Universal scaling properties of superconductors in magnetic fields

Abstract

Based on renormalization group arguments we establish that for a superconductor in the presence of a weak magnetic field the dependence on the magnetic induction, B, and the deviation from the zero-field critical temperature, τ, of a thermodynamic quantity, P, takes the scaling form P= τ θ X( B/ Φ 0 τ −2 ν , qτ − νω e ), where θ and ν are XY exponents, q is the scaled electromagnetic coupling and νω e is the associated crossover exponent. For q/ τ νω e ≪1, 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 Δ= x 0( B/ Φ 0) 1/2 ν and that the singular part of the free energy at the critical temperature takes the form F sing= c( d)( B/ Φ 0) d/2 where c( d) is a universal amplitude. A one loop approximation in three dimensions gives c(3)∼0.22. The results presented here should have equal applicability to the nematic to smectic-A transition.