Long-range order for the spin-1 Heisenberg model with a small antiferromagnetic interaction

TL;DR

This paper proves the existence of long-range order in the spin-1 Heisenberg model with a small antiferromagnetic interaction, using reflection positivity and infrared bounds, highlighting the robustness of nematic order.

Contribution

It introduces a matrix representation approach to establish long-range order in the SU(2) invariant spin-1 Heisenberg model with small antiferromagnetic perturbations.

Findings

Long-range order exists in the model with purely nematic interaction.

Small antiferromagnetic interactions do not destroy the long-range order.

The matrix representation clarifies key identities in the analysis.

Abstract

We look at the general SU(2) invariant spin-1 Heisenberg model. This family includes the well known Heisenberg ferromagnet and antiferromagnet as well as the interesting nematic (biquadratic) and the largely mysterious staggered-nematic interaction. Long range order is proved using the method of reflection positivity and infrared bounds on a purely nematic interaction. This is achieved through the use of a type of matrix representation of the interaction making clear several identities that would not otherwise be noticed. Using the reflection positivity of the antiferromagnetic interaction one can then show that the result is maintained if we also include an antiferromagnetic interaction that is sufficiently small.