Parquet Graph Resummation Method for Vortex Liquids

We present in detail a nonperturbative method for vortex liquid systems. This method is based on the resummation of an infinite subset of Feynman diagrams, the so-called parquet graphs, contributing to the four-point vertex function of the Ginzburg-Landau model for a superconductor in a magnetic field. We derive a set of coupled integral equations, the parquet equations, governing the structure factor of the two-dimensional vortex liquid system with and without random impurities and the three-dimensional system in the absence of disorder. For the pure two-dimensional system, we simplify the parquet equations considerably and obtain one simple equation for the structure factor. In two dimensions, we solve the parquet equations numerically and find growing translational order characterized by a length scale $R_c$ as the temperature is lowered. The temperature dependence of $R_c$ is obtained in both pure and weakly disordered cases. The effect of disorder appears as a smooth decrease of $R_c$ as the strength of disorder increases.