Renormalization-group study of the fixed points and of their stability for phase transitions with four-component order parameters.

Abstract

The renormalization-group recursion relations are solved for the effective Hamiltonians relative to phase transitions with four-component order parameters. For this value of n there are 22 types of Hamiltonians which can be classified into two categories according to the action of their normalizer ${G}_{N}$ on the corresponding parameter space (${G}_{N}$ is the symmetry group leaving globally invariant this space). In the first case, ${G}_{N}$ generates a finite number of isolated fixed points whose characteristics can be deduced from the detailed investigation of five Hamiltonians only.In the second category, for which ${G}_{N}$ is a continuous group, there are, in addition to isolated fixed points, continuous manifolds of physically equivalent fixed points (the dimension of the manifolds is either one or three). In the search for a stable fixed point, the continuous manifolds can be ignored, while the isolated points are related to the five former Hamiltonians. For n=4, it is necessary to solve the recursion relations to two-loop order. The only possible stable ones among the fixed points then arise from a splitting of points which coincide, to one-loop order, with the isotropic fixed point. Extending, to two-loop order, a result recently established to the preceding order, we show that if a stable fixed point exists, it is unique. For n=4, the stable fixed point has one of three possible symmetries : di-icosahedral, hypercubic, or dicylindrical.Despite this anisotropy of the critical fluctuations, the exponents associated with any of the stable fixed points are identical to order ${\ensuremath{\epsilon}}^{2}$ to the ``isotropic'' exponents corresponding to n=4. The cubic point is destabilized by any operator of symmetry lower than cubic. The dicylindrical one remains stable with respect to certain anisotropies of lower symmetry. We examine the available experimental data in light of the preceding theoretical results concerning the critical behavior and the thermodynamic order of the transitions. On the other hand, we establish two general symmetry conditions relative to the stable fixed points determined by the renormalization-group equations. The first one specifies group theoretically, for each value of n, the possible symmetries ${G}_{i}^{\mathrm{*}}$ of the stable fixed points. The second one formulates a necessary condition for the occurrence of an anisotropic stable fixed point: The normalizer ${G}_{N}$ of the considered parameter space must fulfill the condition ${G}_{N}$\ensuremath{\subset}@;${\mathrm{BG}}_{i}^{\mathrm{*}}$ for one, at least, of the former ${G}_{i}^{\mathrm{*}}$ groups. These rules are shown to be very restrictive for n=4: On the basis of symmetry the lack of a stable fixed point can be asserted for 10 Hamiltonians out of 22, without solving the fixed-point equations.