Improved perturbation theory in the vortex liquids state of type II superconductors

We develop an optimized perturbation theory for the Ginzburg - Landau description of thermal fluctuations effects in the vortex liquids. Unlike the high temperature expansion which is asymptotic, the optimized expansion is convergent. Radius of convergence on the lowest Landau level is $a_T=-3$ in 2D and $a_T=-5$ in 3D. It allows a systematic calculation of magnetization and specific heat contributions due to thermal fluctuations of vortices in strongly type II superconductors to a very high precision. The results are in good agreement with existing Monte Carlo simulations and experiments. Limitations of various nonperturbative and phenomenological approaches are noted. In particular we show that there is no exact intersection point of the magnetization curves both in 2D and 3D.