Abstract
A lattice homogeneous under the modular group γ of fractional linear transformations is constructed. The generating function for close packed dimer configurations on this infinite lattice is found directly, without doing it first for a finite lattice, using the Pfaffian method. This requires orienting the lattice. The group SL(2, ℤ) is used to this end. Computation of the generating function is reduced to a particular case of the problem of finding the number of words reducible to the identity for a group which is the free product of two cyclic groups. Solution of this problem gives the dimer generating function as the solution of an algebraic equation. Considered as a function of the activities, the free energy has a logarithmic singularity.
Next an Ising model is built on the same lattice. The free energy per spin is evaluated by solving a dimer problem on an associated lattice following the general prescription of Fisher. It is a rational function of the solution of a system of two algebraic equations.
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Kasteleyn, P.W.: J. Math. Phys.4, 1332 (1963)
McCoy, B., Wu, T.T.: The two-dimensional Ising Model. Cambridge, Mass.: Harvard University Press (1973)
Fisher, M.E.: J. Math. Phys.7, 1776 (1966)
The standard reference concerning the modular group is Klein, F., Fricke, R.: Vorlesungen über die Theorie der elliptischen Modulfunktionen. Leipzig: Teubner 1890. A concise summary of its main properties is given in Chapter I of Gunning, R.C.: Lectures on modular forms. Princeton: Princeton University Press 1962. Facts concerning fractional linear transformations, hyperbolic geometry and discontinuous groups may be found in Chapter 3 of Siegel, C.L.: Topics in complex function theory. London: Wiley 1971
See Magnus, W.: Non-euclidean tesselations and their groups. New York-London: Academic Press 1973
Lund, F.: Ph. D. Thesis. Princeton University (unpublished)
Onsager, L.: Phys. Rev.65, 117 (1944)
For this and other topics concerning combinatorial group theory, see Magnus, W., Karrass, A., Solitar, D.: Combinatorial group theory. London: Wiley 1966. In particular the fundamental problem of deciding in a finite number of steps whether a word in a group defines the identity element of the group itself, was formulated by Dehn, M.: Math. Ann.71, 116 (1911)
Kasteleyn, P.W.: Physica27, 1209 (1961)
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Communicated by G. Gallavotti
Research sponsored in part by National Science Foundation Grant No. GP 30799 X to Princeton University
On leave from the Politecnico of Turin, Italy
Research sponsored in part by National Science Foundation Grant No. 40768X
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Lund, F., Rasetti, M. & Regge, T. Statistical mechanics models and the modular group. Commun.Math. Phys. 51, 15–40 (1976). https://doi.org/10.1007/BF01609049
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DOI: https://doi.org/10.1007/BF01609049