Duality of the 2D Nonhomogeneous Ising Model on the Torus

Duality relations for the 2D nonhomogeneous Ising model on the finite square lattice wrapped on the torus are obtained. The partition function of the model on the dual lattice with arbitrary combinations of the periodical and antiperiodical boundary conditions along the cycles of the torus is expressed through some specific combination of the partition functions of the model on the original lattice with corresponding boundary conditions. It is shown that the structure of the duality relations is connected with the topological peculiarities of the dual transformation of the model on the torus. PACS numbers: 05.50.+q E-mail address: abugrijgluk.apc.org The duality relation for the two-dimensional Ising model was discovered by Kramers and Wannier [1]. In their work, Kramers and Wannier showed the correspondence between the partition function of the model in low-temperature phase and the partition function of the model on the dual lattice in high-temperature phase and vice versa: (cosh 2K̃)) Z̃(K̃) = (cosh 2K)Z(K) (1) sinh 2K · sinh 2K̃ = 1. Using this self-duality property, the critical temperature of the 2D Ising model was determined [1] before Onsager had obtained the exact solution [2]. Kadanoff and Ceva [3] generalized the Kramers-Wannier duality relation (1) to the nonhomogeneous case (the coupling constants are arbitrary functions of lattice site coordinates) with spherical boundary conditions ∏ r̃,μ (cosh 2K̃μ(r̃)) Z̃[K̃] = ∏ r,μ (cosh 2Kμ(r)) Z[K], (2) sinh 2Kx(r) · sinh 2K̃y(r̃) = 1, sinh 2Ky(r) · sinh 2K̃x(r̃) = 1. (3) Here μ = x, y and r, r̃, Kμ(r), K̃μ(r̃) are coordinates and coupling constants on the original and dual lattices respectively. Since the Kadanoff-Ceva relation (2) defines the connection between functionals, this relation is very useful for analysis of the thermodymamic phases of the model. Thus, for example, this relation allows one to define correctly the disorder parameter μ, to obtain the duality relation connecting correlation functions on the original and dual lattices, to define ”mixed” correlation functions 〈σ(ri) . . . σ(rj)μ(rk) . . . μ(rl)〉 and so on (see Ref. [3]). As was already mentioned in Ref. [1,3], relations (1) and (2) can not be understood literally. So, for example, using the method of comparing highand low-temperature expansions for deriving the duality relation (1) in the case of the periodical boundary conditions, it is hard to take into account and to compare the graphs wrapping up the torus. In fact Eq. (1) is correct in the thermodynamic limit (for the specific free energy). However for the nonhomogeneous case the procedure of thermodynamic limit is rather ambiguous. In Ref. [3] the duality relation (3) was obtained for spherical (nonphysical for the square lattice) boundary conditions. Since duality is a popular method of nonperturbative investigation in field theory and statistical mechanics (for review see Ref. [4]), it is important to formulate a duality transformation for finite systems. Recently, we have suggested [5,6] the exact duality relations for the nonhomogeneous Ising model on a finite square lattice of size N = n×m wrapped on the torus: ∏ r̃,μ (cosh 2K̃μ(r̃)) −1/2 Z̃[K̃] = ∏ r,μ (cosh 2Kμ(r)) T̂Z[K]. (4)