Discrete Quantum Geometry and Intrinsic Spin Hall Effect*

The intrinsic anomalous (spin) Hall effect originates from the topological property of the Fermi Sea, and it can be evaluated based on the integral of the Berry curvature among the occupied states. The numerical evaluation using Wannier interpolation meets a difficulty of the singularities caused by band crossings. Here, we show that the quantum geometry of the Fermi surface can be numerically described by a 3-dimensional discrete quantum manifold, which not only avoids singularities in the Fermi Sea, but also enables the precise computation of the intrinsic Hall conductivity resolved in spin, as well as any other local properties of the Fermi surface. Numerical accuracy is assured even when singularities is arbitrarily close to the Fermi level, and this method remains robust with Kramers degeneracy. We demonstrated this approach by calculating the anomalous Hall and spin Hall conductivities in a two-band model of Weyl semimetal and a full-band ab-initio model of zinc-blende GaAs.