Weyl semimetal and topological numbers

Abstract

Generalized Dirac monopoles in momentum space are constructed in even [Formula: see text] dimensions from the Weyl Hamiltonian in terms of Green’s functions. In [Formula: see text] 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 and Lorentz invariant field theory action is studied 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 Pfaffian invariant [Formula: see text]. Relevance of monopoles and Chern numbers for the semiclassical chiral kinetic theory is also discussed.