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

TL;DR

This paper evaluates the invaded cluster algorithm's effectiveness in accurately determining critical properties of 2D Ising models on various periodic and aperiodic graphs, including new quasiperiodic tilings, and explores its generalization to models with non-uniform couplings.

Contribution

The paper demonstrates the invaded cluster algorithm's high accuracy for critical temperature estimation on diverse graphs and introduces a generalized version for models with variable couplings.

Findings

Accurately reproduces known critical temperatures on multiple graphs

Determines critical temperatures for new quasiperiodic tilings

Finds deviations from universality in certain non-uniform models

Abstract

We demonstrate that the invaded cluster algorithm, recently introduced by Machta et al, is a fast and reliable tool for determining the critical temperature and the magnetic critical exponent of periodic and aperiodic ferromagnetic Ising models in two dimensions. The algorithm is shown to reproduce the known values of the critical temperature on various periodic and quasiperiodic graphs with an accuracy of more than three significant digits. On two quasiperiodic graphs which were not investigated in this respect before, the twelvefold symmetric square-triangle tiling and the tenfold symmetric T\"ubingen triangle tiling, we determine the critical temperature. Furthermore, a generalization of the algorithm to non-identical coupling strengths is presented and applied to a class of Ising models on the Labyrinth tiling. For generic cases in which the heuristic Harris-Luck criterion predicts deviations from the Onsager universality class, we find a magnetic critical exponent different from the Onsager value. But also notable exceptions to the criterion are found which consist not only of the exactly solvable cases, in agreement with a recent exact result, but also of the self-dual ones and maybe more.