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

TL;DR

This paper derives a Ginzburg-Landau free energy for a $p+ip$ superconductor with spin-orbit coupling, revealing mixed gradient terms and the influence of anisotropy and spin-orbit effects on the free energy parameters.

Contribution

It introduces a detailed derivation of the free energy including mixed gradient terms and spin-orbit effects, expanding previous models with additional phenomenological parameters.

Findings

Mixed gradient terms appear naturally due to symmetry.

Spin-orbit coupling modifies the coefficients of the free energy.

Fermi surface anisotropy effects are enhanced by spin-orbit interactions.

Abstract

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.