Ising model on the aperiodic Smith hat

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 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 $T_c^{\ast }/J=2.143\pm 0.0005$. Moreover, we confirmed the duality relation between the critical temperatures on the dual pair of aperiodic lattices, $\sinh (2J/T_c)\sinh (2J/T_c^{\ast })=1.000\pm 0.001$. We also checked the duality relation for the nearest-neighbor correlation at the critical temperature, essentially the energy, $\epsilon (T_c)/\coth (2J/T_c)+\epsilon (T_c^{\ast })/\coth (2J/T_c^{\ast })=1.000\pm 0.001$.

arXiv commentaccepted publication in J. Phys. A

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Cited by in corpus (5)

• Monte Carlo study on critical exponents of the classical Heisenberg model in ferromagnetic icosahedral quasicrystal
• Beating the aliasing limit with aperiodic monotile arrays
• Family of Aperiodic Tilings with Tunable Quantum Geometric Tensor
• Critical behavior of the Ising model on square-triangle tilings
• Chiral Diffraction from Aperiodic Monotile Lattice