Three-dimensional resonating-valence-bond liquids and their excitations

Abstract

We show that there are two types of resonating-valence-bond (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 nonbipartite face-centered 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.