Characterization of topological insulators based on the electronic polarization with spiral boundary conditions

We introduce the electronic polarization originally defined in one-dimensional lattice systems to characterize two-dimensional topological insulators. The main idea is to use spiral boundary conditions which sweep all lattice sites in one-dimensional order. We find that the sign of the polarization changes at topological transition points of the two-dimensional Wilson-Dirac model (the lattice version of the Bernevig-Hughes-Zhang model) in the same way as in one-dimensional systems. Thus the polarization plays the role of ``order parameter'' to characterize the topological insulating state and enables us to study topological phases in different dimensions in a unified way.