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

George E. Andrews, R. J. Baxter, P. J. Forrester

Research output: Contribution to journalArticle

633 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)193-266
Number of pages74
JournalJournal of Statistical Physics
Volume35
Issue number3-4
DOIs
StatePublished - May 1 1984

Fingerprint

Solid Model
Ramanujan
apexes
Vertex of a graph
Hexagon
Vertex Model
Integer
Elliptic function
hexagons
Critical Behavior
Square Lattice
Ludwig Boltzmann
Model
Critical Exponents
sublattices
integers
Analogue
elliptic functions
Evaluate
Range of data

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Andrews, George E. ; Baxter, R. J. ; Forrester, P. J. / Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities. In: Journal of Statistical Physics. 1984 ; Vol. 35, No. 3-4. pp. 193-266.
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Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities. / Andrews, George E.; Baxter, R. J.; Forrester, P. J.

In: Journal of Statistical Physics, Vol. 35, No. 3-4, 01.05.1984, p. 193-266.

Research output: Contribution to journalArticle

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