Scaling of the quasiparticle spectrum for d-wave superconductors

In finite magnetic field H, the excitation spectrum of the low energy quasiparticles in a 2 dimensional d-wave superconductor exhibits a scaling with respect to $H^{1/2}$. This property can be used to calculate scaling relations for various physical quantities at low temperature T. As examples, we make predictions for the scaling behavior of the finite magnetic field specific heat, quasiparticle magnetic susceptibility, optical conductivity tensor, and thermal conductivity tensor. These predictions are compatible with existing experimental data. Most notably, the thermal Hall coefficient $\kappa_{xy}$ measured by Krishana et al. in YBCO is found to scale as $\kappa_{xy} \sim T^2 F(\alpha T/H^{1/2})$ for $T \lesssim 30 K$ in agreement with our predictions, where $\alpha$ is a constant and F is a scaling function.