Derivation of a Ginzburg-Landau free energy density of a $p+ip$ superconductor from spin-orbit coupling with mixed gradient terms

A Ginzburg-Landau free energy for a superconducting chiral $p$-wave order parameter is derived from a two-dimensional tight-binding lattice model with weak spin-orbit coupling included as a general symmetry-breaking field. Superconductivity is accounted for by a BCS-type nearest-neighbor opposite-spin interaction where we project the potential onto the $p$-wave irreducible representation of the square lattice symmetry group and assume this to be the dominating order. The resulting free energy contains kinetic terms that mix components of the order parameter as well as directional gradients—so-called mixed gradient terms—as a virtue of the symmetry of the order parameter. Spin-orbit coupling and electron-hole anisotropy lead to additional contributions to the coefficients of these terms, increasing the number of necessary phenomenological parameters by one compared to previous work and leading to an increase in the coefficient measuring Fermi-surface anisotropy for Rashba spin-orbit coupling in the continuum limit.