First-principles analysis of in-plane anomalous Hall effect using symmetry-adapted Wannier Hamiltonians and multipole decomposition

TL;DR

This work develops a microscopic framework that combines time-reversal-symmetric Wannier functions with a symmetry-adapted multipole basis to decompose first-principles Hamiltonians into electric, magnetic, magnetic toroidal, and electric toroidal channels, enabling rank-by-rank analysis of the in-plane anomalous Hall effect (IAHE). By applying this to body-centered cubic Fe, the authors show that high-rank magnetic and magnetic toroidal multipoles contribute as strongly as magnetic dipoles, with magnetic-toroidal 16-poles sometimes acting with opposite sign, and that uniaxial strain along [103] can markedly reshape and even invert the IAHE. The TRS-Wannier plus SAMB approach provides a quantitative measure of Hamiltonian symmetry fidelity (via $\Delta_{\mathrm{energy}}$) and a physically interpretable map from multipoles to IAHE, including a minimal $p_z$-$d_{xy}$ model that captures strain-induced valley features. Overall, the work demonstrates multipole-resolved Hamiltonian engineering as a practical route to predict and control IAHE in simple ferromagnets, with potential implications for magnetoelastic spintronic design.

Abstract

The in-plane anomalous Hall effect occurs when magnetization lies within the same plane as the electric field and Hall current, and requires magnetic point groups lacking rotational or mirror symmetries. While it is observed in both Weyl semimetals and elemental ferromagnets, the microscopic role of higher-order multipoles remains unclear. Here, we develop a microscopic framework that combines time-reversal-symmetric Wannier functions with a symmetry-adapted multipole basis to decompose the first-principles Wannier Hamiltonian into electric, magnetic, magnetic toroidal, and electric toroidal multipoles. This approach allows us to rotate the magnetization rank by rank and quantify how each multipole affects the conductivity. Applying this framework to body-centered cubic iron, we find that high-rank magnetic and magnetic toroidal multipoles contribute with magnitudes comparable to magnetic dipoles, while magnetic toroidal 16-poles act with the opposite sign. Furthermore, based on this multipole analysis, we apply uniaxial strain along the [103] direction to control the dominant multipoles contributing to the conductivity. The strain substantially modifies its angular dependence, demonstrating that multipole-resolved Hamiltonian engineering and magnetoelastic control serve as practical routes to predict and tune the in-plane anomalous Hall conductivity in simple ferromagnets.

Figures (17)

• Figure 1: Decomposition of the anomalous Hall vector $\bm{\sigma}$ into the components parallel to the magnetization ($\sigma_\parallel$), in-plane perpendicular ($\sigma_\perp$), and out-of-plane ($\sigma_{\bm n}$). The gray plane represents the plane formed by rotating $\bm{H}$.
• Figure 2: Schematic definitions of the magnetization rotation planes and the azimuthal angle $\psi$.
• Figure 3: Crystal structures of body-centered cubic iron (bcc $\ce{Fe}$).
• Figure 4: Dependence of the band structure on the bond range. Considering neighbors up to the 35th-nearest neighbors leads to convergence with the DFT results.
• Figure 5: Top 20 coefficients in multipole decomposition of TRS-Wannier Hamiltonian.