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

TL;DR

The work addresses soft breaking of the discrete $\mathbb{Z}_2$ symmetry in a two-dimensional $\phi^4$-type theory by introducing a hyperbolic generalizing function $f(\phi)$ in the kinetic term. It develops a noncanonical framework, derives the BPS equations $\phi' = \pm W_\phi/f$ with $V(\phi) = W_\phi^2/(2f)$, and demonstrates equivalence to a canonical formulation via a field redefinition. Through analytical and numerical analysis of a hyperbolically deformed $\phi^4$ model (with $W(\phi)=\sqrt{\lambda}(\nu^2\phi-\phi^3/3)$ and $f(\phi)$ controlled by parameters $p,z,q$), the paper shows the emergence of symmetric and asymmetric double-kink/antikink configurations, preserved translational zero modes, and rich stability spectra. It also connects these solitonic structures to Su–Schrieffer–Heeger (SSH) domain walls in dimerized polymers, suggesting potential applications to charge transport and topological materials in low dimensions.

Abstract

We consider a two-dimensional scalar field theory that modifies the standard $φ^4$ 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.

Figures (19)

• Figure 1: Potential $V(\phi)$ vs. $\phi$ varying the parameter $z$ while keeping $p$ constant. In all case, we adopt $\lambda=\nu=q=1$.
• Figure 2: Potential $V(\phi)$ vs. $\phi$ varying the parameter $p$ while keeping $z$ constant. In all cases, we adopt $\lambda=\nu=q=1$.
• Figure 3: Field solution $\phi(x)$ vs. $x$ without asymmetry (i.e., $p=0$) and varying $z$. For all cases, we adopt $\lambda=\nu=q=1$.
• Figure 4: Field solution $\phi(x)$ vs. $x$ with left-handed asymmetry (i.e., $p=1$) and varying $z$. For all cases, we use $\lambda=\nu=q=1$.
• Figure 5: Field solution $\phi(x)$ vs. $x$ with right-handed asymmetry (i.e., $p=1$) and varying $z$. For all cases, we use $\lambda=\nu=q=1$.