Z_2 topological invariant and solutions of the Dirac equation

Details Z_2 topological invariant and solutions of the Dirac equation Z_2 topological invariant and solutions of the Dirac equation

2010-09-28

arxiv.org/abs/1009.5502v1

SPIN 1, 33 (2011)

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

4 pages

Scientific paper

10.1142/S2010324711000057

We present a general description of topological insulators from the point of view of the Dirac equation. The Z_2 index for the Dirac equation is always zero, and thus the Dirac equation is topologically trivial. After the quadratic Bp^2 term in momentum p is introduced to correct the band gap mv^2 of the Dirac equation (v has the dimension of speed), the Z_2 index is modified as 1 for a dimensionless parameter 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 the band gap mv^2 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 an explicit relation between the Dirac equation and topological insulators.

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