Evolution of high-order Van Hove singularities away from cupratelike 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 cupratelike electronic band dispersions. Numerical and analytic results give insight into the quantized value of the VHS power-law exponent ${p}_{V}$ and the transition between hoVHSs with different values of ${p}_{V}$. The cuprates are found to lie in the parameter regime where the amplitude of the hoVHS is not too large. Our study supports the idea that superconductivity in the cuprates occurs near a crossover between two different competing orders (antiferromagnetism and the density wave associated with the hoVHS), which is why high-temperature superconductivity is so rare.