The joint action of a magnetic field and of interactions is crucial for the appearance of exotic quantum phenomena, such as the quantum Hall effect. Owing to their rich nuclear structure, equivalent to an additional synthetic dimension, one-dimensional alkaline-earth(-like) fermionic gases with synthetic gauge potential and atomic contact repulsion may display similar related properties. Here we show the existence and the features of a hierarchy of fractional insulating and conducting states by means of analytical and numerical methods. We demonstrate that the gapped states are characterized by density and magnetic order emerging solely for gases with effective nuclear spin larger than 1/2, whereas the gapless phases can support helical modes. We finally argue that these states are related to an unconventional fractional quantum Hall effect in the thin-torus limit and that their properties can be studied in state-of-the-art laboratories.

We consider a quantum spin Hall system in a two-terminal setup, with an extended tunneling contact connecting upper and lower edges. We analyze the effects of this geometry on the backscattering current as a function of voltage, temperature, and strength of the electron interactions. We find that this configuration may be useful to confirm the helical nature of the edge states and to extract their propagation velocity. By comparing with the usual quantum point contact geometry, we observe that the power-law behaviors predicted for the backscattering current and the linear conductance are recovered for low enough energies, while different power-laws also emerge at higher energies.Comment: 8 pages, 6 figures, published versio

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 enhance the chiral currents. Our numerical simulations are performed for atoms with two and three internal spin states.

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