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.

Aperiodic Ising quantum chains

“…(3.15) and below], we shall restrict ourselves to eigenvalues that vanish in the limit of an infinite system size. 3 Note that this labeling of the fields differs from that used in Eqs. (2.1) and (3.1) in that the field variables are now labeled in the same way as the coupling constants (i.e., as the bonds) rather then by the sites of the chain.…”

Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models

“…For the self-dual choices (#3, #4), again, the Onsager exponent was found in all cases. The critical temperature, however, was not the chosen one for even values of k (2, 4), but agreed with it for odd ones (1,3,5) and probably also in the random version. (For the latter, the fluctuations in β est were too large to state this unambiguously.)…”

“…The first two choices (#1, #2) are on the exactly solvable submanifold, i.e. we chose three of the coupling strengths (J aa , J ab , J ba ) arbitrarily and the inverse critical temperature to be the one of the square lattice Ising model, β c = arsinh(1)/2 ≃ 0.44069, and determined the other five couplings by numerically solving the five equations [3] sinh(2β c J xy ) sinh(2β cJxy ) = 1, x, y ∈ A,…”

“…in each step it is applied to each letter of the word. This yields a (semi-) infinite limiting word w. For k = 1, we have the silver mean chain treated in [3]. Each letter is assigned an interval of a specific length.…”

“…Part of the Labyrinth tiling corresponding to k = 1 is shown in figure 4. On a 4-dimensional submanifold of the 8-dimensional coupling space, the Ising model is exactly solvable and shows critical behaviour of Onsager type regardless of the underlying word w [3]. To see what the Harris-Luck criterion [16,22] predicts, we have to look at the fluctuation exponent ω, which describes how the deviations of the mean coupling strength in a finite patch, compared to the infinite volume mean, scale with the size of the patch.…”