  We show that indirect spin-spin interactions between effective spin-1/2
systems can be realized in two parallel 1D optical lattices loaded with polar
molecules and/or Rydberg atoms. The effective spin can be encoded into
low-energy rotational states of polar molecules or long-lived states of Rydberg
atoms, tightly trapped in a deep optical lattice. The spin-spin interactions
can be mediated by Rydberg atoms, placed in a parallel shallow optical lattice,
interacting with the effective spins by charge-dipole (for polar molecules) or
dipole-dipole (for Rydberg atoms) interaction. Indirect XX, Ising and XXZ
interactions with interaction coefficients $J^{\bot}$ and $J^{zz}$ sign varying
with interspin distance can be realized, in particular, the $J_{1}-J_{2}$ XXZ
model with frustrated ferro-(antiferro-)magnetic nearest (next-nearest)
neighbor interactions.
