Fractionally quantized Berry phase, adiabatic continuation, and edge states

Abstract

Symmetry protected quantization of the Berry phase is discussed in relation to edge states. Assuming the existence of an adiabatic process that protects the quantization of the Berry phase, a nontrivial Berry phase $\ensuremath{\gamma}\ensuremath{\propto}2\ensuremath{\pi}\ensuremath{\rho}$ ($\ensuremath{\rho}$ is the local filling of particles) for the bulk suggests appearance of edge states with boundaries. We have applied this generic consideration for the Bloch states of a two-dimensional model with massless Dirac fermions where $\ensuremath{\gamma}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ implies the edge states. The entanglement entropy is evaluated for the models and its relation to the bulk-edge correspondence of Dirac fermions is discussed as well.