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

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.