Three dimensional resonating valence bond liquids and their excitations

We show that there are two types of RVB liquid phases present in three-dimensional quantum dimer models, corresponding to the deconfining phases of U(1) and Z_2 gauge theories in d=3+1. The former is found on the bipartite cubic lattice and is the generalization of the critical point in the square lattice quantum dimer model found originally by Rokhsar and Kivelson. The latter exists on the non-bipartite face-centred cubic lattice and generalizes the RVB phase found earlier by us on the triangular lattice. We discuss the excitation spectrum and the nature of the ordering in both cases. Both phases exhibit gapped spinons. In the U(1) case we find a collective, linearly dispersing, transverse excitation, which is the photon of the low energy Maxwell Lagrangian and we identify the ordering as quantum order in Wen's sense. In the Z_2 case all collective excitations are gapped and, as in d=2, the low energy description of this topologically ordered state is the purely topological BF action. As a byproduct of this analysis, we unearth a further gapless excitation, the pi0n, in the square lattice quantum dimer model at its critical point.