### Abstract

The eight-vertex model is equivalent to a "solid-on-solid" (SOS) model, in which an integer height l_{i} is associated with each site i of the square lattice. The Boltzmann weights of the model are expressed in terms of elliptic functions of period 2 K, and involve a variable parameter η. Here we begin by showing that the hard hexagon model is a special case of this eight-vertex SOS model, in which η=K/5 and the heights are restricted to the range 1≤l_{i}≤4. We remark that the calculation of the sublattice densities of the hard hexagon model involves the Rogers-Ramanujan and related identities. We then go on to consider a more general eight-vertex SOS model, with η=K/r (r an integer) and 1≤l_{i}≤r-1. We evaluate the local height probabilities (which are the analogs of the sublattice densities) of this model, and are automatically led to generalizations of the Rogers-Ramanujan and similar identities. The results are put into a form suitable for examining critical behavior, and exponents β, α, {Mathematical expression} are obtained.

Original language | English (US) |
---|---|

Pages (from-to) | 193-266 |

Number of pages | 74 |

Journal | Journal of Statistical Physics |

Volume | 35 |

Issue number | 3-4 |

DOIs | |

State | Published - May 1 1984 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*35*(3-4), 193-266. https://doi.org/10.1007/BF01014383

}

*Journal of Statistical Physics*, vol. 35, no. 3-4, pp. 193-266. https://doi.org/10.1007/BF01014383

**Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities.** / Andrews, George E.; Baxter, R. J.; Forrester, P. J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities

AU - Andrews, George E.

AU - Baxter, R. J.

AU - Forrester, P. J.

PY - 1984/5/1

Y1 - 1984/5/1

N2 - The eight-vertex model is equivalent to a "solid-on-solid" (SOS) model, in which an integer height li is associated with each site i of the square lattice. The Boltzmann weights of the model are expressed in terms of elliptic functions of period 2 K, and involve a variable parameter η. Here we begin by showing that the hard hexagon model is a special case of this eight-vertex SOS model, in which η=K/5 and the heights are restricted to the range 1≤li≤4. We remark that the calculation of the sublattice densities of the hard hexagon model involves the Rogers-Ramanujan and related identities. We then go on to consider a more general eight-vertex SOS model, with η=K/r (r an integer) and 1≤li≤r-1. We evaluate the local height probabilities (which are the analogs of the sublattice densities) of this model, and are automatically led to generalizations of the Rogers-Ramanujan and similar identities. The results are put into a form suitable for examining critical behavior, and exponents β, α, {Mathematical expression} are obtained.

AB - The eight-vertex model is equivalent to a "solid-on-solid" (SOS) model, in which an integer height li is associated with each site i of the square lattice. The Boltzmann weights of the model are expressed in terms of elliptic functions of period 2 K, and involve a variable parameter η. Here we begin by showing that the hard hexagon model is a special case of this eight-vertex SOS model, in which η=K/5 and the heights are restricted to the range 1≤li≤4. We remark that the calculation of the sublattice densities of the hard hexagon model involves the Rogers-Ramanujan and related identities. We then go on to consider a more general eight-vertex SOS model, with η=K/r (r an integer) and 1≤li≤r-1. We evaluate the local height probabilities (which are the analogs of the sublattice densities) of this model, and are automatically led to generalizations of the Rogers-Ramanujan and similar identities. The results are put into a form suitable for examining critical behavior, and exponents β, α, {Mathematical expression} are obtained.

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U2 - 10.1007/BF01014383

DO - 10.1007/BF01014383

M3 - Article

VL - 35

SP - 193

EP - 266

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -