Catastrophe theory classification of Fermi surface topological transitions in two dimensions

TL;DR

This paper uses catastrophe theory to classify all possible singularities in the Fermi surface topology of two-dimensional electronic systems, revealing universal divergence behaviors and symmetry restrictions.

Contribution

It provides a comprehensive classification of Fermi surface topological transitions in 2D systems using catastrophe theory, including universal divergence exponents and symmetry constraints.

Findings

Identifies 17 standard catastrophe types for Fermi surface singularities.

Shows power-law divergence of density of states at singularities.

Determines symmetry restrictions on catastrophe types at high symmetry points.

Abstract

We classify all possible singularities in the electronic dispersion of two-dimensional systems that occur when the Fermi surface changes topology, using catastrophe theory. For systems with up to seven control parameters (i.e., pressure, strain, bias voltage, etc), the theory guarantees that the singularity belongs to to one of seventeen standard types. We show that at each of these singularities the density of states diverges as a power law, with a universal exponent characteristic of the particular catastrophe, and we provide its universal ratio of amplitudes of the prefactors of energies above and below the singularity. We further show that crystal symmetry restricts which types of catastrophes can occur at the points of high symmetry in the Brillouin zone. For each of the seventeen wallpaper groups in two-dimensions, we list which catastrophes are possible at each high symmetry point.