A renormalization-group study of helimagnets in D = 2 + ϵ dimensions

Abstract The nonlinear sigma model O(N) ⊗ O(2)/O(N − 2) ⊗ O (2) describing the phase transition of N-component helimagnets is built and studied up to two-loop order in D = 2 + ϵ dimensions. It is shown that a stable fixed point exists as soon as N is greater than 3 (or equal) in the neighborhood of two dimensions. The critical exponents ν and η are obtained. In the N = 3 case, the symmetry of the system is dynamically enlarged at the fixed point from O(3) ⊗ O(2)/O(2) to O(3) × O(3)/O(3) ∼ O(4)/O(3). We show that the order parameter for Heisenberg helimagnets involves a tensor representation of O(4) and we verify it explicitly at one-loop order on the value of the exponents. We show that for large N and in the neighborhood of two dimensions this nonlinear sigma model describes the same critical theory as the Landau-Ginzburg linear theory. As a consequence, the critical behavior evolves smoothly between D = 2 and D = 4 dimensions in this limit. However taking into account the old results from the D = 4 − ϵ expansion of the linear theory, we show that most likely the nature of the transition must change between D = 2 and D = 4 dimensions for sufficiently small N (including N = 3). The simplest possibility is that there exists a dividing line Nc(D) in the plane (N, D) separating a first-order region containing the Heisenberg point at D = 4 and a second-order region containing the whole D = 2 axis. We conclude that the phase transition of Heisenberg helimagnets in dimension 3 is either first order or second order with O(4) exponents involving a tensor representation or tricritical with mean-field exponents.