Models in Condensed Matter Near Criticality

The Landau-Ginzburg free energy expansion of the superconducting order parameter in the presence of a magnetic vector potential has been used as a basis for the analysis of magnetic field penetration in superconductors. Several specific cases have been examined in one and two dimensions in order to solve the complicated system of coupled nonlinear partial differential equations that describe interactions between the magnetic field and superconducting charge density. Exact solutions are found at the critical temperature, while accurate series expansions are used close to it. Oscillatory damped profiles of the magnetic field penetration are found. In addition, periodic patterns of magnetic induction and a phase-shifted superconducting charge density are obtained. Other approximate methods were used to examine the behavior of the superconducting system for arbitrary temperatures below the critical temperature. For two dimensions, two types of solutions were obtained; vortices which exist below the critical temperature and spirals which exist at the critical temperature. In two dimensions, the symmetries that were considered so as to reduced the equations to nonlinear ordinary differential equations gave us in some cases, equations which have no known analytical solutions. A method was developed to find analytical solutions to some of these equations which are of the particular form; f ^{''} + A(f)f^' + B(f)(f ^')^2 + C(f) = 0. The method consists of converting the ordinary differential equation into a system of coupled algebraic equations which in principal can be solved using a symbolic solver program on a computer. Also, solutions were found for seven special cases, one of which is the solutions to the nonlinear ordinary differential equation; f^{'' } + B(f)(f^')^2 + C(f) = 0. Several examples of the method are given as a demonstration of the power of the method. The last part consists of investigating a generalized Hamiltonian of two interacting quasi-particles. The method of coherent structures is developed for systems with two different types of interacting particles. The general formalism is worked out leading to coupled field equations relevant for all three combinations of quantum statistics. The Frohlich Hamiltonian for electron-phonon coupling in metals is analyzed as an example.