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 $\ensuremath{\xi}$ 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.
The (1+1)-dimensional sine-Gordon model is studied in the framework of the variational Gaussian wave functionals. The renormalized mass depends on two dimensionless parameters. In the phase diagram a first-order phase transition will replace the second-order phase transition for large bare mass. The spectrum of the two-particle states consists of a continuum of scattering states and the isolated values connected to the excitonlike bound states. The energy of the bound states and the phase shifts of the scattering states are explicitly calculated.