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

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 $L_{c}$.