  We propose to simulate the anisotropic and chiral Dzyaloshinskii-Moriya (DM)
interaction with Rydberg atom arrays. The DM Hamiltonian is engineered in a
one-dimensional optical lattice or trap array with effective long-range Rydberg
spins, interacting indirectly via a mobile mediator Rydberg atom. A host of XXZ
and DM Hamiltonians can be simulated with out-of-phase sign periodic coupling
strengths; for initial states in a stationary condensate, the DM interaction
vanishes. This theory allows for determination of the DM interaction (DMI)
vector components from first principles. The inherent anisotropy of the
Rydberg-Rydberg interactions, facilitates the DMI coupling to be tuned so as to
be comparable to the XXZ interaction. Our results make plausible the formation
of non-trivial topological spin textures with Rydberg atom arrays.
