Models for phase transitions with continuous constant energy surfaces

We have studied models of phase transitions of a type introduced by Brasovskii. These models involve an infinite number of degenerate order parameters, each associated with a finite nonzero wave vector. Brasovskii discussed the case of a spherical constant-energy surface in reciprocal space. We extend the Brasovskii model to different spatial dimensions $d$ and to different continuous constant-energy surfaces of dimension $m$. The ring model, the model discussed in most detail, has a continuous constant-energy surface described by a two-dimensional circle embedded in a $d$ dimensional space. The order of the phase transition to the nonuniform state is shown to depend on the number of independent directions orthogonal to the constant-energy surface in reciprocal space. When there are more than four directions perpendicular to the constant-energy surface ($d>\mathrm{}5\mathrm{}$ for the ring model), fluctuations are unimportant and the transition is continuous with mean-field exponents. However, when there are less than four directions ($d<\mathrm{}5\mathrm{}$ for the ring model), the phase transition to a nonuniform state is shown to be first order, rather than second order, as predicted by mean-field theory. We also present the analogous results for the more general models.