Project 4

First and second order phase transition in the Ising model of ferromagnetism

Write a program for the 2D Ising model on a $64\times 64$ square lattice with periodic boundary conditions which is placed in an applied external magnetic field $H$. The energy of a particular state of the system is given by:

$E=-J\sum _{\langle i,j\rangle }S_{i}S_{j}-H\sum _{i}S_{i}$

where $\langle i,j\rangle$ means that the sum runs over all pairs of nearest neighbor spins (attached to the lattice sites), and $J$ is the exchange coupling. Positive $J>0$ interaction between the spins leads to ferromagnetism, will negative $J>0$ gives rise to antiferromagnetism in the Ising model. Choose units so that $|J|=1$ and the Boltzmann constant is $k_{B}=1$. The free parameters of the model are then just the temperature $T$ and the applied field $H$.

Parth I