Vacuum Structure of Two-Dimensional $\phi^4$ Theory on the Orbifold $S^{1}/Z_{2}$

TL;DR

This paper investigates the vacuum structure and quantum corrections of two-dimensional $^4$ theory on an orbifold, revealing a phase transition at a critical size and differences in fermionic zero modes.

Contribution

It provides a detailed analysis of phase transitions, vacuum degeneracy, and quantum corrections in $^4$ theory on $S^{1}/Z_{2}$, including supersymmetric cases.

Findings

Phase transition at critical length $L_c=2\pi/m$.

Degenerate vacua for $L>L_c$.

Quantum corrections cancel singularities at $L=0$.

Abstract

We consider the vacuum structure of two-dimensional $\phi^4$ theory on $S^{1}/Z_{2}$ both in the bosonic and the supersymmetric cases. When the size of the orbifold is varied, a phase transition occurs at $L_{c}=2\pi/m$, where $m$ is the mass of $\phi$. For $L<L_{c}$, there is a unique vacuum, while for $L>L_{c}$, there are two degenerate vacua. We also obtain the 1-loop quantum corrections around these vacuum solutions, exactly in the case of $L<L_{c}$ and perturbatively for $L$ greater than but close to $L_{c}$. Including the fermions we find that the "chiral" zero modes around the fixed points are different for $L<L_{c}$ and $L>L_{c}$. As for the quantum corrections, the fermionic contributions cancel the singular part of the bosonic contributions at L=0. Then the total quantum correction has a minimum at the critical length $L_{c}$.