  Rydberg atoms with dipole-dipole interactions provide intriguing platforms to
explore exotic quantum many-body physics. Here we propose a novel scheme with
laser-assisted dipole-dipole interactions to realize synthetic magnetic field
for Rydberg atoms in a two-dimensional array configuration, which gives rise to
the exotic bosonic topological states. In the presence of an external effective
Zeeman splitting gradient, the dipole-dipole interaction between neighboring
Rydberg atoms along the gradient direction is suppressed, but can be assisted
when Raman lights are applied to compensate the energy difference. With this
scheme we generate a controllable uniform magnetic field for the complex
spin-exchange coupling model, which can be mapped to hard core bosons coupling
to an external synthetic magnetic field. The highly tunable flat Chern bands of
the hard core bosons are then obtained and moreover, the bosonic fractional
quantum Hall states can be achieved with experimental feasibility. This work
opens an avenue for the realization of the highly-sought-after bosonic
topological orders using Rydberg atoms.
