Ising model on the aperiodic Smith hat

Smith et al discovered an aperiodic monotile of 13-sided shape in 2023. It is called the `Smith hat' and consists of 8 kites. We deal with the statistical physics of the lattice of the kites, which we call the `Smith-kite lattice'. We studied the Ising model on the aperiodic Smith-kite lattice and the dual Smith-kite lattice using Monte Carlo simulations. We combined the Swendsen-Wang multi-cluster algorithm and the replica exchange method. We simulated systems up to the total spin number \(939201\). Using the finite-size scaling analysis, we estimated the critical temperature on the Smith-kite lattice as \(T_c/J=2.405 \pm 0.0005\) and that of the dual Smith-kite lattice as \(T^{*}_{c}/J=2.143 \pm 0.0005\). Moreover, we confirmed the duality relation between the critical temperatures on the dual pair of aperiodic lattices, \(\sinh(2J/T_c) \sinh(2J/T^{*}_{c}) = 1.000 \pm 0.001\). We also checked the duality relation for the nearest-neighbor correlation at the critical temperature, essentially the energy, \(\epsilon(T_c)/\coth(2J/T_c) + \epsilon(T^{*}_c)/\coth(2J/T^{*}_c) = 1.000 \pm 0.001\).