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