Effects of van-Hove singularities on the upper critical field

We present a study of the superconducting pairing susceptibility $K_T(r)$ for a two-dimensional isotropic system with a strong power-law divergence in the density of states $N(ϵ)\sim {ϵ}^{-1+\frac{1}{b}}$, $b>1$. We show that the pair propagator has the scaling form, $K_T(r)=r^{b-3}F(T^{1\mathrm{/}b}r)$. An anomalous short-range behavior is found, leading straightforwardly to positive curvature in the upper critical field, for $b\lesssim 2$ and to a zero temperature divergence, $H_{c2}\sim T^{-2+\frac{4}{b}}$, for $b>2$.