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

Using Cellular Automata, we simulate spin systems corresponding to $2d$ Ising model with various kinds of boundary conditions (bcs). The appearance of spontaneous magnetization in the absence of magnetic field is studied with a $64\times64$ square lattice with five different bcs, i.e., periodic, adiabatic, reflexive, fixed ($+1$ or $-1$) bcs with three initial conditions (all spins up, all spins down and random orientation of spins). In the context of $2d$ Ising model, we have calculated the magnetisation, energy, specific heat, susceptibility and entropy with each of the bcs and observed that the phase transition occurs around $T_c$ = 2.269 as obtained by Onsager. We compare the behaviour of magnetisation vs temperature for different types of bcs by calculating the number of points close to the line of zero magnetisation after $T>T_c$ at various lattice sizes. We observe that the periodic, adiabatic and reflexive bcs give closer approximation to the value of $T_c$ than fixed +1 and fixed -1 bcs with all three initial conditions for lattice size less than $70\times70$. However, for lattice size between $70\times70$ and $100\times100$, fixed +1 bc and fixed -1 bc give closer approximation to the $T_c$ with initial conditions all spin down configuration and all spin up configuration respectively.