Topological properties of diagrammatic perturbation theory for anyon system

Abstract

Perturbation theory is a main tool to understand the effect of complicated interactions in physical system. The perturbative expansion of physical observable in fermionic and bosonic syste can be described by Feynman diagrams. According to the linked cluster theorem, one of the basic rule of the quantum field theory, only the connected diagrams affect to physical observable, and the disconnected diagram cancelled each other. On the other hand, anyons are quasiparticles in two dimensions, not belonging to the bosons and fermions, and obeying fractional statistics. Anyons are classified according to their braiding properties, Abelian anyon or non-Abelian anyon. Due to the braiding properties, it is believed as key element of the topological quantum computation. Despite of a much effort, there is no unequivocal evidence of anyon in experimentally. In this dissertation, we apply the diagrammatic perturbation theory to the anyon system. We find that the anyon braiding results in 'topological connection' which has been never discussed before. We show that a disconnected diagram for bosons and fermions can be topologically connected and contribute to physical observable for anyons. We propose an interferometer setup operating in the fractional quantum Hall system to detect such a `topologically connected diagram' or equivalently fractional statistics.