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Beyond the two-dimensional (2D) saddle-point Van Hove singularities (VHSs)with logarithmic divergences in the density of states (DOS), recent studieshave identified higher-order VHSs with faster-than-logarithmic divergences thatcan amplify electron correlation effects. Here we show that the cuprate high-Tcsuperconductors harbor high-order VHSs in their electronic spectra and unveil anew correlation that the cuprates with high-order VHSs display higher Tc. Ouranalysis indicates that the normal and higher-order VHSs can provide astraightforward new marker for identifying propensity of a material toward theoccurrence of correlated phases such as excitonic insulators and supermetals.Our study opens up a new materials playground for exploring the interplaybetween high-order VHSs, superconducting transition temperatures and electroncorrelation effects in the cuprates and related high-Tc superconductors.

High-$T_c$ superconductors as a New Playground for High-order Van Hove singularities and Flat-band Physics