In a general Landau-Ginzburg-Wilson Hamiltonian, symmetry allows a term linear in the gradient and trilinear in the order parameter in many cases. General renormalization-group equations to $O\left(\epsilon \right)$ are given for such terms, and applied to the antiferrodistortive phase transition in calcite ($R\bar{3}c$). The commensurate transition is continuous in mean-field theory, but in the renormalization-group theory, this new term is relevant near all fixed points to $O\left({\epsilon }^2\right)$ and causes a fluctuation-induced first-order transition.

• Received 30 April 1984

DOI:https://doi.org/10.1103/PhysRevLett.53.2425