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

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Abstract

The eight-vertex model is equivalent to a “solid-on-solid” (SOS) model, in which an integer heightl i is associated with each sitei of the square lattice. The Boltzmann weights of the model are expressed in terms of elliptic functions of period 2K, 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 exponentsβ, α,\(\bar \alpha \) are obtained.

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Authors and Affiliations

  1. Department of Mathematics, McAllister Building, Pennsylvania State University, 16802, University Park, Pennsylvania

    George E. Andrews

  2. Research School of Physical Sciences, The Australian National University, G.P.O. Box 4, 2601, Canberra, Australia

    R. J. Baxter & P. J. Forrester

Authors
  1. George E. Andrews
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  2. R. J. Baxter
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  3. P. J. Forrester
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Additional information

Supported by the Guggenheim Foundation and in part by the National Science Foundation, grant MCS 8201733.

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Andrews, G.E., Baxter, R.J. & Forrester, P.J. Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities. J Stat Phys 35, 193–266 (1984). https://doi.org/10.1007/BF01014383

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  • DOI: https://doi.org/10.1007/BF01014383

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