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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.

We revisit the issue of thermally assisted quantum annealing by a detailed study of the dissipative Landau-Zener problem in presence of a Caldeira-Leggett bath of harmonic oscillators, using both a weak-coupling quantum master equation and a quasi-adiabatic path-integral approach. Building on the known zero-temperature exact results (Wubs et al., PRL 97, 200404 (2006)), we show that a finite temperature bath can have a beneficial effect on the ground-state probability only if it couples also to a spin-direction that is transverse with respect to the driving field, while no improvement is obtained for the more commonly studied purely longitudinal coupling. In particular, we also highlight that, for a transverse coupling, raising the bath temperature further improves the ground-state probability in the fast-driving regime. We discuss the relevance of these findings for the current quantum-annealing flux qubit chips., Comment: 10 pages, 6 figures

Synthetic ladders realized with one-dimensional alkaline-earth(-like) fermionic gases and subject to a gauge field represent a promising environment for the investigation of quantum Hall physics with ultracold atoms. Using density-matrix renormalization group calculations, we study how the quantum Hall-like chiral edge currents are affected by repulsive atom-atom interactions. We relate the properties of such currents to the asymmetry of the spin resolved momentum distribution function, a quantity which is easily addressable in state-of-art experiments. We show that repulsive interactions significantly stabilize the quantum Hall-like helical region and enhance the chiral currents. Our numerical simulations are performed for atoms with two and three internal spin states., 16 pages, 8 figures, 1 appendix

The joint action of a synthetic gauge potential and of atomic contact repulsion in a one-dimensional alkaline-earth(-like) fermionic gas with nuclear spin I leads to the existence of a hierarchy of fractional insulating and conducting states with intriguing properties. We unveil the existence and the features of those phases by means of both analytical bosonization techniques and numerical methods based on the density-matrix renormalization group algorithm. In particular, we show that the gapless phases can support helical modes, whereas the gapped states, which appear under certain conditions, are characterised both by density and magnetic order. Several distinct features emerge solely for spin I larger than 1/2, thus making their study with cold-atoms unique. We will finally argue that these states are related to the properties of an unconventional fractional quantum Hall effect in the thin-torus limit. The properties of this hierarchy of states can be experimentally studied in state-of-the-art cold-atom laboratories., (8 pages, 5 figures; Supplementary Material: 4 pages, 5 figures)

We consider a superconductor-two dimensional topological insulator- superconductor junction (S-2DTI-S) and study how the 2{\pi}- and 4{\pi}-periodic Josephson currents are affected by the electron-electron interaction. In the long-junction limit the supercurrent can by evaluated by modeling the system as a helical Luttinger liquid coupled to superconducting reservoirs. After having introduced bosonization in the presence of the parity constraint we turn to consider the limit of perfect and poor interfaces. For transparent interfaces, where perfect Andreev reflections occur at the boundaries, the Josephson current is marginally affected by the interaction. On the contrary, if strong magnetic scatterers are present in the weak link, the situation changes dramatically. Here Coulomb interaction plays a crucial role both in low and high temperature regimes. Furthermore, a phase-shift of Josephson current can be induced by changing the direction of the magnetization of the impurity.

The combination of interactions and static gauge fields plays a pivotal role in our understanding of strongly-correlated quantum matter. Cold atomic gases endowed with a synthetic dimension are emerging as an ideal platform to experimentally address this interplay in quasi-one-dimensional systems. A fundamental question is whether these setups can give access to pristine two-dimensional phenomena, such as the fractional quantum Hall effect, and how. We show that unambiguous signatures of bosonic and fermionic Laughlin-like states can be observed and characterized in synthetic ladders. We theoretically diagnose these Laughlin-like states focusing on the chiral current flowing in the ladder, on the central charge of the low-energy theory, and on the properties of the entanglement entropy. Remarkably, Laughlin-like states are separated from conventional liquids by Lifschitz-type transitions, characterized by sharp discontinuities in the current profiles, which we address using extensive simulations based on matrix-product states. Our work provides a qualitative and quantitative guideline towards the observability and understanding of strongly-correlated states of matter in synthetic ladders. In particular, we unveil how state-of-the-art experimental settings constitute an ideal starting point to progressively tackle two-dimensional strongly interacting systems from a ladder viewpoint, opening a new perspective for the observation of non-Abelian states of matter., 19 pages, 17 figures. Updated version after publication in Phys. Rev. X