Thermal Fluctuations and Vortex Lattice Melting in Anisotropic Superconductors.

Because of the combination of high transition temperatures, strong anisotropy, and short coherence lengths which occurs in high temperature superconductors, strong thermal fluctuations are present over a wide temperature interval in these materials. Thermal fluctuations are especially important in a magnetic field where they are responsible for the melting of the Abrikosov vortex lattice at temperatures below the mean-field critical temperature giving rise to a vortex liquid state. In this work, we carry out a systematic study of the thermal fluctuation and vortex lattice melting problems for anisotropic superconductors in strong magnetic fields using realistic Ginzburg-Landau and Lawrence-Doniach models within the framework of the lowest Landau level (LLL) approximation. We start with the two dimensional system and express the problem in a new way by introducing a novel quantity which we call the Abrikosov-ratio entropy. We show that the thermodynamics of vortex states are solely determined by this quantity. We study it analytically by perturbation expansions and numerically by Monte Carlo simulation methods and show that there apparently exists a finite temperature weakly first order phase transition associated with the distributions of vortices due to the peculiar properties of Abrikosov-ratio entropy. We then introduce a quantity proportional to the Fourier transform of the superfluid density to study vortex states in detail. We derive for the first time a formal closed form expression for the density-density correlation function and evaluate it exactly up to 12-th order by a high temperature perturbation calculation. Low temperature correlation functions, obtained using Pade approximants, are in good agreement with Monte Carlo simulation results and show that the vortex-liquid becomes strongly correlated at temperatures well above the vortex solidification temperature. We have also evaluated the high-temperature expansion for the free energy to 13 -th order, two-orders further than in previous work. By Monte Carlo simulation methods, we are able to efficiently study this quantity to establish strong evidence for a weakly first order melting phase transition from low-temperature vortex solid state characterized by power-law positional correlations in the superfluid density to high-temperature vortex liquid state. Finally we study the Lawrence-Doniach model for layered superconductors using Monte Carlo simulation methods by calculating various correlation functions. We find a first order melting transition from a 3D vortex solid state to vortex liquid state. We construct the phase diagram in terms of dimensionless parameters characterizing intra-layer and inter-layer couplings and discuss the dependence on measurable parameters of real materials. The melting phase diagram in terms of temperature and magnetic field for real materials can be constructed in terms of measurable parameters only.