Numerical Renormalization, Doped Antiferromagnets, and Statistics - Topics in Low Dimensional Physics
The first part of the thesis deals with so called numerical renormalization group methods, and in particular the density-matrix renormalization group (DMRG). This method performs extremely well for large one-dimensional quantum systems like spin-chains and Hubbard models. We discuss general features of the quantum states obtained by using this method for large systems. One finds that the method works better, i.e. faster convergence and higher accuracy, when applied to systems having an energy gap between the ground state and the first excited state. An investigation of the behavior of the DMRG for a simple gapless system is done, mainly focusing on the properties of correlation functions.
The second part of the thesis deals with topics related to superconductivity. It starts with a discussion of magnetic Josephson junctions and the physical processes governing the behavior of such a system. We also introduce and discuss models of high-temperature superconductors, in particular the t-J model and its relation to stripes. Starting from the t-J model on a two dimensional square lattice, we discuss an approximation where the spin-configuration is assumed static. We discuss the properties of the fictitious magnetic field experienced by the electrons due to the static spin-configuration. Configurations being close to ferromagnetic and antiferromagnetic turn out to have very different properties. Furthermore, we derive an effective model for this system and look for different phases of this model. In particular we investigate whether the system takes advantage of a spin-generated fictitious flux. Uniform phases are discussed in a Hartree-Fock approximation and we also consider stripes between antiferromagnetically ordered regions, these being modeled in a self-consistent Hartree approximation.
Finally, and inspired by the fictitious fluxes, there is a chapter discussing the statistics of particles living on a circle. It is shown that there are two free parameters in the quantization of such a system and we also sort out the relation between fermions and hard-core bosons in one dimension.