Topological insulator and the Dirac equation

Abstract

We present a general description of topological insulators from the point of view of Dirac equations. The Z_{2} index for the Dirac equation is always zero, and thus the Dirac equation is topologically trivial. After the quadratic B term in momentum is introduced to correct the mass term m or the band gap of the Dirac equation, the Z_{2} index is modified as 1 for mB>0 and 0 for mB<0. For a fixed B there exists a topological quantum phase transition from a topologically trivial system to a non-trivial one system when the sign of mass m changes. A series of solutions near the boundary in the modified Dirac equation are obtained, which is characteristic of topological insulator. From the solutions of the bound states and the Z_{2} index we establish a relation between the Dirac equation and topological insulators.