Resonating Valence Bond States in an Electron-Phonon System

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 ${\mathbb{Z}}_2$ topological order (on the triangular lattice) and a multi-critical 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 (e.g. valence-bond crystals), 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 pseudo-spin $SU(2)$ symmetry that implies an exact constraint on the superconducting order parameters.