High-temperature expansion for Ising models on quasiperiodic tilings

Repetozwicz, Przemyslaw; Grimm, Uwe and Schreiber, Michael (1999). High-temperature expansion for Ising models on quasiperiodic tilings. Journal of Physics A: Mathematical and General, 32(24) pp. 4397–4418.

DOI: https://doi.org/10.1088/0305-4470/32/24/306

Abstract

We consider high-temperature expansions for the free energy of zero-field Ising models on planar quasiperiodic graphs. For the Penrose and the octagonal Ammann-Beenker tiling, we compute the expansion coefficients up to 18th order. As a by-product, we obtain exact vertex-averaged numbers of self-avoiding polygons on these quasiperiodic graphs. In addition, we analyse periodic approximants by computing the partition function via the Kac-Ward determinant. It turns out that the series expansions alone do not yield reliable estimates of the critical exponents. This is due to the limitation on the order of the series caused by the number of graphs that have to be taken into account, and, more seriously, to rather strong fluctuations in the behaviour of the coefficients. Nevertheless, our results are compatible with the commonly accepted conjecture that the models under consideration belong to the same universality class as those on periodic two-dimensional lattices.