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 eight 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 <?CDATA $939\,201$?> 939 201 . Using the finite-size scaling (FSS) analysis, we estimated the critical temperature on the Smith-kite lattice as <?CDATA Tc/J = 2.405 ± 0.0005?> T c / J = 2.405 ± 0.0005 and that of the dual Smith-kite lattice as <?CDATA T^\,*c/J = 2.143 ± 0.0005?> T c ∗ / J = 2.143 ± 0.0005 . Moreover, we confirmed the duality relation between the critical temperatures on the dual pair of aperiodic lattices, <?CDATA sinh(2J/Tc) sinh(2J/T^\,*c) = 1.000 ± 0.001?> sinh ( 2 J / T c ) sinh ( 2 J / T c ∗ ) = 1.000 ± 0.001 . We also checked the duality relation for the nearest-neighbor correlation at the critical temperature, essentially the energy, <?CDATA ε(Tc)/coth(2J/Tc) + ε(T^\,*c)/coth(2J/T^\,*c) = 1.000 ± 0.001?> ϵ ( T c ) / coth ( 2 J / T c ) + ϵ ( T c ∗ ) / coth ( 2 J / T c ∗ ) = 1.000 ± 0.001 .