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

A three-dimensional anisotropic sine-Gordon model, derived as the spin-wave approximation to the biaxial ($m=2\mathrm{}$) 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 $\xi$ and power-law decay of correlations, and controlled by a stable fixed line, from one with finite $\xi$ and exponential decay. As the critical surface is approached from the latter phase, $\xi$ diverges as exp ($\sigma t^{-\nu }$) where $\nu =1\mathrm{}$ is a universal number, $t$ measures the distance from the critical surface, and $\sigma$ is nonuniversal. On the critical surface correlations decay like $r^{-\eta }{\left(\ln r\right)}^{-\tilde{\eta }}$, where $\eta =4\mathrm{}$ and $\tilde{\eta }=0.88\mathrm{}\cdots$. Speculations on the occurrence of an infinite-order transition in liquid-crystal mixtures exhibiting nematic, smectic-$A$, and smectic-$C$ phases are advanced.