Laughlin proposed a Hall cylinder. On Laughlin’s proposed Hall cylinder, the magnetic field piece through the surface of the cylinder. We could also realize the Hall cylinder using synthetic dimension. Here, we could use the internal states of atoms such as the hyperfine spin state of alkaline atom or nuclear spin state of alkaline-earth like atom as the synthetic dimension $w$. We could realize a closed loop with finite momentum transfer. Combined with a physical dimension $y$, this corresponds to a cylinder geometry with magnetic flux penetrating the surface of the cylinder.
Comparing to Laughlin’s Hall cylinder, the major difference is that we use discrete site in the synthetic dimension instead of a continuous model. However, there are also interesting physics brought by the finite number of site, that is, there would be an emergent lattice in the physical dimension.