Catalog of noninteracting tight-binding models with two energy bands in one dimension

We classify Hermitian tight-binding models describing noninteracting electrons on a one-dimensional periodic lattice with two energy bands. To do this, we write a generalized Rice-Mele model with two orbitals per unit cell, including all possible complex-valued long-range hoppings consistent with Hermicity. We then apply different forms of time-reversal, charge-conjugation and chiral symmetry in order to constrain the parameters, resulting in an array of possible models in different symmetry classes. For each symmetry class, we define a single, canonical form of the Hamiltonian and identify models that are related to the canonical form by an off-diagonal unitary transformation in the atomic basis. The models have either symmorphic or nonsymmorphic nonspatial symmetries (time $T$, chiral and charge-conjugation). The nonsymmorphic category separates into two types of state of matter: an insulator with a $\mathbb{Z}_2$ topological index in the absence of nonsymmorphic time-reversal symmetry or, in the presence of nonsymmorphic time-reversal symmetry, a metallic state. The latter is an instance of Kramer's degeneracy with one degeneracy point in the Brillouin zone as opposed to no degeneracy points in symmorphic systems with $T^2 = 1$ and two in symmorphic systems with $T^2 = - 1$.