Topological Concepts for the Weyl Hamiltonians with the Berry Gauge Field

TL;DR

This paper explores the topological invariants of Weyl Hamiltonians in higher dimensions, linking winding numbers, Chern numbers, and Berry fields, with implications for physical theories like chiral kinetic theory.

Contribution

It provides explicit calculations of topological invariants for higher-dimensional Weyl Hamiltonians and clarifies their physical significance.

Findings

Winding numbers equal to topological charge and Chern numbers

Explicit calculations for 3+1 and 5+1 dimensions

Relevance to semiclassical chiral kinetic theory

Abstract

The winding numbers for the even d+1 spacetime dimensional Weyl Hamiltonians are calculated in terms of the related Green's functions. It is shown that these winding numbers result in the divergence of the Dirac monopole fields, hence they are equal to the unit topological charge. It is demonstrated that the winding numbers are also equal to the Chern numbers which are expressed as the integral of the Berry field strength. Explicit calculations are presented for the 3+1 and 5+1 dimensional cases. Relevance of these topological invariants for the physical systems like the semiclassical chiral kinetic theory are discussed.