Anomalous Hall Effect and Skyrmion Number in Real- and Momentum-space

TL;DR

This paper clarifies the anomalous Hall effect in the double exchange model, exploring the roles of lattice structure, skyrmion number, real- and momentum-space duality, and disorder, and classifies different regimes of AHE mechanisms.

Contribution

It provides a comprehensive classification of AHE mechanisms based on exchange coupling, scattering, and spin texture length scales, resolving key unresolved issues.

Findings

Identifies two distinct AHE mechanisms: real-space skyrmion-number and momentum-space skyrmion-density.

Classifies regimes in parameter space where each mechanism dominates.

Clarifies the roles of lattice structure, disorder, and duality in AHE.

Abstract

We study the anomalous Hall effect (AHE) for the double exchange model with the exchange coupling $|J_H|$ being smaller than the bandwidth $|t|$ for the purpose of clarifying the following unresolved and confusing issues: (i) the effect of the underlying lattice structure, (ii) the relation between AHE and the skyrmion number, (iii) the duality between real and momentum spaces, and (iv) the role of the disorder scatterings; which is more essential, $\sigma_H$ (Hall conductivity) or $\rho_H$ (Hall resistivity)? Starting from a generic expression for $\sigma_H$, we resolve all these issues and classify the regimes in the parameter space of $J_H \tau$ ($\tau$: elastic-scattering time), and $\lambda_{s}$ (length scale of spin texture). There are two distinct mechanisms of AHE; one is characterized by the real-space skyrmion-number, and the other by momentum-space skyrmion-density at the Fermi level, which work in different regimes of the parameter space.