Existence of N\'eel order in the S=1 bilinear-biquadratic Heisenberg model via random loops

TL;DR

This paper proves the existence of Ne9el order in a spin-1 Heisenberg model with bilinear and biquadratic interactions in three or more dimensions, using a novel random loop representation and mathematical techniques.

Contribution

It introduces a random loop model for the spin-1 Heisenberg system and establishes Ne9el order for a broad parameter range in higher dimensions.

Findings

Ne9el order exists in 3D and higher for certain interaction ratios.

A new relation between spin and loop correlations is established.

The proof employs reflection positivity and infrared bounds.

Abstract

We consider the general spin-1 SU(2) invariant Heisenberg model with a two-body interaction. A random loop model is introduced and relations to quantum spin systems is proved. Using this relation it is shown that for dimensions 3 and above N\'eel order occurs for a large range of values of the relative strength of the bilinear ($-J_1$) and biquadratic ($-J_2$) interaction terms. The proof uses the method of reflection positivity and infrared bounds. Links between spin correlations and loop correlations are proved.