Low-field anomalous Hall effect in nonmagnetic metals by Wannier interpolation.

In ferromagnets, the Bloch states acquire a Berry curvature that produces a Hall effect at B = 0: the anomalous Hall effect (AHE). In nonmagnetic metals the Hall effect only appears at linear order in B. Interestingly, the low-field Hall conductivity of nonmagnetic metals has an anomalous (Berry-curvature) contribution in addition to the ordinary (Lorentz-force) one. In noncentrosymmetric crystals it goes as σ_AHE ∝ ∫ d³k Ω_k (m_k°B)f₀', where Ω_k and m_k are the Berry curvature and intrinsic magnetic moment (spin plus orbital) imparted on the Bloch states by the broken inversion symmetry; since Ω_k and m_k are both odd under time reversal, k and −k contribute equal amounts to σ_AHE. In centrosymmetric cystals the bands are Kramers degenerate, and the expression for σ_AHE involves the trace of the product of 2 × 2 matrices describing the Berry currvature and magnetic moment of the degenerate states; in this case TrΩ_k = Trm_k = 0, but Tr(Ω_km_k) = Tr(Ω_−km_−k) ≠ 0. Working in this non-Abelian setting, we develop a Wannier-interpolation scheme to calculate the low-field anomalous Hall conductivity from first principles. As a by-product, we obtain the anomalous g factors of the Bloch states. The low-field AHE is present in all metals, but is more pronounced in Weyl and Dirac semimetals.