Finance

Combinatorics

Physics

Topology (electrical circuits)

Symmetry (geometry)

Mathematics

Asymmetry

Phase transition

Critical exponent

Philosophy

Economics

Critical point (mathematics)

Divergence (linguistics)

Geometry

Twist

Linguistics

Quantum mechanics

Momentum (technical analysis)

Scaling

Physical review. B./Physical review. B

A critical point of the energy dispersion is the momentum where electron velocity vanishes. At the corresponding energy, the density of states (DOS) exhibits non-analyticity such as divergence. Critical points can be first classified as ordinary and high-order ones, and the ordinary critical points have been studied thoroughly by Léon van Hove. In this work, we describe and classify high-order critical points based on topology, scaling and symmetry, which are beyond Léon van Hove's framework. We show that high-order critical points can have power-law divergent DOS with particle-hole asymmetry, and can be realized at generic or symmetric momenta by tuning a few parameters such as twist angle, strain, pressure and/or external fields.

Classification of critical points in energy bands based on topology, scaling, and symmetry