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 non-analyticity such as divergence.
Critical points can be rst classied as ordinary and high-order ones, and the ordinary critical
points have been studied thoroughly by Leon 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 elds.