LAUSR

We study a simple electron-phonon model on square and triangular versions of the Lieb lattice using an asymptotically exact strong coupling analysis. At zero temperature and electron density n=1 (one electron per unit cell), for various ranges of parameters in the model, we exploit a mapping to the quantum dimer model to establish the existence of a spin-liquid phase with Z_{2} topological order (on the triangular lattice) and a multicritical line corresponding to a quantum critical spin liquid (on the square lattice). In the remaining part of the phase diagram, we find a host of charge-density-wave phases (valence-bond solids), a conventional s-wave superconducting phase, and with the addition of a small Hubbard U to tip the balance, a phonon-induced d-wave superconducting phase. Under a special condition, we find a hidden pseudospin SU(2) symmetry that implies an exact constraint on the superconducting order parameters.