# Star-triangle and star-star relations in statistical mechanics

## Abstract

The homogeneous three-layer Zamolodchikov model is equivalent to a four-state model on the checkerboard lattice which closely resembles the four-state critical Potts model, but with some of its Boltzmann weights negated. Here the author shows that it satisfies a star-to-reverse-star (or simply star-star) relation, even though they know of no star-triangle relation for this model. For any nearest-neighbor checkerboard model, they show that this star-star relation is sufficient to ensure that the decimated model (where half the spins have been summed over) satisfies a twisted Yang-Baxter relation. This ensures that the transfer matrices of the original model commute in pairs, which is an adequate condition for solvability.

- Authors:

- Australian National Univ., Canberra (Australia)|[Northeastern Univ., Boston, MA (United States). Physics Dept.

- Publication Date:

- OSTI Identifier:
- 462614

- Report Number(s):
- CONF-9603223-

Journal ID: IJPBEV; ISSN 0217-9792; TRN: IM9719%%84

- Resource Type:
- Journal Article

- Journal Name:
- International Journal of Modern Physics B

- Additional Journal Information:
- Journal Volume: 11; Journal Issue: 1-2; Conference: Exactly soluble models in statistical mechanics: Current status and historical perspectives, Boston, MA (United States), Mar 1996; Other Information: PBD: 20 Jan 1997

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 66 PHYSICS; STATISTICAL MECHANICS; MATHEMATICAL MODELS; MATRICES; MATHEMATICS; NUMERICAL SOLUTION; TRANSFORMATIONS

### Citation Formats

```
Baxter, R.J..
```*Star-triangle and star-star relations in statistical mechanics*. United States: N. p., 1997.
Web. doi:10.1142/S0217979297000058.

```
Baxter, R.J..
```*Star-triangle and star-star relations in statistical mechanics*. United States. doi:10.1142/S0217979297000058.

```
Baxter, R.J.. Mon .
"Star-triangle and star-star relations in statistical mechanics". United States. doi:10.1142/S0217979297000058.
```

```
@article{osti_462614,
```

title = {Star-triangle and star-star relations in statistical mechanics},

author = {Baxter, R.J.},

abstractNote = {The homogeneous three-layer Zamolodchikov model is equivalent to a four-state model on the checkerboard lattice which closely resembles the four-state critical Potts model, but with some of its Boltzmann weights negated. Here the author shows that it satisfies a star-to-reverse-star (or simply star-star) relation, even though they know of no star-triangle relation for this model. For any nearest-neighbor checkerboard model, they show that this star-star relation is sufficient to ensure that the decimated model (where half the spins have been summed over) satisfies a twisted Yang-Baxter relation. This ensures that the transfer matrices of the original model commute in pairs, which is an adequate condition for solvability.},

doi = {10.1142/S0217979297000058},

journal = {International Journal of Modern Physics B},

number = 1-2,

volume = 11,

place = {United States},

year = {1997},

month = {1}

}

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