Ising model on the aperiodic Smith hat (2402.11331v2)

Abstract: 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 theSmith-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, $\epsilon(T_c)/\coth(2J/T_c) + \epsilon(T^{}_c)/\coth(2J/T^{}_c) = 1.000 \pm 0.001$.