# First- and second-order transitions in models with a continuous set of energy minima

## Abstract

A class of models having an n-component order parameter, and whose excitation spectrum ω(q→) has a continuous set {q→} of minima, is studied in d=5-ε dimensions using the renormalization group (RG). It has been argued, at least for small n, that models of this type exhibit fluctuation-induced first-order phase transitions. For the case in which the set of energy minima consists of a single square in reciprocal space, the absence of a stable fixed point for small ε implies that the phase transition is first order for all finite n, in agreement with n=1 results of Mukamel and Hornreich. For n= a stable fixed point does exist; its domain of attraction includes part of the space of physically allowed coupling constants. The transition may, therefore, be either first or second order, depending on the particular choice of coupling constants. An exact solution of the n= single-square model shows that these RG results hold for all d between 3 and 5. The exact solution also determines the structure of the ordered phase; for some choices of the coupling constants, for example, the ordered phase is a spiral, the order parameter behaving as φ→(x→)∼e^1cos(q→•x→)+e^2sin(q→• x→), where q→ lies on the square and e^1•e^2=0. Under the assumption that the critical properties of the square model accurately approximate those of the more physically relevant ring model, we argue that the n-component single-ring model has a first-order transition for all finite n, and a second-order transition for n=. An exact calculation at n= for the single-ring model shows that the transition into a spiral-ordered phase is indeed continuous. © 1981 The American Physical Society.