Hopf Term for a Two-Dimensional Electron Gas

TL;DR

This paper discusses the derivation of the Hopf term in a two-dimensional electron gas, emphasizing its connection to topological invariants and the quantized Hall conductivity, and critiques previous derivations for shortcomings.

Contribution

It provides a microscopic derivation of the Hopf term and clarifies its relation to topological invariants and Hall conductivity, addressing gaps in prior work.

Findings

Hopf term prefactor expressed via topological invariant

Relation between Hopf term and quantized Hall conductivity

Critique of previous derivation methods

Abstract

In this Comment on the paper by W. Apel and Yu. A. Bychkov, cond-mat/9610040 and Phys. Rev. Lett. 78, 2188 (1997), we draw attention to our prior microscopic derivations of the Hopf term for various systems and to shortcomings of the Apel-Bychkov derivation. We explain how the value of the Hopt term prefactor $\Theta$ is expressed in terms of a topological invariant in the momentum space and the quantized Hall conductivity of the system. (See also related paper cond-mat/9703195)