Renormalization, duality, and phase transitions in two- and three-dimensional quantum dimer models

We derive an extended lattice gauge theory type action for quantum dimer models and relate it to the height representations of these systems. We examine the system in two and three dimensions and analyze the phase structure in terms of effective theories and duality arguments. For the two-dimensional case we derive the effective potential both at zero and finite temperature. The zero-temperature theory at the Rokhsar-Kivelson (RK) point has a critical point related to the self-dual point of a class of $Z_N$ models in the $N\to \infty$ limit. Two phase transitions featuring a fixed line are shown to appear in the phase diagram, one at zero temperature and at the RK point and another one at finite temperature above the RK point. The latter will be shown to correspond to a Kosterlitz-Thouless (KT) phase transition, while the former will be governed by a KT-like universality class, i.e., sharing many features with a KT transition but actually corresponding to a different universality class. On the other hand, we show that at the RK point no phase transition happens at finite temperature. For the three-dimensional case we derive the corresponding dual gauge theory model at the RK point. We show in this case that at zero temperature a first-order phase transition occurs, while at finite temperatures both first- and second-order phase transitions are possible, depending on the relative values of the couplings involved.

Figure 1

(Color online) Schematic representation of the main different phases arising in two dimensions and at zero temperature. There is numerical evidence for a mixed phase between the VBS and the plaquette states (not shown here) (Ref. 22). The resulting phase transition is discontinuous.

Figure 2

(Color online) Schematic representation of the phases of the two- dimensional QDM at the RK point. The zero temperature KT (quantum) critical point is given by ${\widehat{K}}_c=8/\pi$. Note that for $z>0$ [Panel (a)], the KT-like transition is from a plaquette to an RVB state, which is a liquid of valence bonds. In panel (b) the RVB state emerges out of a VBS state at $z<0$.

Figure 3

(Color online) Schematic phase diagram of the two-dimensional QDM in the $\rho T$-plane. For large $\rho$, the KT temperature is high and varies linearly with $\rho$ with a slope of $\left(2/\pi \right)$ [Eq. (42)]. Here we have also indicated the mixed phase (Refs. 22, 25). Temperature effects melt the dimerized phases to a valence-bond (VB) liquid phase, which is actually the same as a thermally induced RVB state.