  An ultracold gas of heteronuclear alkali dimer molecules with hyperfine
structure loaded into a one-dimensional optical lattice is investigated. The
\emph{Hyperfine Molecular Hubbard Hamiltonian} (HMHH), an effective low-energy
lattice Hamiltonian, is derived from first principles. The large permanent
electric dipole moment of these molecules gives rise to long range
dipole-dipole forces in a DC electric field and allows for transitions between
rotational states in an AC microwave field. Additionally, a strong magnetic
field can be used to control the hyperfine degrees of freedom independently of
the rotational degrees of freedom. By tuning the angle between the DC electric
and magnetic fields and the strength of the AC field it is possible to control
the number of internal states involved in the dynamics as well as the degree of
correlation between the spatial and internal degrees of freedom. The HMHH's
unique features have direct experimental consequences such as quantum
dephasing, tunable complexity, and the dependence of the phase diagram on the
molecular state.
