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Boundary Topological Superconductors

For strongly anisotropic time-reversal invariant (TRI) insulators in two and three dimensions, the band inversion can occur respectively at all TRI momenta of a high symmetry axis and plane. Although these classes of materials are topologically trivial as the strong and weak $Z_{2}$ indices are all trivial, they can host an even number of unprotected helical gapless edge states or surface Dirac cones on some boundaries. We show in this work that when the gapless boundary states are gapped by $s_{\pm}$-wave superconductivity, a boundary time-reversal invariant topological superconductor (BTRITSC) characterized by a $Z_{2}$ invariant can be realized on the corresponding boundary. Since the dimension of the BTRITSC is lower than the bulk by one, the whole system is a second-order TRI topological superconductor. When the boundary of the BTRITSC is further cut open, Majorana Kramers pairs and helical gapless Majorana modes will respectively appear at the corners and hinges of the considered sample in two and three dimensions. Furthermore, a magnetic field can gap the helical Majorana hinge modes of the three-dimensional second-order TRI topological superconductor and lead to the realization of a third-order topological superconductor with Majorana corner modes. Our proposal can potentially be realized in insulator-superconductor heterostructures and iron-based superconductors whose normal states take the desired inverted band structures.