# Anisotropic sine-Gordon model and infinite-order phase transitions in three dimensions

## Abstract

A three-dimensional anisotropic sine-Gordon model, derived as the spin-wave approximation to the biaxial (m=2) Lifshitz point problem in a uniform magnetic field, is shown to possess [in close analogy to the isotropic two-dimensional (2D) sine-Gordon theory which is well known to describe the critical behavior of the 2D XY model], a surface of infinite-order phase transitions. This critical surface separates a phase characterized by infinite correlation length and power-law decay of correlations, and controlled by a stable fixed line, from one with finite and exponential decay. As the critical surface is approached from the latter phase, diverges as exp (t-) where =1 is a universal number, t measures the distance from the critical surface, and is nonuniversal. On the critical surface correlations decay like r-(lnr)-, where =4 and =0.88. Speculations on the occurrence of an infinite-order transition in liquid-crystal mixtures exhibiting nematic, smectic-A, and smectic-C phases are advanced. © 1981 The American Physical Society.