Gauge theories and topological models (anyons and topological order)

Author(s)

Miguel Jorge Bernabé Ferreira

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 2016-08-12

Examining board members

Paulo Teotonio Sobrinho; Joao Carlos Alves Barata; Valdecir Marvulle; Roldão da Rocha Junior; Luis Gregorio Godoy de Vasconcellos Dias da Silva

• Advisor: Paulo Teotonio Sobrinho

Abstract

One of the most interesting properties of quantum particles is the indistinguishability of particles of the same kind (as for example two electrons). On three dimensions these particles are known to be either fermions or bosons depending on their statistical behaviour. The reason for that is topology, in other words these two possible statistics are due to the space topology. However, on two dimensions there are particles called anyons which are neither fermion nor boson; they may obey a fractional statistic or a even more weird non-abelian statistic - where a single exchange of two identical anyons a unitary transformation on the wave function instead of just acquiring a phase factor. As well as the usual fermionic and bosonic statistic, the anyonic statistic depends strongly on the space topology and thus anyonic systems are often used to describe topological phases of matter of two dimensional systems. In this work we are going to show some general (model free) aspects of anyonic systems and also analyse some many body systems that describe anyons as quasi-particle excitations. We will mostly study a class of model called quantum double models (QDMs). Quantum double models are (2+1)D models where the degrees of freedom are elements of a group G living on the edges of lattice and the dynamic is given by a many body hamiltonian. The QDM is a well known and studied model on the literature, however in this work we are going to show an alternative construction for QDMs which will play two very important roles in this thesis. First, it will allows us to obtain the QDMs from deforming a topological invariant, and that helps to easily identify the topological order on this model. Besides, one can also obtain a many body hamiltonian that represents a generalization of the the QDM hamiltonian. Some of these new models describe anyonic systems other than the ones that can be described by usual QDM. In other words, this new construction leads to a many body hamiltonian that can describe both quantum double models and generalizations of it as particular cases. (AU)

• FAPESP's process: 12/21355-9 - Gauge Theories and Topological models
• Grantee: Miguel Jorge Bernabé Ferreira
• Support Opportunities: Scholarships in Brazil - Doctorate