  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.

High Energy Physics - Theory;Mesoscale and Nanoscale Physics;High Energy Physics - Phenomenology