  We derive a novel lattice Hamiltonian, the \emph{Molecular Hubbard
Hamiltonian} (MHH), which describes the essential many body physics of
closed-shell ultracold heteronuclear molecules in their absolute ground state
in a quasi-one-dimensional optical lattice. The MHH is explicitly
time-dependent, making a dynamic generalization of the concept of quantum phase
transitions necessary. Using the Time-Evolving Block Decimation (TEBD)
algorithm to study entangled dynamics, we demonstrate that, in the case of hard
core bosonic molecules at half filling, the MHH exhibits an emergent time scale
over which spatial entanglement grows, crystalline order appears, and
oscillations between rotational states self-damp into an asymptotic
superposition. We show that this time scale is a non-monotonic function of the
physical parameters describing the lattice. We also point out that experimental
mapping of the static phase boundaries of the MHH can be used to measure the
molecular polarizability tensor.
