Real space Berry curvature of itinerant electron systems with spin-orbit interaction

TL;DR

This paper derives a generalized real space Berry curvature formula for itinerant electrons with spin-orbit coupling, explaining the topological Hall effect in complex magnetic textures beyond traditional models.

Contribution

It introduces a new, covariant expression for Berry curvature that extends the understanding of topological effects in magnetic materials with spin-orbit interactions.

Findings

Generalized Berry phase formula for systems with SOC

Connection between magnetic textures and topological Hall effect

Application to collinear and coplanar antiferromagnetic phases

Abstract

By considering an extended double-exchange model with spin-orbit coupling (SOC), we derive a general form of the Berry phase $\gamma$ that electrons pick up when moving around a closed loop. This form generalizes the well-known result valid for SU(2) invariant systems, $\gamma=\Omega/2$, where $\Omega$ is the solid angle subtended by the local magnetic moments enclosed by the loop. The general form of $\gamma$ demonstrates that collinear and coplanar magnetic textures can also induce a Berry phase different from 0 or $\pi$, smoothly connecting the result for SU(2) invariant systems with the well-known result of Karplus and Luttinger for collinear ferromagnets with finite SOC. By taking the continuum limit of the theory, we also derive the corresponding generalized form of the real space Berry curvature. The new expression is a generalization of the scalar spin chirality, which is presented in an explicitly covariant form. We finally show how these simple concepts can be used to understand the origin of the spontaneous topological Hall effect that has been recently reported in collinear and coplanar antiferromagnetic phases of correlated materials.