Thermodynamic Fluctuations of the Order Parameter in Type-II Superconductors near the Upper Critical FieldHc2

In this paper we investigate the mean-square deviation of the order parameter from its equilibrium for type-II superconductors, assuming that the Gibbs free energy is given by a Ginzburg-Landau-type functional. It turns out that the equilibrium function, which is already varying with position because of the penetrating magnetic flux, resists distortion much less than the constant order parameter in type-I superconductors. Therefore, the mean-square deviation becomes equal in magnitude to the square of the equilibrium order parameter in the magnetic field range $\frac{(H_{c2\mathrm{}}-B)}{H_{c2\mathrm{}}\lesssim {10}^{-4\mathrm{}}} \mathrm{to} {10}^{-5\mathrm{}}$. This means that the range of magnetic fields in which Ginzburg-Landau-type descriptions should fail and a singular behavior might show up seems not entirely out of reach of present experimental techniques, in contrast to the situation in type-I superconductors. As a necessary byproduct of our investigation, we prove that in the limit as $B$ goes to $H_{c2\mathrm{}}$, the equilibrium function corresponding to a triangular fluxoid lattice gives a local minimum of the free energy in function space; and we obtain an orthogonal set of fluctuations of the equilibrium function which are normal modes in the sense that the second-order increment of the free energy does not mix their amplitudes. The special properties of the most important fluctuations suggest that the vortex crystal melts slightly below $H_{c2\mathrm{}}$, but that the identity of the vortices is maintained.