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

TL;DR

This paper proves a mathematical property of maximum-density dimer packings on site-diluted kagome and related lattices, showing how the structure of unmatched vertices depends on the cluster size parity, which supports the stability of certain quantum spin liquids.

Contribution

It provides a rigorous inductive proof of the structure of maximum matchings in site-diluted kagome lattices and related structures, extending previous numerical findings.

Findings

Maximum matchings have at most one monomer in connected components.

Odd-sized clusters have exactly one monomer in their maximum matchings.

Even-sized clusters admit perfect matchings with no monomers.

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.