Low-temperature specific heat and thermal Hall conductivity in a vortex state of d-wave superconductors

We analyze the mixed state of $d$-wave lattice superconductors focusing on the quasiparticle contribution to the specific heat and the thermal Hall conductivity at intermediate magnetic fields $H_{c1}\ll H \ll H_{c2}$. In the ultra-low temperature regime $T\ll T_0\approx v_D^2/(v_F l)$ the specific heat follows a general scaling form $C[T, H=hc/el^2] =(T/v_F l) \Phi[ v_F/(Tl), v_F/v_D, k_F l]$. In this regime the specific heat exhibits oscillatory behavior as a $2\pi$-periodic function of $k_F l$: in general it has an activated form $C\propto \exp(-\Delta_m/T)$ except for a discrete set of $k_Fl$ where $\Delta_m=0$ and $C\propto T^2$. At temperatures $T_0\ll T\ll \Delta$, the $k_Fl$-oscillations become unobservable due to thermal broadening and the Simon-Lee scaling is recovered. The results of the analysis of the thermal Hall conductivity are similar: in particular, at the lowest temperatures, $\kappa_{xy}$ is an oscillating $2\pi$-periodic function of $k_Fl$. We calculate the scaling functions numerically and compare our results with the existing experimental data on the specific heat and thermal Hall conductivity.