High-order Van Hove singularities and their connection to flat bands

The flattening of single-particle band structures plays an important role in the quest for novel quantum states of matter due to the crucial role of interactions. Recent advances in theory and experiment made it possible to construct and tune systems with nearly flat bands, ranging from graphene multilayers and moire' materials to kagome' metals and ruthenates. While theoretical models predict exactly flat bands under certain ideal conditions, evidence was provided that these systems host high-order Van Hove points, i.e., points of high local band flatness and power-law divergence in energy of the density of states. In this review, we examine recent developments in engineering and realising such weakly dispersive bands. We focus on high-order Van Hove singularities and explore their connection to exactly flat bands. We provide classification schemes and discuss interaction effects. We also review experimental evidence for high-order Van Hove singularities and point out future research directions.