Soft breaking of the $\mathbb{Z}_2$ symmetry in the $\phi^4$ theory

Abstract

We consider a two-dimensional scalar field theory that modifies the standard $\phi^4$ model by introducing a smooth breaking of translational invariance through a hyperbolic generalizing function. This function explicitly breaks the $\mathbb{Z}_2$ symmetry; however, it also introduces a mechanism capable of generating new energy minima (vacua) and of localizing or delocalizing field fluctuations around these vacua. Thus, this mechanism enables the continuous transformation of the kink/antikink-like configurations into compacted asymmetric double-kink/antikink structures. Accordingly, this transformation gives rise to new classes of configurations resembling asymmetric non-topological solitons, characterized by a soft breaking of the $\mathbb{Z}_2$ symmetry. These findings are particularly compelling, as they are consistent with results describing Su-Schrieffer-Heeger (SSH) domain walls in dimerized polymer systems.