Correlated phases of bosons in tilted, frustrated lattices*

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 measuring the boson density. We extend a previous study (Phys. Rev. B {\bf 66}, 075128 (2002)) of cubic lattices to a variety of lattices and tilt directions in 2 dimensions: square, triangular, decorated square, and kagome, while noting the significance of three-body interactions. We find quantum phases with Ising density wave order, with superfluidity transverse to the tilt direction, a sliding Luttinger liquid phase, and quantum liquid states with no broken symmetry. Some cases map onto effective quantum dimer models.