Interaction effects in one-dimensional helical liquids

Helical liquids are one-dimensional quantum many-body systems where the low-energy excitations have spin and momentum degrees of freedom locked together. They can be realized in several different setups such as semiconducting wires, carbon nanotubes, and ultracold atoms trapped in optical lattices. They also appear as edge states of two-dimensional topological insulators. As a result of spin-momentum locking, they exhibit peculiar transport properties and represent promising platforms for the realization of Majorana fermions when coupled to superconductors; moreover, in the presence of particle-particle interactions, they can enter a fractional helical phase whose low-energy excitations have a fractional charge and a fractional Abelian statistics. In this thesis we study various aspects of interacting one-dimensional helical liquids. Firstly, we consider a one-dimensional chain of alkaline earth-(like) atoms hosting low-energy helical excitations and, by means of the synthetic dimension framework, we show that it is equivalent to an interacting fermionic ladder pierced by a constant magnetic field which displays features related to the quantum Hall effect. In these synthetic ladders, we investigate how helical excitations analogous to the chiral edge modes of the quantum Hall effect are affected by repulsive interactions and we find the existence of a hierarchy of fully gapped phases characterized by peculiar charge and spin orders. Then, we study the charge and the spin patterns of a quantum dot embedded into a spin-orbit coupled quantum wire subject to a magnetic field and we explore how the 2π-periodic and the 4π-periodic Josephson currents in a superconductor-helical liquidsuperconductor hybrid junction are affected by interactions. Finally, we show that, by entangling two identical particles with a qubit, we can transmute the quantum statistics of particles in a scattering experiment which could be realized using the helical edge states of a two-dimensional topological insulator.