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A comparative study of $2d$ Ising model at different boundary conditions using non-deterministic Hexagonal Cellular Automata

The spin system of the $2d$ Ising model having a hexagonal-lattice is simulated using non-deterministic Cellular Automata. The method to implement this program is outlined and our results show a good approximation to the exact analytic solution. The phase transition in $2d$ Ising model is studied with a $40\times40$ hexagonal-lattice with five different boundary conditions (bcs) i.e., adiabatic, periodic, reflexive, fixed $+1$ and fixed $-1$ with random orientation of spins as initial conditions in the absence of an external applied magnetic field. The critical temperature below which the spontaneous magnetization appears as well as other physical quantities such as the magnetization, energy, specific heat, susceptibility and entropy with each of the bcs are calculated. The phase transition occurs around $T^H_c$ = 1.5 which approximates well with the result obtained from exact analytic solution by Wannier and Houtappel. We compare the behavior of magnetisation per cell for five different types of bcs by calculating the number of points close to the line of zero magnetization for $T>T^H_c$. We find that the periodic, adiabatic and reflexive bcs give closer approximation to the value of $T^H_c$ than fixed $+1$ and fixed $-1$ bcs with all three initial conditions for lattice size less than $50\times50$. However, for lattice size between $50\times50$ and $200\times200$, fixed $+1$ bc and fixed $-1$ bc give closer approximation to the $T^H_c$ with initial conditions in which all spins are in down configuration and all spins are in up configuration respectively.