Catastrophe theory classification of Fermi surface topological transitions in two dimensions

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 one of seventeen standard types known as catastrophes. 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 for 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.