Symmetry and stationary points of a free energy: The case of superfluidHe3

The symmetry of the free energy of superfluid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is used to investigate stationary points which correspond to possible superfluid phases. For this purpose a complete classification of both continuous and discrete subgroups of the symmetry of the free-energy functional describing p-wave superfluidity of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, G=SO(3)×SO(3)×U(1), is presented. The corresponding order parameters of phases with broken symmetry are determined explicitly. It is shown that all superfluid phases previously found in the literature by minimizing the Ginzburg-Landau functional are included in this classification. Hence, in all these phases the symmetry G is only partly broken such that they still contain a residual symmetry. This is true both for inert and noninert states. The concept of broken relative symmetries—of great importance for an understanding of the properties of superfluid ${}_{}{}^3{}_{}{}^{}\mathrm{appears}$ as a very natural feature in such a group-theoretical treatment. The classification is also applied to superfluid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ in a magnetic field and with dipolar coupling. Using the methods of differential topology developed by Michel, we show that the symmetry properties of phases and their energy are related. This is explicitly verified by applying the classification to two systems whose free-energy functional is simpler than that of superfluid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ (e.g., d-wave pairing or superfluidity in neutron stars) and whose minima are known from analytic calculation. Hence, group-theoretical methods prove to be a very valuable tool for investigating the stationary points of complicated free-energy functionals in condensed-matter physics, in particular of those where an analytical minimization seems untractable as in superfluid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$.