Entropy of full covering of the kagome lattice by straight trimers

Abstract

We consider the number of ways all the sites of a kagome lattice can be covered by non-overlapping linear rigid rods where each rod covers 3 sites. We establish a 2-to-1 correspondence between the configurations of trimers on the kagome lattice to the covering by dimers of a related hexagonal lattice to show that entropy of coverings per trimer $s_{\text{tri,kag}}$ equals the entropy per dimer $ s_{\text{dim,hex}} $, and is given by $ s_{\text{tri,kag}} = s_{\text{dim,hex}} = \frac{1}{2 π} \int_0^{ 2 π/3} \log( 2 + 2 \cos k) dk \approx 0.323065947\ldots$.

Figures (4)

• Figure 1: (a) The kagome lattice as a union of non-overlapping equilateral triangles (pink colour) of side length $3$. Each triangle can be independently covered by trimers in two ways. (b) An example of covering of three adjacent triangles by trimers.
• Figure 2: (a) A kagome lattice strip (of width $L=4$) represented as a decorated square lattice. The two subsequent horizontal rows ($j$ and $j+1$) are marked by circles. (b) Each of the three sites of a trimer is given a different label, depending on the inclination of the trimer, and the relative position of the site within the trimer.
• Figure 3: Comparison of five trimer configurations. The only difference between (a) and (b) is at the three trimers bordering the colored hexagon. Configurations (b) and (c) have the same boundary sites (represented by black squares), but with regions A and B exchanged. The defect lines are marked by ticks. Panel (d) shows our reference configuration, while (e) illustrates the case with many defect lines.
• Figure 4: The kagome lattice (solid) and the corresponding hexagonal lattice (dashed lines). The sites belonging to the horizontal rows of the kagome lattice, shown as open symbols, are common to the hexagonal lattice also. Dimers are represented by blue bonds, the defect line is marked by ticks, and the boundary sites are denoted by squares.