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

## 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.

- Authors:

- Department of Mathematics, University of Warwick, Coventry CV4 7AL (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 22306041

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 55; Journal Issue: 9; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANTIFERROMAGNETISM; FERROMAGNETISM; HEISENBERG MODEL; HEISENBERG PICTURE; INTERACTIONS; SPIN

### Citation Formats

```
Lees, Benjamin, E-mail: b.lees@warwick.ac.uk.
```*Long-range order for the spin-1 Heisenberg model with a small antiferromagnetic interaction*. United States: N. p., 2014.
Web. doi:10.1063/1.4895758.

```
Lees, Benjamin, E-mail: b.lees@warwick.ac.uk.
```*Long-range order for the spin-1 Heisenberg model with a small antiferromagnetic interaction*. United States. doi:10.1063/1.4895758.

```
Lees, Benjamin, E-mail: b.lees@warwick.ac.uk. Mon .
"Long-range order for the spin-1 Heisenberg model with a small antiferromagnetic interaction". United States. doi:10.1063/1.4895758.
```

```
@article{osti_22306041,
```

title = {Long-range order for the spin-1 Heisenberg model with a small antiferromagnetic interaction},

author = {Lees, Benjamin, E-mail: b.lees@warwick.ac.uk},

abstractNote = {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.},

doi = {10.1063/1.4895758},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 9,

volume = 55,

place = {United States},

year = {2014},

month = {9}

}

DOI: 10.1063/1.4895758

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