Bipartite Rokhsar–Kivelson points and Cantor deconfinement

Abstract

Quantum dimer models on bipartite lattices exhibit Rokhsar--Kivelson points with exactly known critical ground states and deconfined spinons. We examine generic, weak perturbations around these points. In $d=2+1$ we find a first-order transition between a ``plaquette'' valence bond crystal and a region with a devil's staircase of commensurate and incommensurate valence bond crystals. In the part of the phase diagram where the staircase is incomplete, the incommensurate states exhibit a gapless photon and deconfined spinons on a set of finite measure, almost but not quite a deconfined phase in a compact $U(1)$ gauge theory in $d=2+1$! In $d=3+1$, we find a continuous transition between the $U(1)$ resonating valence bond phase and a deconfined staggered valence bond crystal. In an appendix, we comment on analogous phenomena in quantum vertex models, most notably the existence of a continuous transition on the triangular lattice in $d=2+1$.