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

A critical point of the energy dispersion is the momentum where electron velocity vanishes. At the corresponding energy, the density of states (DOS) exhibits nonanalyticity 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 L\'eon van Hove, whose DOS is particle-hole symmetric and logarithmically divergent. In this work, we describe and classify high-order critical points based on topology, scaling, and symmetry. 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.