Random geometry of maximum-density dimer packings of the site-diluted kagome lattice

TL;DR

The paper addresses how vacancy disorder on the site-diluted kagome lattice constrains maximum-density dimer packings through the Gallai-Edmonds decomposition. It develops an inductive proof showing that odd-parity clusters $C_{2n-1}$ are spanned by a single ${\mathcal{R}}$-type region with exactly one monomer, while even-parity clusters $C_{2n}$ admit a single ${\mathcal{P}}$-type region with no monomers, and extends this framework to other line-graph lattices. The approach leverages a local connectivity property and the structure theory of Gallai-Edmonds, revealing a robust geometric picture relevant to vacancy-induced local moments in sRVB spin liquids. The results bridge mathematical graph theory with condensed-matter applications and underline how local monomer geometry influences low-energy physics in diluted frustrated magnets.

Abstract

Recent work that analyzed the effect of vacancy disorder on a short-range resonating valence bond spin liquid state of kagome-lattice antiferromagnets argued that such spin liquids are stable to vacancy disorder. The argument relied crucially on a numerical study that identified the following property of the site-diluted kagome lattice: maximum-density dimer packings (maximum matchings) of any connected component of such site-diluted kagome lattices have at most one unmatched vertex that hosts a monomer. Here, we provide an inductive proof of a stronger result that implies this property: If a connected cluster of such a lattice has an odd number of vertices, its Gallai-Edmonds decomposition~\cite{Lovas_Plummer_1986} has exactly one ${\mathcal R}$-type region that spans the entire connected cluster and hosts a single monomer of any maximum-density dimer packing. If on the other hand it has an even number of sites, it admits perfect matchings (fully-packed dimer coverings with no monomers) and its Gallai-Edmonds decomposition consists of a single ${\mathcal P}$-type region that spans the entire cluster. Our proof also applies to the site-diluted Archimedean star lattice, the site-diluted pyrochlore lattice (corner-sharing tetrahedra), the site-diluted hyperkagome lattice, and, more generally, to any lattice satisfying a certain local connectivity property. It does not apply to bond-diluted versions of such lattices.

Figures (4)

• Figure 1: The above figure shows all possible even parity clusters of size two and four on the kagome lattice. Notice that a single site, when attached to these regions, makes them a ${\mathcal{R}}$-type region.
• Figure 2: The above figure shows all possible odd parity clusters of size three and five on the kagome lattice. Notice that these motifs are ${\mathcal{R}}$-type regions with one monomer. a) and b) can be generated by attaching a single site to a) of Fig. \ref{['Fig:P_regions']} and similarly c), d), e) and f) can be generated by attaching a single site to b) and c) of Fig. \ref{['Fig:P_regions']}.
• Figure 5: The largest blossom in the ${\mathcal{R}}$-type region that spans the largest connected cluster of a site-diluted kagome lattice occupies a nonzero fraction of the cluster, which grows rapidly as the dilution $n_v$ is reduced
• Figure 6: The number density of odd-type sites in the ${\mathcal{R}}$-type region that spans the largest connected cluster of a site-diluted kagome lattice goes to zero rapidly as the dilution $n_v$ is reduced.

Theorems & Definitions (2)

• Theorem 1
• proof