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

Thomas Jackson

TL;DR

This work addresses long-range antiferromagnetic order and ground-state degeneracy of the AKLT model on trees and tree-like graphs. It develops a transfer-operator framework anchored in a single-site map with transfer function $F_d(t)$ and shows that Cayley trees with degree $d\ge 5$ admit non-unique ground states; it then extends the analysis to treelike graphs via a cell-based transfer function $F_{\Lambda}(t)$, deriving growth-conditions for irregular and bilayer trees that determine order vs. uniqueness. The results include a decorational scaling for decorated Cayley trees and a path-weight condition for rooted quasi-Cayley trees, plus irregular-tree growth criteria and a fixed-point analysis for bilayer trees that broaden the Fannes–Nachtergaele–Werner paradigm to more complex graph structures. Overall, the paper demonstrates that local degree and volumetric growth govern ground-state uniqueness and long-range order in AKLT models on a wide class of graphs.

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.

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

TL;DR

This work addresses long-range antiferromagnetic order and ground-state degeneracy of the AKLT model on trees and tree-like graphs. It develops a transfer-operator framework anchored in a single-site map with transfer function and shows that Cayley trees with degree admit non-unique ground states; it then extends the analysis to treelike graphs via a cell-based transfer function , deriving growth-conditions for irregular and bilayer trees that determine order vs. uniqueness. The results include a decorational scaling for decorated Cayley trees and a path-weight condition for rooted quasi-Cayley trees, plus irregular-tree growth criteria and a fixed-point analysis for bilayer trees that broaden the Fannes–Nachtergaele–Werner paradigm to more complex graph structures. Overall, the paper demonstrates that local degree and volumetric growth govern ground-state uniqueness and long-range order in AKLT models on a wide class of graphs.

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.

Paper Structure

This paper contains 11 sections, 17 theorems, 110 equations, 6 figures.

Table of Contents

  1. The AKLT Hamiltonian and ground state on a general tree
  2. The Weyl Representation and Uniqueness of Finite-Volume States
  3. AKLT model transfer operator on a single site
  4. The transfer function $F_d(t)$
  5. Cayley trees: two proof methods
  6. Tree-like graphs: the graphical picture
  7. Examples
  8. Decorated Cayley trees
  9. Rooted Quasi-Cayley trees
  10. Irregular Trees
  11. Bilayer trees

Key Result

Theorem 1.1

klt Let $\psi\in{\mathcal{H}}$ such that $H\psi=0$; then there exists a unique $\phi$ such that $\phi$ is a polynomial in $u_x,v_x$ for $x\in\partial\Lambda$ and such that Conversely if the last equation holds then $H\psi=0$

Figures (6)

  • Figure 1: Vertex of degree $d=7$ with one outgoing index and six ingoing indices
  • Figure 2: First two layers of Cayley tree of degree $d=5$
  • Figure 3: First and second layers of the quasi-Cayley tree generated by the cell on the left with root $0$ and boundary $\{1,2,3\}$.
  • Figure 4: First two layers of the quasi-Cayley tree generated from the tree on the left; note this is also the degree $d=2$ Cayley tree with decoration number $g=1$
  • Figure 5: First two layers of the tree from Figure $2$ with decoration number $g=2$; note this is also the Cayley tree with degree $d=2$ and decoration number $g=5$.
  • ...and 1 more figures

Theorems & Definitions (35)

  • Theorem 1.1
  • proof
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • ...and 25 more