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

TL;DR

The paper tackles the challenge of predicting anomalous Hall conductivity (AHC) by exploring a geometry-driven descriptor of the Fermi surface, $\mathbb{H}_F$, which quantifies the concentration of hyperbolic regions on the FS. It demonstrates a striking $R^2 \approx 0.97$ correlation between $\mathbb{H}_F$ and intrinsic AHC across 16 materials, and shows a linear relation $\sigma^{AHC} = m\,\mathbb{H}_F + \sigma_0$ with $m \approx 1573~(\Omega\mathrm{cm})^{-1}$, indicating a practical upper bound for single-EBR FS. The work further shows that multi-EBR FSs can exceed this limit, and that the FS-based approach aligns with SHC trends, offering a computationally cheaper alternative to Berry-curvature-based Kubo calculations and enabling high-throughput screening. It argues that FS geometry provides insights into quantum geometry and proposes connections to the quantum geometric tensor, suggesting future directions to unify geometric and topological perspectives in quantum transport, with potential extensions to 3D bands and bosonic excitations.

Abstract

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.

Figures (7)

• Figure 1: (color online):Local Fermi surface geometry and Fermi surface orbits in a magnetic field. A. Three possible local geometric types (left to right): elliptic, planar, and hyperbolic. For the hyperbolic case, the orbit's projection onto the local coordinate system $(x_1,x_2)$ around the point of splitting orbits is shown on the right. B. An example of a fully hyperbolic surface.
• Figure 2: (color online):A. Schematic image of the $\mathbb{H}_F$ calculation in the direction of Hall measurement. B. Correlation graph of the predicted SHC values via the Kubo formalism vs $\mathbb{H}_F$, as defined in the text. C. Correlation graph top: experimentally determined intrinsic AHC vs $\mathbb{H}_F$ for various 2D or layered materials ((l) identifies layered structures). Bottom: experimentally determined intrinsic AHC vs predicted values of AHC via the Kubo formalism.
• Figure 3: (color online):A. Bandstrucure of $CrPt_3$. Blue and yellow colors represent two topologically disconnected (having different EBRs) sets of bands crossing the Fermi level. These sets are disconnected by the continuous gap present between them; i.e. true semimetallic behavior. B. Graph of energy resolved AHC predicted in two different ways: red dashed line is the Kubo based prediction, black dashed line stems from the linear correlation between $H_F$ and AHC calculated separately for FS contributions from each set of bands, then summed together for total $\mathbb{H}_F$.
• Figure 4: (color online):$\mathbb{H}_F$ dependence of k-mesh density for Fe and Co$_2$FeSi. y-axis is the calculated total $\mathbb{H}_F$, and the x-axis is k-points cubed.
• Figure 5: (color online): Fermi surface of Ni: A. conduction band, B. valence band. Blue represents hyperbolic points with low mean curvature (smooth), and red represents hyperbolic points with giant mean curvature (singular).