Coupled order parameters, symmetry-breaking irrelevant scaling fields, and tetracritical points

Abstract

The phase diagrams of systems described by a Hamiltonian containing an anisotropic quadratic term of the form $\frac{1}{2}g\Sigma {\alpha =1\mathrm{}}^n{c}_{\alpha }\int {\overrightarrow{\mathrm{x}}}^{}{S}_{\alpha }^2\left(\overrightarrow{\mathrm{x}}\right)$, and a cubic anisotropic term $\nu \Sigma {\alpha =1\mathrm{}}^n\int {\overrightarrow{\mathrm{x}}}^{}{S}_{\alpha }^4\left(\overrightarrow{\mathrm{x}}\right)$, are studied using mean-field theory, scaling theory, and expansions in $\epsilon \mathrm{}(=4-d)\mathrm{}$ and $\frac{1}{n}$. Here, $S_{\alpha }\left(\overrightarrow{\mathrm{x}}\right)$ ($a=1, \dots , n$) is a local $n$-component ordering variable. Systems to which the analysis is applicable include perovskite crystals, stressed along the [100] direction ($n=3\mathrm{}$), anisotropic antiferromagnets in a uniform field, uniaxially anisotropic ferromagnets, ferroelectric ferromagnets and crystalline ${}_{}{}^4{}_{}{}^{}\mathrm{He}\mathrm{}(n=2\mathrm{})\mathrm{}$. When $g=0\mathrm{}$ and $T=T_c$ these systems undergo a phase transition that may be associated (for small $n$) with the Heisenberg fixed point (${\nu }^{*}=0\mathrm{}$) or (otherwise) with the cubic fixed point (${\nu }^{*}>0\mathrm{}$) of the renormalization group. Although $\nu$ is an "irrelevant variable" in the former case, it is found to have important effects. For $\nu <0\mathrm{}$, the point $g=0\mathrm{}$, $T=T_c$ represents a bicritical point in the $g-T$ plane, at which a first-order "spin-flop" line (separating two distinct ordered phases) meets two critical lines. For $\nu >0\mathrm{}$, the "flop" line splits into two critical lines, associated with transitions between each of the ordered phases and a new intermediate phase; the point $T=T_c$, $g=0\mathrm{}$ is then tetracritical. The shape of the boundary of the intermediate phase is given by $T=T_2(g, \nu )$ with $[T_c-T_2(g, \nu \left)\right]\sim {\left(\frac{g}{\nu }\right)}^{\frac{1}{{\psi }_2}}$, where ${\psi }_2={\phi }_g-{\phi }_{\nu }$ (if the tetracritical point is Heisenberg-like) or ${\psi }_2={\phi }_g^C$ (if it is cubic). Here, ${\phi }_g$, ${\phi }_{\nu }$, and ${\phi }_g^C$ are appropriate crossover exponents associated with the two symmetry-breaking perturbations. The phase diagram of [111] -stressed perovskites is also discussed and the experimental situation briefly reviewed.