Topological system with a twisting edge band: A position-dependent Hall resistance

We study a [nu] = 1 topological system with one twisting edge-state band and one normal edge-state band. For the twisting edge-state band, Fermi energy goes through the band three times, thus having three edge states on one side of the sample, while the normal edge band contributes only one edge state on the other side of the sample. In such a system, we show that it consists of both topologically protected and unprotected edge states, and as a consequence, its Hall resistance depends on the location where the Hall measurement is done even for a translationally invariant system. This unique property is absent in a normal topological insulator.