Dimers on the Triangular Kagome Lattice

TL;DR

This paper provides exact analytical results for close-packed dimers on the triangular kagome lattice, revealing short-range correlations, a constant entropy, and deconfined monomers, indicating a short-ranged spin liquid phase.

Contribution

It derives exact solutions for dimer configurations and correlations on the TKL, and connects these to the properties of the quantum and classical Heisenberg models on the same lattice.

Findings

Dimer-dimer correlations are short-ranged and vanish beyond a certain distance.

The entropy per site is 1/3 ln2, independent of bond weights.

Monomers are deconfined, indicating a short-range spin liquid phase.

Abstract

We derive exact results for close-packed dimers on the triangular kagome lattice (TKL), formed by inserting triangles into the triangles of the kagome lattice. Because the TKL is a non-bipartite lattice, dimer-dimer correlations are short-ranged, so that the ground state at the Rokhsar-Kivelson (RK) point of the corresponding quantum dimer model on the same lattice is a short-ranged spin liquid. Using the Pfaffian method, we derive an exact form for the free energy, and we find that the entropy is 1/3 ln2 per site, regardless of the weights of the bonds. The occupation probability of every bond is 1/4 in the case of equal weights on every bond. Similar to the case of lattices formed by corner-sharing triangles (such as the kagome and squagome lattices), we find that the dimer-dimer correlation function is identically zero beyond a certain (short) distance. We find in addition that monomers are deconfined on the TKL, indicating that there is a short-ranged spin liquid phase at the RK point. We also find exact results for the ground state energy of the classical Heisenberg model. The ground state can be ferromagnetic, ferrimagnetic, locally coplanar, or locally canted, depending on the couplings. From the dimer model and the classical spin model, we derive upper bounds on the ground state energy of the quantum Heisenberg model on the TKL.