One-dimensional topological superconductors with nonsymmorphic symmetries

Abstract: We present example four-band Hermitian tight-binding Bogoliubov-de-Gennes (BdG) Hamiltonians and Kramer's degenerate Hamiltonians in one dimension. Starting from a generalized Rice-Mele model, we incorporate superconducting terms to obtain a four-band BdG Hamiltonian with intrinsic charge-conjugation symmetry, and constrain it using symmorphic or nonsymmorphic time-reversal symmetries. In position space we find that each form of time-reversal symmetry, when applied to random BdG matrices, results in a unique block diagonalization of the Hamiltonian when translational symmetry is also enforced. We provide representative models in all relevant symmorphic symmetry classes, including the non-superconducting CII class. For nonsymmorphic time-reversal symmetry, we identify a $\mathbb{Z}_4$ topological index with two phases supporting Majorana zero modes and two without, and study disorder effects in the presence of topological solitons. We further generalize a winding-number method, previously applied only to $\mathbb{Z}_2$ invariants without Kramer's degeneracy, to compute indices for both the $\mathbb{Z}_4$ model and a non-superconducting AII model with nonsymmorphic chiral symmetry and Kramer's degeneracy. We propose topolectric circuit implementations of the charge-density-wave and $\mathbb{Z}_4$ models which agree with the topological calculations. Finally, we show that, in one dimension, nonsymmorphic unitary symmetries do not produce new topological classifications beyond $\mathbb{Z}$ or $\mathbb{Z}_2$ indices.