Half-Quantum Vortices in the Superfluid Phases of $^3$He Confined in Nafen

Based on a strong-coupling Ginzburg-Landau theory that accounts for the relative stability of the confined equal-spin pairing superfluid phases in $^3$He, we report theoretical calculations for the structure of half-quantum vortices (HQVs) in the confined phases of $^3$He. We predict the regions of stability for two different ground state vortex phases within these confined phases, and provide an identification of the topological defects order parameter, mass currents, and spin-polarization. We investigate a pair of polar HQVs that are stabilized in the polar phase, and a newly discovered pair of polar-distorted chiral HQVs in the polar-distorted chiral A phase. We lastly calculate the confined superfluid phase diagram in a magnetic field, $H = 0 - 370G {\bf{Ω}}$, and obtain excellent agreement with the experimentally reported phase transitions observed in the equal-spin pairing phases of $^3$He.