ISSN:
1572-9613
Keywords:
Statistical mechanics
;
lattice statistics
;
number theory
;
eight-vertex model
;
solid-on-solid model
;
hard hexagon model
;
Rogers-Ramanujan identities
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01014383
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