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Superfluid 3He in a restricted geometry with a perpendicular magnetic field

We theoretically investigate the role of surface Andreev bound states (SABSs) on the phase diagram and spin susceptibilities of superfluid 3He confined to a restricted geometry. We first explicitly derive the dispersion of the SABS in 3He-B in the presence of a magnetic field, where the Majorana Ising spin and the spin susceptibility contributed from the SABS are associated with the SO(3) order parameter manifold. Subsequently, based on the quasiclassical Eilenberger theory with Fermi liquid corrections, we discuss the nonlinear effect of a magnetic field on the SABS, where the magnetic field is perpendicular to the specular surface. It is directly demonstrated that a gapped SABS strongly enhances the magnetization density and spin susceptibility at the surface, compared with that in the normal 3He. To capture the characteristics of the SABS, we show the field- and temperature-dependences of the spatially averaged susceptibility which is detectable through NMR experiments. It turns out that the contribution of the SABS leads to nonmonotonic temperature-dependence of the spin susceptibility. Furthermore, we present the superfluid phase diagram, where the B-phase undergoes a first-order (second-order) phase transition to A-phase or planar phase at low (high) temperatures.