Topology and interaction effects in one-dimensional systems

With the discovery of the integer quantum Hall effect by von Klitzing and collaborators in 1980,
the mathematical field of topology entered the world of condensed matter physics. Almost three
decades later, this eventually led to the theoretical prediction and the experimental realization
of many intriguing topological materials and topology-based devices. In this Ph.D. thesis, we
will study the interplay between topology and another key topic in condensed matter physics,
namely the study of inter-particle interactions in many-body systems. This interplay is analyzed
from two different perspectives.

Firstly, we studied how the presence of electron-electron interactions affects single-electron
injection into a couple of counter-propagating one-dimensional edge channels. The latter appear
at the edges of topologically non-trivial systems in the quantum spin Hall regime and they can
also be engineered by exploiting the integer quantum Hall effect. Because of inter-channel
interactions, the injected electron splits up into a couple of counter-propagating fractional
excitations. Here, we carefully study and discuss their properties by means of an analytical
approach based on the Luttinger liquid theory and the bosonization method. Our results are
quite relevant in the context of the so-called electron quantum optics, a fast developing field
which deeply exploits the topological protection of one-dimensional edge states to study the
coherent propagation of electrons in solid-state devices. As an aside, we also showed that
similar analytical techniques can also be used to study the time-resolved dynamics of a Luttinger
liquid subject to a sudden change of the interaction strength, a protocol known as quantum
quench which is gaining more and more attention, especially within the cold-atoms community.

Secondly, we study how inter-particle interactions can enhance the topological properties
of strictly one-dimensional fermionic systems. More precisely, the starting point is the seminal
Kitaev chain, a free-fermionic lattice model which hosts exotic Majorana zero-energy modes at
its ends. The latter are extremely relevant in the context of topological quantum computation
because of their non-Abelian anyonic exchange statistics. Here we show that, by properly
adding electron-electron interactions to the Kitaev chain, it is possible to obtain lattice models
which feature zero-energy parafermionic modes, an even more intriguing generalization of
Majoranas. To this end, we develop at first an exact mapping between Z4 parafermions and
ordinary fermions on a lattice. We subsequently exploit this mapping to analytically obtain an
exactly solvable fermionic model hosting zero-energy parafermions. We study their properties
and numerically investigate their signatures and robustness even when parameters are tuned
away from the exactly solvable point.