Análise fractal das propriedades de transporte quântico no regime universal

Abstract

The dimensionality of a quantum system influences many properties, we can observe different behaviors if the same system is two-dimensional or three-dimensional. There are already many papers in the literature that investigate these properties in integer dimensions, however little is known about the behavior of these systems in fractional dimensions. This master thesis has the purpose of studying the phenomenon of quantum transport in a scattering region with fractional dimension in the universal regime, to achieve this goal, we performed many computer simulations using a scattering region in the shape of a fractal called the Sierpinski carpet, which has a fractal dimension of 1.8928. We determined the configuration in which the system reaches the universal regime in each of the ensembles of random matrix theory, in the orthogonal circular ensemble, unitary circular ensemble and symplectic circular ensemble. Finally, we calculated the fractal dimension of the conductance and shot noise curves in each ensemble using the boxcounting technique.