Competition of vortex core structures in superfluid $^3$He-B

Among vortex structures identified so far in superfluid $^3$He-B, the most common are the A-phase-core vortex and the double-core vortex. According to earlier numerical calculations, the double-core vortex is energetically favored nearly everywhere in the $p$-$T$ phase diagram. Nevertheless, in experiments the A-phase-core vortex has been observed down to temperatures of $0.6T_{\mathrm{c}}$ at high pressures. We use the Ginzburg-Landau formalism to calculate the energies of the two vortex structures in the experimentally relevant magnetic field as well as the energy barrier for the transition between the two structures. Assigning vanishing barrier as the boundary of the metastability region of the A-phase-core vortex, we reproduce the experimentally measured vortex phase diagram and provide an explanation for the reappearance of the double-core vortex near the critical temperature $T_{\mathrm{c}}$ at low pressures: The difference in Zeeman energy between the two vortex structures becomes relatively more important close to $T_{\mathrm{c}}$, and the A-phase-core vortex becomes unstable. In contrast to the equilibrium vortex structures, we suggest that the vortex nucleation process favors the A-phase-core vortex over the double-core vortex. Our approach can be used to analyze competition between different vortex structures in other unconventional superfluids and superconductors.