Topological Numbers and the Weyl Semimetal

TL;DR

This paper explores the topological properties of Weyl semimetals, linking monopoles, Chern numbers, and transport phenomena like the chiral magnetic and anomalous Hall effects through a topological field theory framework.

Contribution

It introduces a topological field theory for Weyl semimetals that connects monopoles, Chern numbers, and observable transport effects, emphasizing the role of topology in these materials.

Findings

Monopoles in momentum space are characterized by Chern and winding numbers.

The effective action predicts the chiral magnetic and anomalous Hall effects.

Chern numbers influence conductivity coefficients in Weyl semimetals.

Abstract

Generalized Dirac monopoles in momentum space are constructed in even d+1 dimensions from the Weyl Hamiltonian in terms of Green's functions. In 3+1 spacetime dimensions, the (unit) charge of the monopole is equal to both the winding number and the Chern number, expressed as the integral of the Berry curvature. Based on the equivalence of the Chern and winding numbers, a chirally coupled field theory action is proposed for the Weyl semimetal phase. At the one loop order, the effective action yields both the chiral magnetic effect and the anomalous Hall effect. The Chern number appears as a coefficient in the conductivity, thus emphasizes the role of topology. The anomalous contribution of chiral fermions to transport phenomena is reflected as the gauge anomaly with the topological term $(\bm{E}\cdot\bm{B})$. Relevance of monopoles and Chern numbers for the semiclassical chiral kinetic theory is also discussed.