Abstract
In this paper, we provide the exact expression for the coefficients in the low-temperature series expansion of the partition function of the two-dimensional Ising model on the infinite square lattice. This is equivalent to exact determination of the number of spin configurations at a given energy. With these coefficients, we show that the ferromagnetic–to–paramagnetic phase transition in the square lattice Ising model can be explained through equivalence between the model and the perfect gas of energy clusters model, in which the passage through the critical point is related to the complete change in the thermodynamic preferences on the size of clusters. The combinatorial approach reported in this article is very general and can be easily applied to other lattice models.
Highlights
We provide the exact expression for the coefficients in the low-temperature series expansion of the partition function of the two-dimensional Ising model on the infinite square lattice
We show that the ferromagnetic–to–paramagnetic phase transition in the square lattice Ising model can be explained through equivalence between the model and the perfect gas of energy clusters model, in which the passage through the critical point is related to the complete change in the thermodynamic preferences on the size of clusters
The combinatorial approach reported in this article is very general and can be applied to other lattice models
Summary
The main idea behind this paper is that the low temperature series expansion of the partition function, Z(x), of any lattice model can be obtained from the low temperature series expansion of the corresponding free energy, f(x). Which are given by the N-th complete Bell polynomials, YN({an}), stand for the number of spin configurations with energy 2JN above the ground state. −βf (x) = − ln x + x4 + 2x6 + 9 x8 + 12x10 + 112 x12 + 130x14 + 1961 x16 + Up to this point our considerations were exact and concentrated on the bulk case of the infinite square-lattice Ising model. The presented results may provide an approximate formulae for the coefficients g(N, V) in the low-temperature series expansion of the partition function for the Ising model on a finite square lattice of the size V, i.e. where F(x, V) stands for the free energy. Since the N-th Bell polynomial depends only on the first N variables, cf. Eqs (4) and (5), it is true that for N ≪V:
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