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.
Highlights
Valence electrons in crystalline solids are described by Bloch states with a dispersion relation n(k) between energy and crystal momentum k, with n denoting a set of discrete indices such as band, spin, etc
We present a classification of the singularities allowed at the high-symmetry points in the Brillouin zones corresponding to the seventeen wallpaper groups
We presented a classification of possible Fermi surface topological transitions in two spatial dimensions that take place via higher-order singularities
Summary
Valence electrons in crystalline solids are described by Bloch states with a dispersion relation n(k) between energy and crystal momentum k, with n denoting a set of discrete indices such as band, spin, etc. The higher-order singularities are indexed by three positive integers: the corank, determinacy and codimension The classification by these numbers is unique except for certain degenerate cases, which we show that in two-dimensions can be further distinguished by the winding, i.e., the number of times the electronic dispersion changes sign along a contour encircling the critical point. Catastrophes consistent with the symmetry can occur at such a point Another feature of high-symmetry points in the Brillouin zone is that they can host otherwise atypical higher-order singularities that are not part of the standard seventeen. II we introduce the language of catastrophe theory through a simple example of a tight-binding model that displays a higher-order singularity VII by briefly summarizing the scope of the work and setting the context for future work on the treatment of line singularities and the effects of interaction
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