  We study the `tilting' of Mott insulators of bosons into metastable states.
These are described by Hamiltonians acting on resonant subspaces, and have rich
possibilities for correlated phases with non-trivial entanglement of pseudospin
degrees of freedom encoded in the boson density. We extend a previous study
(arXiv:cond-mat/0205169) of cubic lattices to a variety of lattices and tilt
directions in 2 dimensions: square, decorated square, triangular, and kagome.
For certain configurations three-body interactions are necessary to ensure that
the energy of the effective resonant subspace is bounded from below. We find
quantum phases with Ising density wave order, with superfluidity transverse to
the tilt direction, and a quantum liquid state with no broken symmetry. The
existence of the quantum liquids state is shown by an exact solution for a
particular correlated boson model. We also find cases for which the resonant
subspace is described by effective quantum dimer models.
