Exact low-temperature series expansion for the partition function of the zero-field Ising model on the infinite square lattice

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.

By substituting and p = p(θ 1 , θ 2 ) = cos θ 1 + cos θ 2 , into Eq. (6) in the main paper, the bulk free energy per site in the square lattice Ising model can be written as: Next, the integrand function in Eq. (5) can be decom-posed into a Taylor series as: L n (−2p, 2·2!, 2p·3!, 4!) x n n! (6) where the so-called logarithmic polynomials have been a) Electronic mail: siudem@if.pw.edu.pl b) Electronic mail: agatka@if.pw.edu.pl c) Electronic mail: fronczak@if.pw.edu.pl used, which are defined as (see Eq. (5a), p. 140 in 1 ): where B n,k ({g i }) represent partial Bell polynomials, see Eq. (4) in the main paper. Now, substituting Eq. (7) to (5) one gets the general expression for the low temperature series expansion of the bulk free energy per site (cf. Eq. (6) in the main paper): where the expansion coefficients are given by: Eq. (11) can be further simplified by using the explicit formula for partial Bell polynomials, Eq. (4) in the main paper, according to which the polynomial B n,k in Eq. (11) can be written as: where the summation takes place over all integers and Now, after using Eqs. (13) and (14) in Eq. (11) one gets the following expression for a n : where the explicit summation over k was omitted due to the fact that it is already included in the summation over the variables d 1 , d 3 , d 3 , d 4 which now must only satisfy Eq. (15). The last step towards the final expression for a n is to show that the double integral in Eq. (16) simplifies to: where p is given by Eq. (2). (For reasons of clarity, the detailed calculations leading to Eq. (17) are not discussed here, but will be discussed in Sect. II of this document.) Finally, by inserting Eq. (17) into (16) one gets Eqs. (7) and (8) which are in use in the primary article. For odd values of n: and for even values of n: a n = n! 2 d1,d2,d3,d4 where the summation takes place over all quadruple numbers d 1 , d 2 , d 3 , d 4 ≥ 0, which satisfy conditions d 1 + 2d 2 + 3d 3 + 4d 4 = n and d 1 + d 3 is even.

II. DETAILED CALCULATIONS LEADING TO EQ. (17)
The double integral in Eq. (16) can be transformed as follows: where the integral φ l satisfies the below expression which leads to the following recursive equation: and φ l = 2π (l − 1)!! l!! for even l = 0, 2, 4, . . . , Eq. (23) can be further simplified to: where the assumption that l is even has been used. Eq. (35) exactly corresponds to Eq. (17).

III. ASYMPTOTIC BEHAVIOUR OF THE COEFFICIENTS a2n/(2n)!
As we argued in the main paper the coefficients in the low temperature series expansion of −βf (x) (see Eqs. (10), (18), and (19)) have the asymptotic behaviour which is given by Eq. (13) in the main paper: where C is a positive constant, and (cf. Eq. (20), in the main paper) The log-linear plot of the coefficients a 2n /(2n)! vs 2n, which is shown in Fig. 1 illustrates this behaviour. The logarithm of α: corresponds to the slope of the line, a = 0.3807,which is fitted to the results.

IV. EXACT ENERGY DISTRIBUTION P (N, x) FOR THE SQUARE LATTICE ISING MODEL
Eq. (21) in the main paper, which is exact in the limit of infinite lattice size, i.e. for V → ∞, provides an excellent testbed for comparison of the exact infinite-volume results and the results of finite-size Monte Carlo methods (see e.g. [2][3][4] ).
In Fig. 2, the exact energy distribution P (V, x), Eq. (21) in the main paper, is shown for three lattices of size: V = 256, 512, 1024, and two different temperatures: x = e −2βJ = 0.36 and 0.41 (let us note that x c 0.414).

V. DETAILED CALCULATIONS LEADING TO EQ. (21) IN THE MAIN PAPER
By using Eq. (20) in the main paper and substituting r for x xc , the numerator in Eq. (19) in the main paper can be written as follows: where the expression (18) in the main paper has been used. Then, since the partial Bell polynomials with the coefficients: 1!, 2!, 3! . . . are equal to Lah numbers, Eq. (41) can be further simplified: is the so-called confluent hypergeometric function of the first kind [26], which is defined as: where (a) k and (b) k are Pochhammer symbols.