SU(2)-invariant spin-12Hamiltonians with resonating and other valence bond phases

We construct a family of rotationally invariant, local, $S=1∕2$ Klein Hamiltonians on various lattices that exhibit ground-state manifolds spanned by nearest-neighbor valence bond states. We show that with selected perturbations such models can be driven into phases modeled by well-understood quantum dimer models on the corresponding lattices. Specifically, we show that the perturbation procedure is arbitrarily well controlled by a new parameter which is the extent of decoration of the reference lattice. This strategy leads to Hamiltonians that exhibit (i) $Z_2$ resonating valence bond (RVB) phases in two dimensions, (ii) $U\left(1\right)$ RVB phases with a gapless “photon” in three dimensions, and (iii) a Cantor deconfined region in two dimensions. We also construct two models on the pyrochlore lattice, one model exhibiting a $Z_2$ RVB phase and the other a $U\left(1\right)$ RVB phase.