Critical temperatures of the Ising model on Sierpiñski fractal lattices

We report our latest results of the spectra and critical temperatures of the partition function of the Ising model on deterministic Sierpiñski carpets in a wide range of fractal dimensions. Several examples of spectra are given. When the fractal dimension increases (and correlatively the lacunarity decreases), the spectra aggregates more and more tightly along the spectrum of the regular square lattice. The single real rootvc, comprised between 0 and 1, of the partition function, which corresponds to the critical temperatureTcthrough the formulavc= tanh(1/Tc), reliably fits a power law of exponentkwherekis the segmentation step of the fractal structure. This simple expression allows to extrapolate the critical temperature fork→ ∞. The plot of the logarithm of this extrapolated critical temperature versus the fractal dimension appears to be reliably linear in a wide range of fractal dimensions, except for highly lacunary structures of fractal dimensions close from 1 (the dimension of a quasilinear lattice) where the critical temperature goes to 0 and its logarithm to −∞.


Introduction
Phase transitions on fractals have been investigated since the early applications of fractals in physics, initiated by B. Mandelbrot in three pionnering papers using mainly real space renormalisation group (RSRG) techniques [1][2][3]. Subsequently, the Sierpiñski carpet [4] and its extensions have been used as models to test (i) the congruence between critical exponents calculated by the analytical continuation of ε-expansions of the renormalisation approach in non integer dimensions [5] and their values obtained by direct calculations by different methods (ii) the validity of scaling relations involving the critical exponents when replacing the dimension of space by the fractal dimension.
Actually, Sierpiñsky carpets are infinitely ramified, a condition which is mandatory to get a phase transition at non zero temperature and, by varying parameters, their fractal dimensions may approach any desired value as close as necessary. Furthermore, the Ising model is one of the simplest model of phase transitions This lead many authors to combine Sierpiñsky carpets and the Ising model to investigate phase transitions on fractal lattices.
The most accurate values of critical exponents β and γ have been obtained by the various methods involving Monte Carlo simulations and show that values of critical exponents are different than those obtained by continuation of ε-expansions. So replacing translation invariance by scale invariance changes significantly the critical behavior. But scaling relations -at least those which do not involve α, the estimation of which is unaccurate in fractals [13] -as the Rushbrooke and Josephson's scaling law: dν = γ + 2β are satisfied within the accuracy of these numerical simulations [12,15] However, Pruessner and Loison [16] questioned the relevance of finite size scaling performed on the successive segmentation steps of a single pattern, arguing that the size of the structures is not large enough to exceed the correlation length in the critical region. Alternatively, they based their calculations by reintroducing translation invariance with the juxtaposition of several identical patterns in the two dimensions of space to increase the size of the elementary cell before applying the usual periodic boundary conditions. For a scale invariance n = 3 with a single centered removed site, the values of the critical temperatures obtained after this modification are close to those obtained previously, but significantly different according to their respective accuracy, 1.50 instead of 1.48. This significant discrepancy reveals the difficulty of calculating accurate values of the critical temperature which is crucial for using histogram and finite size scaling method after Monte Carlo simulations.
The aim of this paper is to push forward the method previously introduced [24] to calculate the critical temper-atures the Ising model on Sierpiñski carpets by investigating a larger range of structures and higher segmentation steps to increase the accuracy of results.

The Sierpiñsky carpets
Let us remind the fractal structures corresponding to Sierpiñski carpets. Starting from an initial unit size square, divided into n 2 subsquares of equal size we remove p 2 subsqures in the center and iterate the processus k times. This structure in noted S C(n, p, k). The fractal structure S C(n, p) is the limit of this process when k → ∞. Fig. 1 shows the third segmentation step of S C(3, 1), i.e. S C (3,1,3). Ising spins are located on center of the remaining sites.

Partition functions
In a previous paper [24] we introduced a method to calculate the partition function of the Ising model in planar lacunary and especially fractal lattices, at any segmentation step. We could calculate the full spectrum (i.e. the set of the roots) of this partition function on Sierpiñski carpets and particularly the critical temperature T c which corresponds to the single real value v c of the spectrum comprised between 0 and 1 through the formula The method is an extension of the path counting method used to calculate the partition function of the two dimensional Ising model on square lattice [25]. The aim of this paper is not to expose again the method, but to give more accurate results based on further calculations in the three fields which can be investigated in this way on planar fractal lattices: (i) the partition functions and its spectrum (ii) the critical temperatures on finite values of the segmentation step k and an extended set of fractal dimensions (or equivalently of values of parameters n and p defined above), (iii) the extrapolation from the sequences of critical temperatures at increasing values of the segmentation step k of the critical temperature at the actual fractal limit k → ∞.
Generally, the partition function is expressed by a polynomial P(N, v) which counts the closed pathes on the lattice: where N is the total number of patterns, N z is the number of neighbours, v = tanh(1/T ). For two dimensional lattices, P is obtained as the characteristic polynomial of an appropriate matrix [24]. For example, for S C (3, 1, 1), the first segmentation step of the most simple Sierpiñski carpet, this polynomial is The critical temperature is given by the real positive root of P(v) = P(1, v): The polynomial P(v) can be factorized through rotation invariance in Since all coefficients of Q + are positive, it cannot have positive real roots, so v c is a root of Q − .

Spectra
Some exemples of spectra are shown on Figs. 2-6 for different values of n and p: p = n − 2 (Fig. 2), p = n − 4 (Fig. 3), p = n − 6 (Fig. 5), p = n − 8 (Fig. 6). The two red circles are the location in which the spectrum of the partition function of the Ising model on the square lattice is dense [26].
We observe that more the structure is compact, with a lower lacunarity (increasing values of n − p or increasing concentration n 2 /p 2 ), then the spectrum aggregates more tightly along these two circles. Fig. 4 shows a magnification of the region of the spectrum of S C(3, 1, k) located near the real axis (delimited by the blue frame on the top of Fig. 2), for k = 5 and k = 6. It appears that despite being spectra of scale invariant fractal structures and being themselves likely fractals, these spectra are not scale invariant. S C (3,1,5) S C (4,2,4) S C (6,4,3) S C (10,8,2)    S C (10,4,2) S C (15,9,2) Dotted lines in Fig. 7 correspond to least square linear fits. The values of a(n, p) and b(n, p) are chosen to maximize the determination coefficients R 2 of the least square linear fits for each value of n and p. All R 2 differ from 1 by less than 10 −4 .
All values of b(n, p) are greater than 1, then v c − v c sq converges towards a(n, p) when k → ∞, leading to the extrapolation of v c , and then of T c , when k → ∞. The values of a(n, p), b(n, p), T c (k = ∞) and the maximum value of the segmentation step k max on which extrapolations have been performed, are given in Table 1 for all investigated Sierpiñski carpets up to n = 100. The structures corresponding to the three highest values of n for p = n − 2: n = 25, n = 50 and n = 100 are so lacunary that the accuracy of the extrapolations is not enough to distinguish the critical temperatures from T c = 0. The plots of b(n, p) are given on Fig. 8 (on the right).

Critical temperatures
The values of v c have been calculated for a wide range of structures, up to n = 100 and p = n−2 to p = n−8. We will Table 1. Values of a, b of eq. (1) and of T c (k = ∞) (enlighted in boldfaces), k max is the highest segmentation step used for calculations.