  Ultracold polar molecules trapped on an optical lattice is a many-body system
that, under appropriate conditions, may support collective excitations
reminiscent of excitons in solid state crystals. Here, we discuss the
rotational excitations of molecules on an optical lattice leading to rotational
Frenkel excitons. Apart from solid hydrogen, there is no other natural system
that exhibits rotational excitons. The rotational excitons have unique
properties that can be exploited for tuning non-linear exciton interactions and
exciton-impurity scattering by applying an external electric field. We show
that this can be used to explore the competing role of the dynamical and
kinematic exciton-exciton interactions in excitonic energy transfer and to
study quantum localization in a dynamically tunable disordered potential. The
rotational excitons can also be used as a basis for quantum simulation of
condensed matter models that cannot be realized with ultracold atoms. As an
example, we discuss the possibility of engineering the Holstein model with
polar molecules on an optical lattice.
