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 $\mathrm{XY}$ model], a surface of infinite-order phase transitions. This critical surface separates a phase characterized by infinite correlation length $\ensuremath{\xi}$ and power-law decay of correlations, and controlled by a stable fixed line, from one with finite $\ensuremath{\xi}$ and exponential decay. As the critical surface is approached from the latter phase, $\ensuremath{\xi}$ diverges as exp ($\ensuremath{\sigma}{t}^{\ensuremath{-}\ensuremath{\nu}}$) where $\ensuremath{\nu}=1$ is a universal number, $t$ measures the distance from the critical surface, and $\ensuremath{\sigma}$ is nonuniversal. On the critical surface correlations decay like ${r}^{\ensuremath{-}\ensuremath{\eta}}{(\mathrm{ln}r)}^{\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}}$, where $\ensuremath{\eta}=4$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}=0.88\ensuremath{\cdots}$. Speculations on the occurrence of an infinite-order transition in liquid-crystal mixtures exhibiting nematic, smectic-$A$, and smectic-$C$ phases are advanced.