Entropy of full covering of the kagome lattice by straight trimers

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$.

💡 Research Summary

The paper addresses the long‑standing problem of determining the exact entropy of fully packed linear trimers (rigid rods covering three lattice sites) on the two‑dimensional kagome lattice. While dimer (k = 2) coverings have been solved exactly for many lattices, no exact solution existed for k ≥ 3 on a non‑trivial lattice. The authors exploit a special geometric feature of the kagome lattice to map the trimer covering problem onto a dimer covering problem on a related hexagonal (honeycomb) lattice, establishing a precise two‑to‑one correspondence between trimer configurations and dimer configurations.

The construction begins by introducing a coordinate system for the kagome lattice and labeling each site according to the orientation of the trimer that occupies it. Horizontal rows can be in one of two “reference” states (all sites labeled 2 or all labeled 5). When a region of one reference state meets a region of the other, a narrow defect line appears. The authors show that the allowed motion of this defect line from one row to the next is exactly the same as the world‑line of a particle in a discrete‑time totally asymmetric simple exclusion process (TASEP) moving leftward. Consequently, each admissible trimer tiling can be uniquely described by the space‑time history of an even number of non‑crossing TASEP walkers; there are two trimer tilings for each history, accounting for the overall factor of two.

Next, the authors embed the kagome lattice into a hexagonal lattice by adding extra sites (shown as red circles). In the hexagonal lattice, a dimer occupies a bond that either connects two kagome sites or a kagome site to an added site. By choosing a reference dimer configuration where all dimers attach to the lower kagome site, the defect lines in the dimer picture coincide with the TASEP walkers defined above. Hence each trimer configuration corresponds to exactly two dimer configurations, establishing the 2‑to‑1 mapping.

The dimer problem on the hexagonal lattice is a classic free‑fermion model. Using the transfer‑matrix formalism, the eigenvalues are products of factors (