Fermi Surface Geometry

TL;DR

This paper introduces the Fermi Surface Geometry Effect (FSGE), a novel approach using differential geometry to characterize Fermi surfaces and predict electron transport phenomena with high correlation to experimental data.

Contribution

It develops a new geometric index, $\\mathbb{H}_F$, to describe Fermi surface hyperbolicity and correlates it with the anomalous Hall effect across diverse compounds.

Findings

High correlation (R$^2$=0.97) between geometric index and Hall effect

Universal geometric characterization applicable to various materials

Foundation for a geometrical theory of electronic structures

Abstract

Motivated by the famous and pioneering mathematical works by Perelman, Hamilton, and Thurston, we introduce the concept of using modern geometrical mathematical classifications of multi-dimensional manifolds to characterize electronic structures and predict non-trivial electron transport phenomena. Here we develop the Fermi Surface Geometry Effect (FSGE), using the concepts of tangent bundles and Gaussian curvature as an invariant. We develop an index, $\mathbb{H}_F$, for describing the the "hyperbolicity" of the Fermi Surface (FS) and show a universal correlation (R$^2$ = 0.97) with the experimentally measured intrinsic anomalous Hall effect of 16 different compounds spanning a wide variety of crystal, chemical, and electronic structure families, including where current methods have struggled. This work lays the foundation for developing a complete theory of geometrical understanding of electronic (and by extension magnonic and phononic) structure manifolds, beginning with Fermi surfaces. In analogy to the broad impact of topological physics, the concepts begun here will have far reaching consequences and lead to a paradigm shift in the understanding of electron transport, moving it to include geometrical properties of the E vs k manifold as well as topological properties.