Project 4

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

Write a program which simulates via Metropolis Monte Carlo algorithm the 2D Ising model on a $64\times 64$ square lattice with periodic boundary conditions. The model is assumed to describe a system placed in an applied external magnetic field $H$. The energy of a particular state of this system is then given by:

$E=-J\sum _{⟨i,j⟩}{S}_i{S}_j-H\sum _i{S}_i$

where $⟨i,j⟩$ 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 $H$.

Part I for both PHYS460 and PHYS660 students: First order phase transition in the ferromagnetic Ising model

Consider ferromagnetic Ising model with $J>0$. Start from a case in which the temperature is below the critical point, e.g., $T=1.0$ and the field is large and pointing down, e.g., $H=-5$. Run your code using the initial state of the Ising model in which all spins $S_i=-1$ are parallel to H, so that magnetization per spin is $M=-1$.

Sweep the magnetic field up to a large positive value $H=5$ and then back down to your starting value $H=-5$. Do this in steps of (or smaller if you have the patience!). After each change in , allow enough Monte Carlo trials for the system to reach equilibrium. For each value of , calculate the mean energy per spin mean square energy per spin , mean magnetization per spin and the specific heat (where we use the fluctuation-dissipation theorem to connect the specific heat to the fluctuations in energy . All sums here are over microstates generated by the Monte Carlo simulation.

Part II for both PHYS460 and PHYS660 students: Second order phase transition in the ferromagnetic Ising model

Investigate the behavior of the specific heat as the temperature is increased through the critical value both at (which is a second order or continuous paramagnet-ferromanget phase transition) and first order discontinous phase transtion in non-zero magnetic field . Repeat this calculation for increasing lattice size (for example, L=10,30,50,...).