The molecular Hubbard Hamiltonian (MHH) naturally arises for ultracold polar alkali dimer molecules in optical lattices. We show that, unlike ultracold atoms, differrant molecules display different many-body phases due to intrinsic variances in molecular structure. We also demonstrate a wide variety of experimental controls on molecules via external fields, including applied static electric and magnetic fi elds, an AC microwave field, and the polarization and strength of optical lattice beams. We provide explicit numerical calculations of the parameters of the MHH, including tunneling and direct and exchange dipole-dipole interaction energies, for the molecules 6Li133Cs, 23Na40K, 87Rb133Cs, 40K87Rb, and 6Li23Na in weak and strong applied electric fi elds. As case studies of many-body physics, we use infi nite-size matrix product state methods to explore the quantum phase transitions from the superfluid phase to half fi lled and third filled crystalline phases in one dimension.
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