Prediction of Hall effects from geometrical properties of the Fermi surface

Recently, the intrinsic Hall effects (e. g. spin, anomalous, and planar Hall effects) have been investigated as topological properties stemming from a material's electronic band structure. In our work we show the connection of these properties to the local geometry of the Fermi surface. Specifically we introduce the concept of geodesic flow of the Fermi surface and using this concept we link the Berry curvature term in the semiclassical equation of motion of Bloch electrons with the evolution of the Riemannian metric in the tangent bundle of the Fermi surface. As an example, we consider the comparison of Kubo formalism for the spin Hall effect with the geometrical analysis of the Fermi surface for Pt and Beta-W. Such geometrisation of topological properties can be applied to an algorithmic material search in crystallographic databases, paving the way for high throughput analysis of materials and topologically driven properties.