Singular and Half-Quantum Vortices and Associated Majorana Particles in Superfluid 3He-A between Parallel Plates

Motivated by a recent experiment on superfluid He-3-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.