Composite topological solitons consisting of domain walls, strings, and monopoles in $O(N)$ models

We study various composites of global solitons consisting of domain walls, strings, and monopoles in linear $O(N)$ models with $N=2$ and $3$. Spontaneous symmetry breaking (SSB) of the $O(N)$ symmetry down to $O(N-1)$ results in the vacuum manifold $S^{N-1}$, together with a perturbed scalar potential in the presence of a small explicit symmetry breaking (ESB) interaction. The $O(2)$ model is equivalent to the axion model admitting topological global (axion) strings attached by $N_{\rm DW}$ domain walls. We point out for the $N_{\rm DW} = 2$ case that the topological stability of the string with two domain walls is ensured by sequential SSBs $(\mathbb{Z}_2)^2 \to \mathbb{Z}_2 \to 1$, where the first SSB occurs in the vacuum leading to the topological domain wall as a mother soliton, only inside which the second SSB occurs giving rise to a subsequent kink inside the mother wall. From the bulk viewpoint, this kink is identical to a global string as a daughter soliton. This observation can be naturally extended to the $O(3)$ model, where a global monopole as a daughter soliton appears as a kink in a mother string or as a vortex on a mother domain wall, depending on ESB interactions. In the most generic case, the stability of the composite system consisting of the monopole, string, and domain wall is understood by the SSB $(\mathbb{Z}_2)^3 \to (\mathbb{Z}_2)^2 \to \mathbb{Z}_2 \to 1$, in which the first SSB at the vacuum gives rise to the domain wall triggering the second one, so that the daughter string appears as a domain wall inside the mother wall triggering the third SSB, which leads to a granddaughter monopole as a kink inside the daughter vortex. We demonstrate numerical simulations for the dynamical evolution of the composite solitons.