Topological invariants and edge states in one-dimensional two-tile lattices

The existence of robust conducting edge states is one of the most prominent properties of topological insulator, which is often simply illustrated as a consequence of bulk-boundary correspondence. Then here arises a new question whether similar robust edge states appear in some other topological-trivial systems, or rather, given a general answer of fundamental mathematics such as harmonic analysis or K-theory to this problem, we study one-dimensional two-tile lattices and show that the robust edge states can exist in topological-trivial complex lattices. Under the tight-binding approximation, all kinds of one-dimensional two-tile lattices with staggered hopping matrix elements can be described by the Su-Schrieffer-Heeger model or the Rice-Mele model, depending on their site energy. The site energy values of the Su-Schrieffer-Heeger model are equal, and often assumed to be zero, and the Rice-Mele model is constructed to describe the one-dimensional two-tile lattices having two different site energy values. With the help of the generalized Bloch theorem, the eigen-state problem of electrons in one-dimensional two-tile complex lattices are solved systematically, and the analytical expressions for the wavefunctions of the edge states in the corresponding finite lattice are obtained. The numerical and analytical results show that the edge states can also emerge in any of one-dimensional two-tile lattices beyond the Su-Schrieffer-Heeger lattice, i.e., provided that the magnitude of intracell hopping is less than the intercell hopping, a pair of edge states can also emerge in Rice-Mele lattice. Unlike the Su-Schrieffer-Heeger edge states, the two Rice-Mele edge states are locally distributed at one end of the finite lattice: one at the left and another one at right. The Zak phase is a topological invariant of the Su-Schrieffer-Heeger model, but it is no longer invariant for the Rice-Mele model because of the breaking of spatial inversion symmetry, and therefore the Rice-Mele lattices are topologically trivial. However, the Rice-Mele edge states are also robust to the non-diagonal disorder of the lattice. In addition, it is proven that the winding number can provide a general criterion for the existence of a couple of edge states in any one-dimensional two-tile lattice whether it is the Su-Schrieffer-Heeger lattice or not. These results lead to a conclusion that the topological invariant is not necessary for the robust edge states to occur.