Recent experiments on superfluid ${}^3\mathrm{He}$ in globally anisotropic aerogels have shown realization of the polar superfluid phase and of the half-quantum vortices (HQVs) in this phase upon rotation. To clarify why the HQVs, which had not been detected clearly in the $A$ phase of the bulk liquid, have been realized in the polar phase, we theoretically examine the relative stability of a HQV pair against a single phase vortex in both the bulk $A$ phase and the polar phase in an aerogel. By taking care of important roles of a higher-order gradient term, which assists the stability of HQVs but has never been incorporated so far in the Ginzburg-Landau approach, we find that several consequences, including the extension of the polar phase at lower pressures in the phase diagram, facilitate realization of the HQVs there in contrast to the case of the bulk $A$ phase in a slab geometry.

• Received 7 August 2018
• Revised 12 September 2018

DOI:https://doi.org/10.1103/PhysRevB.98.094524