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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 sizeof 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 quantumcorrections around these vacuum solutions, exactly in the case of $L<L_{c}$ andperturbatively for $L$ greater than but close to $L_{c}$. Including thefermions we find that the "chiral" zero modes around the fixed points aredifferent for $L<L_{c}$ and $L>L_{c}$. As for the quantum corrections, thefermionic contributions cancel the singular part of the bosonic contributionsat L=0. Then the total quantum correction has a minimum at the critical length$L_{c}$.Comment: Revtex, 15 pages, 3 eps figure

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Vacuum structure of two-dimensional<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:math>theory on the orbifold<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mo>/</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>

Vacuum structure of two-dimensionalϕ4theory on the orbifoldS1/Z2