Evolution of high-order Van Hove singularities away from cuprate-like band dispersions and its implications for cuprate superconductivity

We discuss the evolution of high-order Van Hove singularities (hoVHSs) that carry faster-than logarithmic divergences over a wide range of parameters in cuprate-like electronic band dispersions. Numerical analysis gives insight into the quantization of the VHS power-law-exponent pV and into transitions between hoVHSs with different values of pV. The cuprates are found to lie in the parameter regime where the amplitude of the hoVHS is not too large. Our study indicates that the occurrence of high-temperature superconductivity requires simultaneous tuning of two different competing orders (antiferromagnetism and the density wave associated with the hoVHS in cuprates), which is why it is so rare.