Effective field theory of topological insulator and the Foldy-Wouthuysen transformation

Abstract

Employing the Foldy-Wouthuysen transformation it is demonstrated straightforwardly that the first and second Chern numbers are equal to the coefficients of the 2+1 and 4+1 dimensional Chern-Simons actions which are generated by the massive Dirac fermions coupled to the Abelian gauge fields. A topological insulator model in 2+1 dimensions is discussed and by means of a dimensional reduction approach the 1+1 dimensional descendant of the 2+1 dimensional Chern-Simons theory is presented. Field strength of the Berry gauge field corresponding to the 4+1 dimensional Dirac theory is explicitly derived through the Foldy-Wouthuysen transformation. Acquainted with it the second Chern numbers are calculated for specific choices of the integration domain. A method is proposed to obtain 3+1 and 2+1 dimensional descendants of the effective field theory of the 4+1 dimensional time reversal invariant topological insulator theory. Inspired by the spin Hall effect in graphene, a hypothetical model of the time reversal invariant spin Hall insulator in 3+1 dimensions is proposed.

20 pages. Few corrections and Refs. added

References in corpus (7)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Field Theory of Time-Reversal Invariant Insulators
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Cited by in corpus (2)

• Spin Currents of Charged Dirac Particles in Rotating Coordinates
• The Effective field theory of 2+1 dimensional topological insulator in the presence of Rashba spin-orbit interaction