Phase structure of renormalizable four-fermion models in spacetimes of constant curvature.

Abstract

A number of 2D and 3D four-fermion models which are renormalizable, in the 1/N expansion, in a maximally symmetric constant curvature space are investigated. To this purpose, a powerful method for the exact study of spinor heat kernels and propagators on maximally symmetric spaces is reviewed. The renormalized effective potential is found for any value of the curvature and its asymptotic expansion is given explicitly, both for small and for strong curvature. The influence of gravity on the dynamical symmetry-breaking pattern of some U(2) flavorlike and discrete symmetries is described in detail. The phase diagram in ${\mathit{S}}^{2}$ is constructed and it is shown that, for any value of the coupling constant, a curvature exists above which chiral symmetry is restored. For the case of ${\mathit{H}}^{2}$, chiral symmetry is always broken. In three dimensions, in the case of positive curvature, ${\mathit{S}}^{3}$, it is seen that curvature can induce a second-order phase transition. For ${\mathit{H}}^{3}$ the configuration given by the auxiliary fields equated to zero is not a solution of the gap equation. The physical relevance of the results is discussed. \textcopyright{} 1996 The American Physical Society.