Phase Diagram of an Extended Quantum Dimer Model on the Hexagonal Lattice

We introduce a quantum dimer model on the hexagonal lattice that, in addition to the standard three-dimer kinetic and potential terms, includes a competing potential part counting dimer-free hexagons. The zero-temperature phase diagram is studied by means of quantum Monte Carlo simulations, supplemented by variational arguments. It reveals some new crystalline phases and a cascade of transitions with rapidly changing flux (tilt in the height language). We analyze perturbatively the vicinity of the Rokhsar-Kivelson point, showing that this model has the microscopic ingredients needed for the “devil’s staircase” scenario [Eduardo Fradkin et al. Phys. Rev. B 69, 224415 (2004)], and is therefore expected to produce fractal variations of the ground-state flux.

Figure 1

Schematic phase diagram from QMC simulations ($L_x=L_y=60$). The $(v_0,v_3)$ plane is divided into five regions: a staggered phase with the maximal flux ($f=2$), the star and the plaquette phases ($f=0$), the $S_2$ phase ($f=1/2$), and the fan region, containing a cascade of flux sectors $1/2\le f<2$. The plaquette color indicates the dimer density (same scale as in Figs. 2 and 3).

Figure 2

Left: a configuration of three strings and the corresponding dimer covering, with 0- and 3-plaquettes. Left bottom and right: three variational classes of dynamically constrained strings, called $S$, $H$, and $F$ strings. $S$ strings are in a static zigzag configuration, $H$ strings ($F$ strings) are allowed to fluctuate by one row in every second column (in every column). Arrows indicate the fluctuations of the strings, each corresponding to a 3-plaquette flip. Dimer densities are indicated according to the color scale of Fig. 3. For $H$ and $F$ strings, the shown dimer densities correspond to a superposition of the allowed configurations.

Figure 3

Dimer density per hexagon in the fan region, according to the color scale on the right.

Figure 4

Perturbation theory near the RK point. Left: $j$-plaquette densities ${\rho }_j$ as functions of the flux density $f$. Right: ground-state flux density $f$ as a function of the angle $\theta$ that parametrizes the perturbation. The nontrivial region lies between $\pi /2$ (transition out of the staggered phase) and ${\theta }_1\simeq 1.84695$, where $f$ drops discontinuously from $f_1\simeq 0.195654$ to 0. The transition from $f=0$ to $f=2$ occurs at ${\theta }_2\simeq 4.8268$ (not shown).