Surface scattering effect and the stripe order in films of the superfluid $^3$He B phase

Surface scattering effects in thin films of the superfluid 3He B phase have been theoretically investigated, with an emphasis on the stability of the stripe order with spontaneous broken translational symmetry in the film plane and quasiparticle excitations in this spatially inhomogeneous phase. Based on the Ginzburg-Landau theory in the weak coupling limit, we have shown that the stripe order, which was originally discussed for a film with two specular surfaces, can be stable in a film with one specular and one diffusive surfaces which should correspond to superfluid 3He on a substrate. It is also found by numerically solving the Eilenberger equation that due to the stripe structure, a midgap state distinct from the surface Andreev bound state emerges and its signature is reflected in the local density of states.