  We study interacting bosonic or fermionic atoms in a high synthetic magnetic
field in two dimensions spanned by continuous real space and a synthetic
dimension. Here, the synthetic dimension is provided by hyperfine spin states,
and the synthetic field is created by laser-induced transitions between them.
While the interaction is short-range in real space, it is long-range in the
synthetic dimension in sharp contrast with fractional quantum Hall systems.
Introducing an analog of the lowest-Landau-level approximation valid for large
transition amplitudes, we derive an effective one-dimensional lattice model, in
which density-density interactions turn out to play a dominant role. We show
that in the limit of a large number of internal states, the system exhibits a
cascade of crystal ground states, which is known as devil's staircase, in a way
analogous to the thin-torus limit of quantum Hall systems.
