Project 4

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

Introduction

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:

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

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 $H$.

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

Consider the ferromagnetic Ising model with $J>0$. Start from a case in which temperature is below the critical one $T_{c}=2.269$, such as $T=1.0$, and the field is large and pointing down, e.g., $H=-5$. The selected initial state of the Ising model should be the one 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 $\delta H=0.5$. After each change in the magnetic field $H$, allow sufficient Monte Carlo trials for the system to reach the equilibrium and thermalize.

For each value of $H$, calculate the mean energy per spin

$\langle E\rangle ={\frac {1}{N_{MC}}}\sum _{\alpha =1}^{N_{MC}}E_{\alpha }$

the mean square energy per spin

$\langle E^{2}\rangle ={\frac {1}{N_{MC}}}\sum _{\alpha =1}^{N_{MC}}E_{\alpha }^{2}$

the mean magnetization per spin

$\langle M\rangle ={\frac {1}{N_{MC}}}\sum _{\alpha =1}^{N_{MC}}M_{\alpha }$ and the specific heat

$C_{V}={\frac {1}{k_{B}T}}(\langle E^{2}\rangle -\langle E^{2}\rangle )$

where we use the fluctuation-dissipation theorem to connect the specific heat to the fluctuations in energy . All sums in these expressions run are over $N_{MC}$ microstates generated by 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,...).