Magic of high-order van Hove singularity

The van Hove singularity in density of states generally exists in periodic systems due to the presence of saddle points of energy dispersion in momentum space. We introduce a new type of van Hove singularity in two dimensions, resulting from high-order saddle points and exhibiting power-law divergent density of states. We show that high-order van Hove singularity can be generally achieved by tuning the band structure with a single parameter in moire superlattices, such as twisted bilayer graphene by tuning twist angle or applying pressure, and trilayer graphene by applying vertical electric field. Correlation effects from high-order van Hove singularity near Fermi level are also discussed.