of the soliton, the configuration around the vortex is
close to that of a single vortex attached by a soliton.

Figure 8: The left panel shows $\tilde{\phi}(\tilde{x},0)$ for the
configuration shown in the top-left panel of Fig. 2, and the right panel shows
asymptotic behavior of the same solution.

The left panel of Fig. 8 shows a scalar field profile
$\tilde{\phi}(\tilde{x},0)$ at a cross section $y=0$ of the configuration
shown in the top-left panel of Fig. 2 with $\tilde{R}=4.83$. One can see that
apart from the vortex core, the scalar field $\tilde{\phi}$ quickly converges
to a constant in the vacuum. To confirm that the asymptotic behavior is an
exponential tail, we show
$\log\delta\tilde{\phi}=\log(\tilde{v}-|\tilde{\phi}|)$ in the right panel of
Fig. 8. We numerically fit the asymptotic tail and the result is
$\delta\tilde{\phi}(\tilde{x},0)\propto e^{-1.8\tilde{x}}$. Thus, the
amplitude converges exponentially fast to the VEV.

We can confirm the same behaviors in any directions from the vortex center
except for the directions of the soliton within its width. This fact implies
that the vortex tension is finite as in Eq. (2.7), in contrast to an isolated
global vortex whose tension is logarithmically divergent as in Eq. (2.3).
Physically, the logarithmically divergent behavior (of an isolated global
vortex) is replaced by the soliton tension. This point was missed in the
literature Preskill:1992ck ; Vilenkin:2000jqa  in which the vortex tension was
assumed to be logarithmically divergent even when the vortex is attached by a
soliton. This is crucial to evaluate bounce actions in the thin-defect limit
and likewise the decay rates and nucleation probabilities of topological
solitons.

##  Appendix B A derivation of Eqs. (4.8) and (4.16)

In this appendix, we give a derivation of Eqs. (4.8) and (4.16) for the
spatial dimension $d=2,3$. However, we will consider generic $d$ below. Since
the number of codimensions of strings(vortices) is 2 and that of solitons is
1, a soliton is a $(d-1)$-dimensional ball $B^{d-1}$ while the string wrapping
the soliton is a $(d-2)$-dimensional sphere $S^{d-2}$. The volumes of unit
$d$-sphere and $d$-ball are given by

$\displaystyle{\rm
vol}(S^{d-1})=\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}\right)},\qquad{\rm
vol}(B^{d})=\frac{\pi^{d/2}}{\Gamma\left(\frac{d+2}{2}\right)}.$ (B.1)

Therefore, the mass in the thin-defect limit reads

$\displaystyle\tilde{\cal E}(\tilde{R})=\int d^{d}\tilde{x}\,\tilde{\cal
H}_{\rm UV}={\rm vol}(S^{d-2})\tilde{R}^{d-2}\tilde{\mu}+{\rm
vol}(B^{d-1})\tilde{R}^{d-1}\tilde{\sigma},$ (B.2)

with the dimensionless tensions of string and soliton

$\displaystyle\tilde{\mu}=\mu/v^{2},\quad\tilde{\sigma}=\sigma/(mv^{2}),$
(B.3)

respectively. On the other hand, the string world-volume is $S^{d-1}$ and the
soliton world-volume is $B^{d-1}$ for the bounce action, and we have

$\displaystyle\tilde{S}(\tilde{R})\equiv\int d^{d+1}\tilde{x}\,\tilde{\cal
H}_{\rm UV}={\rm vol}(S^{d-1})\tilde{R}^{d-1}\tilde{\mu}+{\rm
vol}(B^{d})\tilde{R}^{d}\tilde{\sigma}.$ (B.4)

Integrating $\tilde{\cal E}(\tilde{R})$, we find

$\displaystyle\int^{\tilde{R}}_{0}d\tilde{r}~{}\tilde{\cal E}(\tilde{r})$
$\displaystyle=$ $\displaystyle\frac{1}{d-1}{\rm
vol}(S^{d-2})\tilde{R}^{d-1}\mu+\frac{1}{d}{\rm
vol}(B^{d-1})\tilde{R}^{d}\sigma$ (B.5) $\displaystyle=$
$\displaystyle\left(\frac{1}{d-1}\frac{{\rm vol}(S^{d-2})}{{\rm
vol}(S^{d-1})}\right){\rm
vol}(S^{d-1})\tilde{R}^{d-1}\mu+\left(\frac{1}{d}\frac{{\rm
vol}(B^{d-1})}{{\rm vol}(B^{d})}\right){\rm
vol}(B^{d})\tilde{R}^{d}\sigma.\;\;\;$

Note that the two ratios are identical as

$\displaystyle\alpha_{d-1}\equiv(d-1)\frac{{\rm vol}(S^{d-1})}{{\rm
vol}(S^{d-2})}=d\frac{{\rm vol}(B^{d})}{{\rm vol}(B^{d-1})},$ (B.6)

and therefore we have

$\displaystyle\tilde{S}(\tilde{R})=\alpha_{d-1}\int^{\tilde{R}}_{0}d\tilde{r}~{}\tilde{\cal
E}(\tilde{r}).$ (B.7)

We assume that this formula is valid not only for the thin-defect limit but
also the case that the defects have regular sizes. We have used
$\alpha_{1}=\pi$ and $\alpha_{2}=4$ in the text.

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