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.