Topological insulator and the Dirac equation

TL;DR

This paper explores the connection between Dirac equations and topological insulators, showing how modifications to the Dirac equation can indicate topological phases and phase transitions.

Contribution

It introduces a modified Dirac equation with a quadratic B term that captures topological phase transitions and boundary states in topological insulators.

Findings

Z_{2} index remains zero for the unmodified Dirac equation.

Adding the quadratic B term changes the Z_{2} index based on the sign of mB.

Boundary solutions indicate topological insulator characteristics.

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.