P -wave superconductivity and the axiplanar phase of triple-point fermions

Abstract

We consider weak-coupling superconductivity in the inversion- and rotation-symmetric system of two pseudospin $s=1$ low-energy fermions of opposite chirality. General contact interactions can lead to Cooper instabilities toward $d$-wave or toward a novel $2\ifmmode\times\else\texttimes\fi{}3 p$-wave matrix order parameter. We compute the Ginzburg-Landau (GL) free energy for the latter. Remarkably, in this case the GL free energy can be minimized exactly, with the resulting ordered state being analogous to the ``axiplanar'' $p$ wave, which exhibits extra degeneracy and generally breaks time-reversal symmetry. Whereas a generic GL free energy for our order parameter has only $U(1)\ifmmode\times\else\texttimes\fi{}SO(2)\ifmmode\times\else\texttimes\fi{}SO(3)$ symmetry, at weak coupling we find it displaying the enlarged $U(1)\ifmmode\times\else\texttimes\fi{}SO(3)\ifmmode\times\else\texttimes\fi{}U(1)\ifmmode\times\else\texttimes\fi{}SO(3)$ symmetry broken down to $SO(2)\ifmmode\times\else\texttimes\fi{}SO(2)$ in the axiplanar ordered phase and leading therefore to six Goldstone bosons. We show how the lattice, once restored, fixes the allowed values of the magnetization in the superconducting state by locking the spatial directions implicit in the order parameter to its high-symmetry axes.