Spontaneously Broken Translational Invariance of Compactified Space

TL;DR

This paper introduces a mechanism for spontaneous breaking of translational invariance in compactified spaces, demonstrated through a real ^4 model on a circle with nontrivial boundary conditions, revealing symmetry breaking at large radii.

Contribution

It presents a novel mechanism for spontaneous translational invariance breaking in compactified spaces using a ^4 model with boundary conditions, highlighting symmetry breaking related to the compactification radius.

Findings

Translational invariance breaks spontaneously when radius exceeds a critical value.

Model behaves like a ^4 theory on a kink background at large radius.

Symmetry breaking involves both translational and global symmetries.

Abstract

We propose a mechanism to break the translational invariance of compactified space spontaneously. As a simple model, we study a real $\phi^4$ model compactified on $M^{D-1}\otimes S^1$ in detail, where we impose a nontrivial boundary condition on $\phi$ for the $S^1$-direction. It is shown that the translational invariance for the $S^1$-direction is spontaneously broken when the radius $R$ of $S^1$ becomes larger than a critical radius $R^*$ and also that the model behaves like a $\phi^4$ model on a single kink background for $R \to \infty$. It is pointed out that spontaneous breakdown of translational invariance is accompanied by that of some global symmetries, in general, in our mechanism.