Coupling to anisotropic elastic media: Magnetic and liquid-crystal phase transitions

A generalization of the Larkin-Pikin-Sak model in which an $n$-component order parameter is coupled to a general anisotropic elastic continuum is studied using the $\epsilon$ expansion. It is found that the fixed-point structure is the same as the isotropic model but that all fixed points are unstable with respect to anisotropic perturbations, independent of external boundary conditions. The compressible smectic-$A$ to smectic-$C$ liquid-crystal transition is also studied. It is found to be unaffected by elastic degrees of freedom and is, therefore, expected to have helium exponents, as previously predicted by de Gennes.