Topological Phases of Interacting Parafermions
Rodriguez Aldavero, Juanjo
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Parafermion chains are one-dimensional, quantum many-body systems which show topological phases possessing intriguing properties. For example, they are able to host quasiparticles with exotic quantum statistics called non-Abelian anyons. Unlike bosons or fermions, the exchange statistics shown by anyons allows them to modify the state of the system they are contained in. While Abelian anyons induce a phase change on the quantum system when they are spatially exchanged, non-Abelian anyons induce unitary transformations, which allows for the development of applications in fields such as quantum computing. In this thesis, the properties of increasingly complex parafermion chains are analysed. The starting point is given by the Kitaev model and its corresponding spin chain, the quantum Ising model. Afterwards, its generalizations lead to the discussion of the 3- state Potts model and the corresponding parafermion chain. Finally, the models are further generalized by adding extension terms on their respective Hamiltonians. Many of these models are very challenging to approach by means of analytical methods due to the presence of strong interactions, so their phase transitions are characterised by means of numerical methods based on tensor network ansätzen such as the Density Matrix Renormalization Group (DMRG) or the Multi-Scale Entanglement Renormalization Ansatz (MERA). The data is then analysed and interpreted using techniques coming from conformal field theory, finite-size scaling or the real-space renormalization group. The aim of this thesis is to characterize a region of the phase diagram of a generalized parafermion chain with two extension terms. A strong emphasis will be placed on the phase transition given between the trivial and ℤ3-ordered topological phases, as well as on a feature shown by the topological phase: a parametrized line on which the system is frustration-free. Along this line, the ground state of the system can be constructed exactly for arbitrarily large systems and exhibits exact degeneracy, even in the presence of strong interactions, paving the way for a more precise analysis of the topological properties of these kinds of models.