Two-dimensional (2D) Van Hove singularities (VHSs) associated with the saddle points or extrema of the energy dispersion usually show logarithmic divergences in the density of states (DOS). However, recent studies find that the VHSs originating from higher-order saddle-points have faster-than-logarithmic divergences, which can amplify electron correlation effects and create exotic states such as supermetals in 2D materials. Here we report the existence of high-order VHSs in the cuprates and related high-Tc superconductors and show that the anomalous divergences in their spectra are driven by the electronic dimensionality of the system being lower than the dimensionality of the lattice. The order of VHS is found to correlate with the superconducting Tc such that materials with higher order VHSs display higher Tcs. We further show that the presence of the normal and higher-order VHSs in the electronic spectrum can provide a straightforward marker for identifying the propensity of a material toward correlated phases such as excitonic insulators or supermetals. Our study opens up a new materials playground for exploring the interplay between high-order VHSs, superconducting transition temperatures and electron correlation effects in the cuprates and related high-Tc superconductors.
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