Dynamics of the quantum dimer model on the triangular lattice: Soft modes and local resonating valence-bond correlations

We report on an exhaustive investigation of the dynamical dimer-dimer correlations in imaginary time for the quantum dimer model on the triangular lattice using the Green’s function Monte Carlo method. We show in particular that soft modes develop upon reducing the dimer-dimer repulsion, indicating the presence of a second-order phase transition into an ordered phase with broken translational symmetry. We further investigate the nature of this ordered phase, for which a 12-site unit cell has been previously proposed, with the surprising result that significant Bragg peaks are only present at two of the three high-symmetry points consistent with this unit cell. We attribute the absence of a detectable peak to its small magnitude due to the nearly uniform internal structure of the 12-site crystal cell.

Figure 1

(Color online) Left: Example of cluster used in the calculations (here the 12-site one). Right: Brillouin zone of the triangular lattice. High-symmetry points are labeled by $K=(4\pi ∕3,0)$, $M=(\pi ,\pi ∕\sqrt{3})$, and $X=(2\pi ∕3,0)$.

Figure 2

(Color online) Typical results for the dynamical dimer-dimer correlations $D(k,\tau )$ for the 432-site cluster at $X=(2\pi ∕3,0)$ (upper panel) and $K=(4\pi ∕3,0)$ (lower panel) for a few selected values of $V∕t$.

Figure 3

(Color online) Lowest-energy excitation spectra for the 432-site cluster for different values of $V∕t$.

Figure 4

(Color online) Size scalings of the gap at the $X$ (upper panel) and $M$ (lower panel) points for different values of $V∕t$.

Figure 5

(Color online) Static dimer-dimer correlation at $V∕t=0.55$ for different cluster sizes (upper panel). Size scaling of the dimer order parameter square $D_k^2$ (calculated at the $X$ point) for different values of $V∕t$ (lower panel).

Figure 6

(Color online) $V∕t$ dependence of the thermodynamic values of $D_k^2$ and of the gap at the $X$ and $M$ points.

Figure 7

(Color online) Mean value of the dimer operator for the 192-site cluster for $V∕t=0.6$ in the presence of the local perturbation (shown in the inset) placed at the rightest site. Dashed lines correspond to correlations lower than the uniform distribution (i.e., $1∕6$) and continuous lines to correlations larger than $1∕6$. The thickness of the bonds indicates how different the amplitude is from the uniform distribution.

Figure 8

(Color online) Static dimer correlations in the trial wave function constructed as the product of uniform RVB superpositions of dimer covering of 12-site hexagons.