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The possible stable singular vortex (SV) and half-quantum vortex (HQV) of the superfluid 3He-A phase confined in restricted geometries are investigated. The associated low-energy excitations are calculated in connection with the possible existence of Majorana zero modes obeying non-Abelian statistics. The energetics between those vortices is carefully examined using the standard Ginzburg--Landau (GL) functional with a strong-coupling correction. The Fermi liquid effect, which is not included in the GL functional, is considered approximately within the London approach. This allows us to determine the stability regions in pressure, temperature, and applied field for SV and HQV. The existence of the Majorana zero mode and its statistics, either Abelian or non-Abelian under braiding of SVs, is studied by solving the Bogoliubov--de Gennes equation for spinful chiral $p$-wave superfluids at sufficiently low temperatures. We determined several conditions controllable external parameters for realizing the non-Abelian statistics of Majorana zero modes e.g., pressure, field direction, and strength.

Motivated by a recent experiment on superfluid 3He-A confined in narrow parallel plates using a rotating cryostat, we explore possible vortices that are stable under a magnetic field applied to an arbitrary angle relative to the plates in order to seek vortices that can accommodate the Majorana zero mode in the core. After proving that the singular vortex with the unit winding number provides the Majorana mode in a spinful situation, we establish the phase diagram in the plane; the rotation frequency $\Omega$ vs the system size $R$ by finding possible order parameter textures within the Ginzburg--Landau framework. We also analyze the stability for a single and a pair of half-quantum vortices, which possess the Majorana mode in their core. It is concluded from the above-mentioned results that the Majorana zero mode can be found in the present on-going experiment performed at ISSP, University of Tokyo.

We investigate the topological aspect of the spin-triplet $f$-wave superconductor UPt3through microscopic calculations of edge- and vortex-bound states based on the quasiclassical Eilenberger and Bogoliubov--de Gennes theories. It is shown that a gapless and linear dispersion exists at the edge of the $ab$-plane. This forms a Majorana valley, protected by the mirror chiral symmetry. We also demonstrate that, with increasing magnetic field, vortex-bound quasiparticles undergo a topological phase transition from topologically trivial states in the double-core vortex to zero-energy states in the normal-core vortex. As long as the $\mbi{d}$-vector is locked into the $ab$-plane, the mirror symmetry holds the Majorana property of the zero-energy states, and thus UPt3preserves topological crystalline superconductivity that is robust against the crystal field and spin--orbit interaction.