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In the last few decades, basic ideas of topology have completely transformedthe prediction of quantum transport phenomena. Following this trend, we godeeper into the incorporation of modern mathematics into quantum materialscience focusing on geometry. Here we investigate the relation between thegeometrical type of the Fermi surface and Anomalous and Spin Hall Effects. Anindex, $\mathbb{H}_F$, quantifying the hyperbolic geometry of the Fermisurface, shows a universal correlation (R$^2$ = 0.97) with the experimentallymeasured intrinsic anomalous Hall conductivity, of 16 different compoundsspanning a wide variety of crystal, chemical, and electronic structurefamilies, including those where topological methods give R$^2$ = 0.52. Thisraises a question about the predictive limits of topological physics and itstransformation into a wider study of bandstructures' and Fermi surfaces'geometries, opening horizon for prediction of phenomena beyond topologicalunderstanding.

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