A Fermi Surface Descriptor Quantifying the Correlations between Anomalous Hall Effect and Fermi Surface Geometry

In the last few decades, basic ideas of topology have completely transformed the prediction of quantum transport phenomena. Following this trend, we go deeper into the incorporation of modern mathematics into quantum material science focusing on geometry. Here we investigate the relation between the geometrical type of the Fermi surface and Anomalous and Spin Hall Effects. An index, $\mathbb{H}_F$, quantifying the hyperbolic geometry of the Fermi surface, shows a universal correlation (R$^2$ = 0.97) with the experimentally measured intrinsic anomalous Hall conductivity, of 16 different compounds spanning a wide variety of crystal, chemical, and electronic structure families, including those where topological methods give R$^2$ = 0.52. This raises a question about the predictive limits of topological physics and its transformation into a wider study of bandstructures' and Fermi surfaces' geometries and relating them to the quantum geometry theory of a more general metric of eigenstates, opening horizon for the prediction of phenomena beyond topological understanding.