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

TL;DR

This paper uses the Foldy-Wouthuysen transformation to connect topological invariants like Chern numbers with Chern-Simons actions in various dimensions, providing a framework for understanding topological insulators and proposing a new 3+1D spin Hall insulator model.

Contribution

It introduces a method to derive effective field theories for topological insulators across multiple dimensions using the Foldy-Wouthuysen transformation, including a novel 3+1D spin Hall insulator model.

Findings

Chern numbers equal to coefficients of Chern-Simons actions in 2+1 and 4+1 dimensions

Explicit derivation of Berry gauge field strength in 4+1 dimensions

Proposal of a 3+1D spin Hall insulator model

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.