Translation symmetry of edge state

Abstract

The interplay of broken translation symmetry due to the edge and the particular properties of the edge states is illustrated considering the simple square lattice. In the case of the most symmetric edge (when its direction coincides with the primitive translation vector) the translation along the edge symmetry enables to transform the two-dimensional (2D) tight-binding method equations into a more simple 1D eigenvalue problem. When the direction of the edge does not coincide with the primitive vector, the above-mentioned symmetry is broken. Nevertheless it can be partially restored enlarging the primitive cell and the number of wave function components, what enables to obtain the above-mentioned effective 1D problem. It is shown that the exact solution of the 1D problem can be obtained by means of the Bethe Ansatz method what was checked by the numerical diagonalization. Using the proposed technique the properties of the edge states were considered. It was shown that there are two reasons for the edge states to appear: the local potential of the edge sites and the modification of the tunnelling amplitudes along the edge. In the case of the most symmetric edge only the second one can lead to the edge state energy dependence differing from the one of the bottom of the continuous band. While in the case of the tilted edge the electron motion along the edge infuences significantly the spectrum of the edge state.