Critical temperatures of the Ising model on Sierpiñski fractal lattices

Abstract

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 root vc, comprised between 0 and 1, of the partition function, which corresponds to the critical temperature Tc through the formula vc = tanh(1/Tc), reliably fits a power law of exponent k where k is the segmentation step of the fractal structure. This simple expression allows to extrapolate the critical temperature for k → ∞. 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 −∞.