String-wall composites winding around a torus knot vacuum in an axionlike model

We study a simple axionlike model with a charged scalar $\phi$ and a double-charged scalar $\zeta$ of global $U(1)$ symmetry. A particular feature of our model is that a vacuum manifold is a torus knot. We consider a hierarchical symmetry-breaking scenario where $\zeta$ first condenses, giving rise to cosmic $\zeta$-strings, and the subsequent condensation of $\phi$ leads to domain-wall formation spanning the $\zeta$-strings. We find that the formation of the walls undergoes two different regimes depending on the magnitude of an explicit breaking term of the relative $U(1)$ between $\zeta$ and $\phi$. One is the weakly interacting regime where the walls are accompanied by another cosmic $\phi$ strings. The other is the strongly interacting regime where no additional strings appear. In both regimes, neither a $\zeta$-string, a $\phi$-string nor a wall alone is topological, but the composite of an appropriate number of strings and walls as a whole is topologically stable, characterized by the fundamental homotopy group of the torus knot. We confirm the formation and the structure of the string-wall system by first-principle cosmological two-dimensional simulations. We find stable string-wall composites at equilibrium, where the repulsive force between $\zeta$-strings and the tension of walls balances, and a novel reconnection of the string-wall composites.