Fluctuations in Superconductors using Landau-Ginzburg Theory

In this paper, we study the fluctuations in superconductors in the framework of Landau-Ginzurg theory. First, we review the formulation of LG theory and derive the LG equations. The we calculate the fluctuation contribution to the saddle point heat capacity, both inside and outside the immediate vicinity of the transition point. Outside the immediate vicinity, we follow the Gaussian integral calculation done in class, and obtained the correction to the heat capacity. In the immediate vicinity of the transition point, instead of using the perturbative renormalization group method introduced in class, we use the multiplicative RG method introduced by Larkin. Through calculating the critical exponents, we prove that the two methods agree with each other to leading order in t. We conclude the paper by a calculation of the fluctuations of the magnetic field, and find that the fluctuations can change the order of the phase transition. 1 Formulation and LG Equations The Landau-Ginzburg formalism is quite general. In class, we discussed the formalism in the context of a magnetic system. It is equally eligible to apply the formalism to a superconducting system, with the following Landau-Ginzburg functional free energy: F [Ψ(r)] = Fn + ∫ dr ( a ∣∣Ψ∣∣2 + b 2 ∣∣Ψ∣∣4 + 1 4m ∣∣∇Ψ∣∣2 + . . .) (1) where Fn is the free energy in the normal phase [6]. The above form is for the case when the external magnetic field is zero. For superconducting states inside an external