Disorder Induced Anomalous Hall Effect in Type-I Weyl Metals: Connection between the Kubo-Streda Formula in the Spin and Chiral basis

TL;DR

This paper develops a systematic method to analyze the anomalous Hall effect in tilted Weyl metals, separating intrinsic, side jump, and skew scattering contributions in the spin basis, and compares results with semiclassical approaches.

Contribution

It introduces a transparent scheme to distinguish Hall effect contributions in the spin basis, simplifying calculations in relativistic systems and connecting diagrammatic and semiclassical methods.

Findings

Side jump contribution dominates at low energies.

The scheme's results agree with Boltzmann equation in leading order.

Intrinsic and skew scattering contributions are quantitatively characterized.

Abstract

We study the anomalous Hall effect (AHE) in tilted Weyl metals with weak Gaussian disorder under the Kubo-Streda formalism in this work. To separate the three different contributions, namely the intrinsic, side jump and skew scattering contribution, it is usually considered necessary to go to the eigenstate (chiral) basis of the Kubo-Streda formula. However, it is more straight-forward to compute the total Hall current in the spin basis. For the reason, we develop a systematic and transparent scheme to separate the three different contributions in the spin basis for relativistic systems by building a one-to-one correspondence between the Feynman diagrams of the different mechanisms in the chiral basis and the products of the symmetric and anti-symmetric part of the polarization operator in the spin basis. We obtain the three contributions of the AHE in tilted Weyl metals by this scheme and found that the side jump contribution exceeds both the intrinsic and skew scattering contribution for the low-energy effective Hamiltonian. We compared the anomalous Hall current obtained from our scheme with the results from the semi-classical Boltzmann equation approach under the relaxation time approximation and found that the results from the two approaches agree with each other in the leading order of the tilting velocity.