Directional phase instability on a cubic compressible lattice near a second-order phase transition with a three-component order parameter

We have examined the question of whether a second-order phase transition can occur in a normal manner for a $n=3\mathrm{}$ component order parameter on a compressible cubic lattice, for which the perovskites are an example. For the general third-order strain coupling, the strain-mediated quartic interactions of the order parameter are found to be controlled by three exponents ${\alpha }_I$, ${\alpha }_{2I}$, and $\alpha _{}^{\prime }{}_{2I}{}^{}$ obtained on the incompressible lattice. Since the specific-heat exponent ${\alpha }_I<0\mathrm{}$ for $n=3\mathrm{}$ in $d=3\mathrm{}$ dimensions, the component of the strain-mediated interactions controlled by ${\mathrm{\alpha }}_{\mathrm{I}}$ does not have the microscopic directional instability discussed by Bergman and Halperin and by de Moura, Lubensky, Imry, and Aharony for the Ising model on an elastically nonisotropic lattice. The expansion in $\epsilon =4-d$ to $O\left({\epsilon }^2\right)$ indicates that ${\alpha }_{2I}$ and $\alpha _{}^{\prime }{}_{2I}{}^{}$ are positive at $d=3\mathrm{}$. The twenty components of the interaction controlled basically by ${\alpha }_{2I}$ or $\alpha _{}^{\prime }{}_{2I}{}^{}$ will thus exhibit the instability and, contrary to the conjecture of de Moura et al., will prevent the cubic or Heisenberg fixed points or the fourteen possible Fisher-renormalized versions thereof from being reached. Stability is not restored if the host lattice is taken to be elastically isotropic but is restored if the strain couples only to the scalar product of order-parameter fields. Otherwise, it appears that the $n=3\mathrm{}$ compressible cubic phase transition cannot be simply second order.