One-dimensional $Z_4$ topological superconductor

We describe the mean-field model of a one-dimensional topological superconductor with two orbitals per unit cell. Time-reversal symmetry is absent, but a nonsymmorphic symmetry, involving a translation by a fraction of the unit cell, mimics the role of time-reversal symmetry. As a result, the topological superconductor has $Z_4$ topological phases, two that support Majorana bound states and two that do not, in agreement with a prediction based on K-theory classification [K. Shiozaki, M. Sato, and K. Gomi, Phys. Rev. B 93, 195413 (2016)]. As with the Kitaev chain, the presence of Majorana bound states gives rise to the $4\pi$-periodic Josephson effect. A random matrix with nonsymmorphic time-reversal symmetry may be block diagonalized, and every individual block has time-reversal symmetry described by one of the Gaussian orthogonal, unitary, or symplectic ensembles. We show how this is manifested in the energy level statistics of a random system in the $Z_4$ class as the spatial period of the nonsymmorphic symmetry is varied from much less than to of the order of the system size.