Correlations and confinement in nonplanar two-dimensional dimer models

We study classical hard-core dimer models on the square lattice with links extending beyond nearest-neighbors. Numerically, using a directed-loop Monte Carlo algorithm, we find that, in the presence of longer dimers preserving the bipartite graph structure, algebraic correlations persist. While the confinement exponent for monomers drifts, the leading decay of dimer correlations remains $1∕r^2$, although the logarithmic peaks present in the dimer structure factor of the nearest-neighbor model vanish. By contrast, an arbitrarily small fraction of next-nearest-neighbor dimers leads to the onset of exponential dimer correlations and deconfinement. We discuss these results in the framework of effective theories, and provide an approximate but accurate analytical expression for the dimer correlations.

Figure 1

Labeling of the links of the $N_1\text{−}{N}_2$ model (a) and the $N_1\text{−}{N}_4$ model (b). (c) shows a step in the directed-loop update for the $N_1\text{−}{N}_2$ model, in which the entrance to the vertex is at site 3 and the exit is at site 8. The solid bond indicates the location of the dimer.

Figure 2

Dimer (a) and monomer (b) correlations along the direction $(r,0)$ for the $N_1\text{−}{N}_2$ model at $w_2=e^{-4}$, compared with those of the pure $N_1$ model. The solid lines show the agreement with the known power law (Ref. 4) for the $N_1$ model. Note that for the pure $N_1$ model $M\left(r\right)=0$ for even $r$. The results were obtained using $L=1024$ lattices.

Figure 3

(Color online) Scaling collapse of monomer (top panel) and dimer (bottom panel) correlations as the fugacity $w_2$ is varied. The two distinct sets of points in the upper panel correspond to even and odd distances.

Figure 4

Various dimer correlations along the direction $(r,0)$ in the $N_1\text{−}{N}_4$ model (on $L=1024$ lattices). The fugacities $w_4=e^{-4}$ and $e^{-3}$ correspond to concentrations $p_4\approx 0.024$ and $0.065$, respectively. The solid line shows the asymptotic form $r^{-2}$.

Figure 5

(Color online) Fourier transform of the dimer-dimer correlation $D_1$ of the $N_1$ model [left, for dimers pointing along $(1,0)$] and $D_4$ of the $N_4$ model [right, for dimers pointing along $(2,1)$] calculated on $L=128$ lattices. In the left graph, the log-divergent peaks at $(\pi ,0)$ and $(\pi ,2\pi )$ have been cut off at the value $2.0$ (for this lattice size the peak value is $\approx 3.12$).

Figure 6

Finite-size scaling of monomer correlations in the $N_1\text{−}{N}_4$ model. (a) The best scaling for three different cases. (b) Failure of a scaling with the pure $N_1$ exponent ${\alpha }_m=1∕2$ for the $N_1\text{−}{N}_4$ model at low fugacity $w_4$.

Figure 7

(Color online) Fourier transform of the dipolar part (i.e., without the dominant logarithmic peaks in $N_1$) of the dimer correlations $D_1$ of the $N_1$ model (left) and $D_4$ of the $N_4$ model (right), calculated from a large-$N$ theory. The overall amplitudes are not given accurately by the theory and are therefore not indicated here.