Long-Range Antiferromagnetic Order in the AKLT Model on Trees and Treelike Graphs

TL;DR

This paper generalizes the understanding of long-range antiferromagnetic order in the AKLT model from Cayley trees to a broader class of tree-like graphs, including various growth conditions and bilayer structures.

Contribution

It extends previous results to include diverse trees and treelike graphs under specific conditions, broadening the applicability of AKLT model analysis.

Findings

Long-range order established on Cayley-like graphs and bilayer Cayley trees.

Simple conditions identified for long-range order on finite subgraph-generated graphs.

Results applicable to trees with specified volume growth rates.

Abstract

We extend the result of Fannes, Nachtergaele, and Werner on long-range order in the AKLT model on Cayley trees to include various trees and tree-like graphs that obey certain conditions. Our examples split into three cases: Cayley-like tree-like graphs generated by a finite subgraph, for which we have a simple condition; arbitrary trees with a prescribed growth rate of their volume; and bilayer Cayley trees.