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 64×64 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 magnetization, energy, specific heat, susceptibility and entropy with each of the bcs and observed that the phase transition occurs around Tc=2.269 as obtained by Onsager. We compare the behavior of magnetization versus temperature for different types of bcs by calculating the number of points close to the line of zero magnetization after T>Tc at various lattice sizes. We observe that the periodic, adiabatic and reflexive bcs give closer approximation to the value of Tc than fixed +1 and fixed −1 bcs with all three initial conditions for lattice sizes less than 70×70. However, for lattice size between 70×70 and 100×100, fixed +1 bc and fixed −1 bc give closer approximation to the Tc with initial conditions all spin down configuration and all spin up configuration, respectively.