| dc.description.abstract | 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. | |
