Parafermionic modes in correlated electron systems - Teixeira, Raphael Levy Ruscio Castro (2023)
Obtaining non-abelian anyons is one of the primary goals within the field of topological states of matter. The simplest of these anyons, Majorana bound states (MBSs), have been proposed as candidates for implementing topological quantum computation relying on its non-abelian statistics. Parafermions bound states (PBSs) can be regarded as ZN generalizations of the Z2-symmetric MBSs. PBS have richer non-abelian exchange statistics compared to MBS and would thus offer advantages for quantum computation. In contrast to MBS, PBS usually require strong interactions between electrons and has been proposed to exist, for instance, in fractional quantum Hall insulators with induced superconductivity. Tight-binding models have also been suggested to host PBSs, although in some cases they are nontopological. Recently, Kondo devices have also been proposed to host parafermionic states. Nonetheless, in all setups, many-body interactions are necessary for the existence of PBS. In this thesis, we are concerned with the properties of PBSs that can be used in experimental setups. While most of the work developed here concerns fermionic models, we also investigate a continuous model of fractional quantum Hall insulators with induced superconductivity. This work is, therefore, divided into two parts. In the first part, we examine the fermionic models using Density Matrix Renormalization Group (DMRG) as well as some properties of Fock-parafermion. We start with a proposal on how to use Quantum dots (QDs) to probe the existence of Z4 parafermions and differentiate it from 2 MBSs (2xZ2). Then, we introduce two fermionic models that host Z3 parafermions. We show the topological equivalence of the models and their properties. In the second part, we use a combination of analytical (semiclassical instanton approximation) and numerical (quantum Monte Carlo simulations) techniques to determine the effective parafermion Hamiltonian and its ground state splitting in a fractional quantum Hall insulators with induced superconductivity.