Universal Scaling Properties of Superconductors in Magnetic Fields

Based on renormalization group arguments we establish that for a superconductor in the presence of a weak external magnetic field, $B$, the dependence on $B$ and the deviation from the critical temperature, $\tau$, of a thermodynamic quantity, $P$, takes the scaling form $P=t^{\theta}X({B\over\Phi_0}\tau^{-2\nu},q\tau^{-\nu\omega_e})$, where $\theta$ and $\nu$ are XY exponents, $q$ is the scaled electromagnetic coupling and $\nu\omega_e$ is the associated crossover exponent. For $q/\tau^{\nu\omega_e}\ll1$, the experimentally accessible region in high-$T_c$ superconductors, there is a reduction to one-variable scaling plus small corrections. In this region we find the shift in the specific heat maximum is given by $\Delta=x_0{(B/\Phi_0)}^{1/2\nu}$ and that the singular part of the free energy at the critical temperature takes the form $F_{sing}=c(d){(B/\Phi_0)}^{d/2}$ where $c(d)$ is a universal amplitude. A one loop approximation in three dimensions gives $c(3)\sim0.22$. The results presented here should have equal applicability to the nematic to smectic-A transition.