A Critical Ising Model on the Labyrinth

Abstract

A zero-field Ising model with ferromagnetic coupling constants on the so-called Labyrinth tiling is investigated. Alternatively, this can be regarded as an Ising model on a square lattice with a quasi-periodic distribution of up to eight different coupling constants. The duality transformation on this tiling is considered and the self-dual couplings are determined. Furthermore, we analyze the subclass of exactly solvable models in detail parametrizing the coupling constants in terms of four rapidity parameters. For those, the self-dual couplings correspond to the critical points which, as expected, belong to the Onsager universality class.

Wavevector-Dependent Susceptibility in Z-Invariant Pentagrid Ising Model

“…The Z-invariant inhomogeneous models are completely integrable (14) even on irregular lattices and their critical exponents are known to be the same as those of homogeneous systems on regular lattices. Thus, the critical behaviors of these quasiperiodic Z-invariant models (14,15,16,17,19,20,21,18) have to be the same, 3 independent of the lattice structure. 4 In such models, the order parameter is the same (14,13) for all sites and it vanishes towards the critical point.…”

“…(55) In this paper, we turn our attention to systems with a quasiperiodic lattice structure. (19,20,21) More specifically we study a Z-invariant Ising model whose spins are on vertices of a Penrose fat-and-skinny rhombus tiling with the pentagrid as its rapidity lines. (5,13,15,16,17) 1.…”

High-temperature expansion for Ising models on quasiperiodic tilings

“…Besides graphical expansions and Monte-Carlo simulations, further methods have been employed to gain information about the critical behaviour of quasiperiodic Ising models. First of all, exactly solvable cases can be constructed as, for instance, the Ising model on the so-called labyrinth tiling [33], see also [16,34] for further examples. These models correspond to particular choices of coupling constants, restricted by the requirement of integrability, and thus might not be representative for the general situation.…”

Coordination sequences for root lattices and related graphs

“…It has been demonstrated recently [2,3] that the location of the critical points for Ising and percolation models on several lattices can be well approximated by empirical functions involving the dimension and the coordination number of the lattice. On the other hand, these quantities alone cannot completely determine the critical point as can be seen from data obtained for different graphs with identical dimension and (mean) coordination number [4,5,6]. This poses the question how to include more detail of the lattice in order to improve the approximation.…”