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