First-order chiral to non-chiral transition in the angular dependence of the upper critical induction of the Scharnberg-Klemm $p$-wave pair state

We calculate the temperature $T$ and angular $(\theta,\phi)$ dependence of the upper critical induction $B_{c2}(\theta,\phi,T)$ for parallel-spin superconductors with an axially symmetric $p$-wave pairing interaction pinned to the lattice and a dominant ellipsoidal Fermi surface (FS). When both parallel-spin states are allowed, the chiral Scharnberg-Klemm state $B_{c2}(\theta,\phi,T)$ exceeds that of the chiral Anderson-Brinkman-Morel state for all FS anisotropies, and exhibits a kink at $\theta=\theta^{*}(T,\phi)$, indicative of a first-order transition from its chiral, nodal-direction behavior to its non-chiral, antinodal-direction behavior. Potential applicability to Sr$_2$RuO$_4$, UCoGe, and topological superconductors is discussed.