Longitudinal and anomalous Hall conductivity of a general two-band model

TL;DR

This paper derives a comprehensive microscopic framework for calculating longitudinal and anomalous Hall conductivities in a general two-band model, applicable to various magnetic and topological systems at finite temperature and scattering rates.

Contribution

It introduces a unified decomposition of conductivities into intraband/interband and symmetric/antisymmetric parts, extending Boltzmann and Berry curvature theories to finite scattering.

Findings

Generalized conductivity formulas including finite scattering effects.

Identification of quantum metric's role in interband corrections.

Application to models like Chern insulators and spiral spin density waves.

Abstract

We derive and analyze the longitudinal and the anomalous Hall conductivity for a general momentum-block-diagonal two-band model. This model captures a broad spectrum of physically very different systems including N\'eel antiferromagnetic and spiral spin density waves as well as models that involve spin-orbit interaction and are known to show topological properties. We present a complete microscopic derivation for finite temperature and constant scattering rate $\Gamma$ that is diagonal and equal, but arbitrarily large for both bands. We identify two criteria that allow for a unique and physically motivated decomposition of the conductivities. On the one hand, we distinguish intraband and interband contributions that are defined by the involved quasiparticle spectral functions. On the other hand, we distinguish symmetric and antisymmetric contributions that are defined by the symmetry under the exchange of the current and the electric field directions. The (symmetric) intraband contributions generalize the formula of standard Boltzmann transport theory, which is valid only in the clean limit (small $\Gamma$), whereas the interband contributions capture interband coherence effects beyond independent quasiparticles. We show that the symmetric interband contribution is a correction only present for finite $\Gamma$ and is controlled by the quantum metric. The antisymmetric interband contributions generalize the formula of the anomalous Hall conductivity in terms of the Berry curvature to finite $\Gamma$. We study the clean (small $\Gamma$) and dirty (large $\Gamma$) limit analytically. The connection between the presented derivation and the Bastin and Streda formalism is given. We apply our results to a Chern insulator, a ferromagnetic multi-d-orbital, and a spiral spin density wave model.