High-temperature expansion for Ising models on quasiperiodic tilings

Abstract

Abstract. We consider high-temperature expansions for the free energy of zerofield 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 analyze periodic approximants by computing the partition function via the Kac-Ward determinant. For the critical properties, we find complete agreement with the commonly accepted conjecture that the models under consideration belong to the same universality class as those on periodic two-dimensional lattices.

Finite-lattice expansion for the Ising model on the Penrose tiling

“…A discussion of the methods of generation and geometrical properties of this tiling can be found in [13], here we only mention a particularly useful feature, namely that embedding numbers of finite patches from this tiling can be calculated exactly and take the form n + mτ where τ = ( √ 2 + 1)/2 is the golden number and n, m are rational numbers. The calculation of the series expansion consists therefore of the following steps:…”

“…Moreover, as opposed to the square lattice graphs in the PT can have different "boundary line fillings", i.e. there are different graphs having the same boundary line [13]. Knowing that the PT contains eight different vertex types, i.e.…”

“…where a = 16, 15, 14, 13, 12, 11, 10, 9}, e = {(9, 10), (10,11), (11,12), (12,13), (12,9), (9, 18)} and s = {12, 11, 10} denote the current and the final boundary lines, the edges and the sites which were not taken into account yet respectively. Can the transfer-matrix formalism (tmf) also be applied to the case of a multicomponent graph?…”

“…Quasiperiodic Ising models were also examined by graphical expansion methods [12,13] and by calculating exact partition functions for PA, obtained from the Kac-Ward determinant [14]. In the first case estimates for T c and critical exponents have not been considerably improved but this approach demonstrated a new feature, a very slow convergence of the partition function (Z) series to its predicted asymptotic form.…”

Wavevector-Dependent Susceptibility in Z-Invariant Pentagrid Ising Model

“…Finite approximants to the aperiodic Penrose tiling have been constructed through periodic pentagrids or projection methods. (78,79,80,73) The mathematician Robinson has brought to our attention an exercise in the book by Grünbaum and Shephard (81) where a tiling with Penrose rhombuses can be cut into patches, and then converted into an aperiodic set of 24 Wang's tiles. We have found that each patch in the Penrose tiling described in the exercise is in fact the image of a parallelogram under the mapping (2.5).…”