Ising model on the aperiodic Smith hat

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