Existence of Néel Order in the S=1 Bilinear-Biquadratic Heisenberg Model via Random Loops

Benjamin Lees

doi:10.1007/s00220-016-2656-1

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.

Highlights

In this work properties of the spin-1 Heisenberg model are deduced using a random loop model first introduced in the work of Nachtergaele [18]
Heisenberg ferromagnet; this improved the bound of Conlon and Solovej [5]
In this article we use the method of reflection positivity and infrared bounds on a random loop model

Summary

Introduction

In this article we use the method of reflection positivity and infrared bounds on a random loop model. Reflection positivity for this quadrant is already known; for J1 < 0 = J2 it was shown in [8] and for J1 = 0 < J2 one can see, for example, [16] lemma 3.4 for an explicit proof It was proved in [8] that Néel order occurs for J1 < 0 = J2; it is clear the result extends to a neighbourhood of the axis J1 < 0 < J2 with J2 sufficiently small. It is impossible to extend the result concerning Néel order any significant amount without some new results This is where the loop model has been essential.

The Random Loop Model

Space-Time Spin Configurations

The Random Loop Representation

J2 4 J2

Occurrence of Néel Order