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 per spin

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

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.

(a) Plot $M$ vs. $H$. You should see a discontinuous change in as H is increased, and a second discontinuous change in $M$ as $H$ is decreased back to its initial value. These discontinuous changes indicate a first order phase transition. Also note that the values of $H$ at which discontinuities occur are not identical. Such behavior is called hysteresis. Perform at least one thousand Monte Carlo time steps, where one time step is one complete pass through the lattice, at each value of the magnetic field.

(b) Repeat the calculations for a temperature $T=4.0$ above the critical point $T_{c}$. Is there a first order phase transition? Is there hysteresis? Provide a physical explanation of any difference in behavior for the two temperatures.

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

Investigate the behavior of the specific heat $C_{V}$ as the temperature is increased through the critical value $T_{c}$ at both $H=0$, in which case we expect a second order or continuous paramagnet-ferromanget phase transition, and $H\neq 0$ , in which case we expect first order discontinous phase transtion. Repeat this calculation for increasing lattice size to see how the peak of $C_{V}$ around $T_{c}$ increases when $H=0$.