Hall effects in the presence of momentum-space and real-space Berry curvatures: A semiclassical analysis

Hall transport in magnetic metals or metal-magnetic insulator heterostructures has contributions from the Lorentz force (ordinary), momentum-space Berry curvature (anomalous), and real-space Berry curvature (topological), which arises from nontrivial magnetic textures such as skyrmions. We have developed a unified semiclassical theory [1, 2] of the three Hall effects taking into account all phase-space Berry curvatures, where we solved the Boltzmann equation perturbatively in the regime where the electronic mean free path ℓ << L_s, the characteristic length scale of the spin texture. Here we present a non-perturbative solution for the Boltzmann equation which is valid for arbitrary values of ℓ/L_s. We discuss the conditions for justifying the commonly used approximation of replacing the inhomogeneous real-space Berry curvature with its spatial average. We show that the topological Hall conductivity evolves from σ_xy^THE∼1/L_s² for ℓ/L_s<<1 to σ_xy^THE∼ L_s² when ℓ/L_s>>1, and demonstrate that the optimal skyrmion size maximizing σ_xy^THE scales as L_s^*∼√ℓ. Our analytical results give insights into these scaling forms that were previously observed only in large-scale numerical simulations [3]. Finally, we show that the widely used empirical rule expressing the total Hall resistivity as a sum of the ordinary, anomalous, and topological components holds only when ℓ≲L_s, but fails outside this regime.

[1] N. Verma, Z. Addison, M. Randeria. Science Adv. 8, eabq2765 (2022)

[2] Z. Addison, L. Keyes, M. Randeria, Phys. Rev. B 12, 014446 (2025).

[3] A. Matsui, T. Nomoto, R. Arita, Phys. Rev. B 104, 174432 (2021).