Novel Phase Structure of Twisted O(N) phi^4 model on M^D-1 \times S^1

TL;DR

This paper explores the complex phase structure of the twisted O(N) phi^4 model on a compactified space, revealing unconventional symmetry breaking patterns influenced by boundary conditions and the size of the compact dimension.

Contribution

It provides a detailed classification of symmetry breaking patterns in the twisted O(N) phi^4 model on M^{D-1} imes S^1, highlighting novel phase structures and dependence on boundary conditions and radius.

Findings

Symmetry breaking patterns depend on boundary conditions and S^1 radius.

Unconventional symmetry breaking patterns are identified.

Spontaneous translational invariance breaking is discussed.

Abstract

We study the O(N) $\phi^4$ model compactified on $M^{D-1}\otimes S^1$, which allows to impose twisted boundary conditions for the $S^1$-direction. The O(N) symmetry can be broken to $H$ explicitly by the boundary conditions and further broken to $I$ spontaneously by vacuum expectation values of the fields. The symmetries $H$ and $I$ are completely classified and the model turns out to have unexpectedly a rich phase structure. The unbroken symmetry $I$ is shown to depend on not only the boundary conditions but also the radius of $S^1$, and the symmetry breaking patterns are found to be unconventional. The spontaneous breakdown of the translational invariance is also discussed.