Quantum nucleation of topological solitons

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.

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 [59] and chiral magnets [60][61][62][63][64][65] with the Dzyaloshinskii-Moriya (DM) interaction [66,67]. The latter has an important nanotechnological application in information processing such as magnetic memory storage devices and magnetic sensors [61]. 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 [68,69] 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 [70][71][72][73]. Since they have been realized in laboratory experiments [74,75], there has been great interests such as an application to information carriers in ultradense memory and logic devices with low energy consumption [76]. The other examples are for instance CP N −1 skyrmion lattices in SU (N ) magnets [77][78][79][80] and 3D skyrmions [81] 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 [82][83][84][85][86] or under rapid rotation [87][88][89], due to a topological term originated from the chiral magnetic effect (CME) [82,90] which is the vector current in the direction of the magnetic field, or chiral vortical effect (CVE) [91][92][93] which is the axial vector current in the direction of the rotation axis, respectively. They also appear with thermal fluctuation [94][95][96] (see also Refs. [97][98][99]). 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 [100][101][102] (see Refs. [2,7,103] 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 [104] (see Ref. [20] as a review, and Refs. [105][106][107] 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 [108,109] and axial domain wall-vortex composites in QCD [110]. 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 [111][112][113] and non-Abelian strings in dense QCD [14,114].
In this paper, we study quantum nucleation of topological solitons through quantum tunneling. For definiteness, we discuss chiral solitons in a complex φ 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 thindefect 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 φ 4 model (the axion model with the domain wall number one) without a topological term. In Sec. 3, we present our model (the complex φ 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.

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 φ 4 model (an axion model with the domain wall number one). The minimal model in 3 + 1 dimensions is with v and m are parameters whose mass dimension is 1, and λ 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 φ → e iη φ spontaneously broken in the homogeneous vacuum |φ| = v.
There is a Nambu-Goldstone (NG) mode and a Higgs mode whose mass is m h = v √ λ. 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 There the vacuum expectation value can be approximated as 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 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 λ → ∞ where the wine bottle potential becomes infinitely steep (m h → ∞), so that the amplitude of φ freezes out as |φ| = v. Writing φ = ve iθ and plugging it into Eq. (2.1), we are lead to the sine-Gordon model 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 θ = 0 with the redundancy of 2πn (n ∈ Z). The ground state energy E (vac) = 0. In addition, there is a sine-Gordon soliton which we take perpendicular to the z-axis without loss of generality: This connects θ = 0 at z → −∞ and θ = 2π at z → ∞. The thickness and the tension, namely the energy per unit area, of the soliton are given by Note that the soliton in the sine-Gordon limit is classically stable but it could be quantummechanically unstable. This is because it can end on a sting which is infinitely thin (δ st → 0) in the λ → ∞ limit, so that holes surrounded by the strings can be created by quantum tunneling effect. The instability of the soliton for a finite λ 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 (φ + φ * ) ∼ 2m 2 v 2 cos θ can pass slip the potential barrier around |φ| = 0 and shrink to the unique vacuum. This is the classical instability of the soliton in the UV theory with finite λ. 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 = 0 from the massless case m = 0 in which the tension is logarithmically divergent as in Eq. (2.3). 1 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 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 δ st δ 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 δ dw , we can use the thin-defect approximation, providing the decay probability [104] where µ is the constant tension of the string and the soliton tension σ is well approximated by Eq.
On the other hand, the classical stability requires v 2 /m 2 1/λ. 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.

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 φ 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.

The models with external fields
We consider the sine-Gordon model under a constant background field B in 3 + 1 dimensions, given by 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 B for either a magnetic field, a rotation or DM term. The Hamiltonian reads 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 θ = 2πn remains the solution of EOM. The energy density is also unchanged from zero. The soliton solution also remains to be the same where we have introduced an arbitrary unit vectorn perpendicular to the soliton. The solution itself is unchanged, however, the soliton tension receives a correction from the background field. Let α be relative angle betweenn and the constant background field B. Then, the additional energy per unit area of the single soliton reads where B = |B|, and we have used the fact that θ increases by 2π when one traverses the soliton alongn. The net tension of the single soliton reads This is minimized when α = 0 (π) for c > 0 (< 0). Namely, the most stable soliton is perpendicular to the external field B. This implies that the soliton is tensionless at the critical value Moreover, it is negative for B > B c . Therefore, the homogeneous configuration θ = 2πn is no longer ground state, but the soliton is the true ground state when B > B 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 c to the one above B c . Is the infinitely large soliton suddenly created at the moment of B = B 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 completion 2 by including a massive degree of freedom (the Higgs mode). We consider a complex φ 4 model (the axion model with the domain wall number one) with a constant background field B:

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.
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 where R is the radius of the soliton, and vol(S d−1 )R d−1 and vol(B d ) are volumes of the unit hypersphere and hyperball, given by (3.10) respectively. Note that the string tension µ 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, µ is always positive. In contrast, the soliton tension σ given in Eq. (3.5) can be either positive or negative. In the absence of the topological term (c = 0), σ 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 = 0, because the soliton tension can be negative σ < 0 for B > B c . Then, a nontrivial stationary point exists at . (3.11) and the nucleation probability can be calculated as (3.12) Since −σ given in Eq. (3.5) is maximized at α = 0 (π) for c > 0 (< 0), the bounce action is minimized there with the negative soliton tension Therefore, the nucleation probability is maximized for the soliton perpendicular to B. Once the disk perpendicular to B is nucleated, it rapidly expands. The thin-defect approximation is justified for R δ dw . This can be rewritten as µ/(−σ ⊥ ) 1/m.

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 c .
The bounce action of a hole on a soliton can be written in the thin-defect limit as with σ 0 = 16mv 2 . Since the second term is positive for B > B 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 c . On the other hand, when we instantaneously decrease B below B c , the stationary point of the bounce action appears at 15) and the value of the action reads (3.16) Comparing this with the bounce action without external field, we find This implies that the decay rate of the soliton is suppressed by the external field. As the external field B increases toward B c from below, the action diverges and the quantum decay is strongly suppressed.

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.

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 Then, we have the Lagrangian whereλ is the unique parameter characterizing solutions. For the (meta-)stable solitons and strings to exist, we need to assumeλ 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 j ·B =Bj z cos α. (4. 3) The Hamiltonian reads

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 Its extremum is given by Eq. (3.12) for d = 2, Note that we have [v] = 1 2 , [λ] = 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 antivortex at its both ends, we can measure the dimensionless radiusR (a half length) of the soliton and evaluate the dimensionless total energyẼ bỹ From this, we can evaluate the bounce action through the following formula with a constant α 1 = π (see Appendix B for a derivation). To understand the formula quickly, let us substitute the energy formulaẼ(R) = 2μ + 2Rσ withμ = µ/v 2 andσ = σ/(mv 2 ) in the thin-defect limit (the soliton of the length 2R 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 µ. If µ 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 The extremum of S is then identified with the zero of E: 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 2R 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 2R 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 2R of the vortices and computeẼ(R).
To be concrete, we takeλ = 100 which is large enough for the soliton and vortices to be classically metastable. The amplitudes (−|φ|) 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 largeR. We evaluateẼ with three different values ofB cos α = {1, 1/2, 1/4} in Eq. (4.4).  The results are shown in Fig. 3. The datas are well fitted by a linear function in the largeR regionẼ = 2Ra + 2b, (4.11) which should be compared with the thin-defect limitẼ = 2Rσ + 2μ. The coefficient a can be either positive or negative because it is related to the soliton tension which depends onB cos α. On the other hand, the constant b should be insensitive onB cos α 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 atR = 0 in Fig. 3. The coefficients read numerically are shown in Table.1. Thus we numerically determine the tensionsμ andσ. Importantly,μ is a constant as we mentioned above, see also Appendix A.
Among the three different choices, onlyB cos α = 1 leads to the negative soliton tension, corresponding to the case of B ≥ B c . The stationary point isR 0 0.75, and the value of the bounce action is aboutS bR 9.0. Hence, the nucleation probability can be estimated as Note that the numerically determined values (R 0 ,σ,μ) = (0.75, −15.6, 11.6) is consistent with the analytic formulaR 0 =μ/(−σ) for the thin-defect limit.

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 and its extremum is given by Eq. (3.12) for d = 3, (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 massẼ for a disk of a soliton of the radius R where the dimensionless HamiltonianH UV is given in Eq. (4.4). Then, we evaluate the bounce action S by the following formula The constant factor is α 2 = 4 whose derivation is given in Appendix B. Again, we should note that this formula is valid for the constant µ. As a quick check of the formula, one can reproduce Eq. (4.13) by substitutingẼ = 2πRμ + πR 2σ . 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 |φ| < 0.9, and the blurred-blue part corresponds to the region for Re[φ] < 0.4. The middle and right panels show −|φ| on thex = 0 andz = 0 cross sections, respectively.
As before we take relatively large valueλ = 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 φ 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 φ 3d (x, y, z) = φ 2d (x cos θ + y sin θ, z) with tan θ = 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 functionẼ(R). Finally, we calculate the bounce action S(R) by plugging it to the formula in Eq. (4.16). The time evolution of the disk soliton Figure 5: The disk soliton bounded by a ring string. The red part shows the region for |φ| < 0.9, and the blurred-blue part corresponds to the region for Re[φ] < 0.4. 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: |φ| < 0.9) and a disk soliton (blurred-blue part: Re[φ] < 0.4). We determine the radiusR of the ring by seeking the point where |φ| = 0. Note that the energy does not immediately vanishes whenR 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.
The numerical results forB cos α = {1, 1/2, 1/4} are shown in Fig. 6. The fact that the energy ofB cos α = 1 is negative for largeR indicates that the soliton tension is negative. The zero ofẼ is found asR 2 which should correspond to the extremum point of the bounce action. The energy ofB cos α = 1/2 is approximately a linear function of R implying the soliton tension vanishes. That ofB cos α = 1/4 grows faster than linear functions, implying that the soliton tension is positive. These are well fitted bỹ E = πR 2 a + 2πRb + c. (4.17) The values of these coefficients are shown in Table 1. The values of a (∼σ) and b (∼μ) are consistent between the left (d = 2) and right (d = 3) tables in Table 1. The last term c corresponds to the remnant energy atR = 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 forB cos α = 1 as Thus, the nucleation probability can be obtained as 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 R 0 =2 0 drẼ 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.

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 φ 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 φ 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 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. 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. [89] that in the case of two-flavor baryonic matter under rotation, non-Abelian solitons with non-Abelian CP N −1 moduli [115,116] 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 [115,117]. 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. [118] was preparing a draft which may have some overlap with our work. vortex attached by a soliton. The left panel of Fig. 8 shows a scalar field profileφ(x, 0) at a One can see that apart from the vortex core, the scalar fieldφ quickly converges to a constant in the vacuum. To confirm that the asymptotic behavior is an exponential tail, we show log δφ = log(ṽ − |φ|) in the right panel of Fig. 8. We numerically fit the asymptotic tail and the result is δφ(x, 0) ∝ e −1.8x . 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 [20,104] 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.
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 (B.1) Therefore, the mass in the thin-defect limit reads with the dimensionless tensions of string and solitoñ µ = µ/v 2 ,σ = σ/(mv 2 ), (B. 3) respectively. On the other hand, the string world-volume is S d−1 and the soliton worldvolume is B d−1 for the bounce action, and we havẽ Note that the two ratios are identical as , (B.6) and therefore we haveS 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 α 1 = π and α 2 = 4 in the text.