On bipartite Rokhsar-Kivelson points and Cantor deconfinement

TL;DR

The paper analyzes quantum dimer models on bipartite lattices near Rokhsar-Kivelson points by mapping to a height-field theory and studying weak perturbations. It finds that RK points correspond to Gaussian fixed points with lattice-dependent anisotropic terms, yielding multicritical structures in 2+1D and a stable fixed point with a single relevant operator in 3+1D. In 2+1D, a devil's staircase of commensurate/incommensurate valence-bond crystals emerges from weak locking, with incommensurate states supporting a gapless photon and deconfined spinons on finite measure (Cantor deconfinement); in 3+1D, there is a continuous transition between a U(1) RVB liquid and a deconfined staggered VBC, with the gauge-theoretic structure playing a crucial role. Across these results, the paper emphasizes departures from Landau theory, highlighting deconfined criticality and the role of dangerously irrelevant operators, and discusses analogous phenomena in quantum vertex models.

Abstract

Quantum dimer models on bipartite lattices exhibit Rokhsar-Kivelson (RK) 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 (RVB) 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.

Figures (1)

• Figure 1: Phase diagram of the square lattice quantum dimer model. In the top part of the figure, the mean tilt of the height surface is plotted as a function of $v/t$ (see Section II). The corresponding dimer phases are sketched on the bottom part. The flat side (columnar and plaquette solids) terminate in the RK critical point ($v/t=1$), which has no dimer long-range order. Here, for the RK quantum dimer model (dashed line), the tilt jumps discontinuously, corresponding to a first order transition into the staggerred solid. Upon inclusion of longer-ranged interactions, however, the tilt will ascend a devil's staircase from the flat region, through a succession of incommensurate and commensurate phases. This phenomenon, (and the resulting structure factor) are discussed in detail in the text.