Long-range and frustrated spin-spin interactions in crystals of cold polar molecules

We describe a simple scheme for the implementation and control of effective spin-spin interactions in self-assembled crystals of cold polar molecules. In our scheme spin states are encoded in two long-lived rotational states of the molecules and coupled via state dependent dipole-dipole forces to the lattice vibrations. We show that by choosing an appropriate time dependent modulation of the induced dipole moments the resulting phonon-mediated interactions compete with the direct dipole-dipole coupling and lead to long-range and tunable spin-spin interaction patterns. We illustrate how this technique can be used for the generation of multi-particle entangled spin states and the implementation of spin models with longe-range and frustrated interactions which exhibit non-trivial phases of magnetic ordering.