Vacuum Structure of Twisted Scalar Field Theories on M^{D-1} \otimes S^1

TL;DR

This paper investigates scalar field theories on a space with a circular dimension, revealing phenomena like critical radii, phase transitions, symmetry breaking, and spontaneous translational symmetry breaking, with detailed analysis of the O(N)  model.

Contribution

It provides a detailed analysis of twisted boundary conditions in scalar theories on M^{D-1}  S^1, highlighting novel features such as critical radii and spontaneous symmetry breaking.

Findings

Identification of critical radii for phase transitions.

Radiative corrections can restore or break symmetries.

Spontaneous breaking of translational invariance along S^1.

Abstract

We study scalar field theories on M^{D-1} \otimes S^1, which allow to impose twisted boundary conditions for the S^1 direction, in detail and report several interesting properties overlooked so far. One of characteristic features is the appearance of critical radii of the circle S^1. A phase transition can occur at the classical level or can be caused by quantum effects. Radiative corrections can restore broken symmetries or can break symmetries for small radius. A surprising feature is that the translational invariance for the S^1 direction can spontaneously be broken. A particular class of coordinate-dependent vacuum configurations is clarified and the O(N) \phi^4 model on M^{D-1}\otimes S^1 is extensively studied, as an illustrative example.