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\noindent\small \textbf{Delft University of Technology}\\
International Course in Computational Physics\\
Supervisor: Dr. J.M. Thijsen \& Dr. A. Akhmerov
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\large{\textbf{Monte Carlo simulations of the 2-Dimensional Ising model}}\\
\normalsize{Emma C. Gerritse \& Sophie L.N. Hermans}
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\begin{abstract}
The Ising model in two dimensions has been simulated using Monte Carlo Methods. Both the Metropolis algorithm (a Markov Chain method) and the Wolff algorithm (a cluster method) were used. Physical properties of the Ising model have been calculated at different system sizes, and their values have been extracted with the help of Finite Scaling Methods.
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\section{The Ising model} The Ising model is a theoretical model of a ferromagnetic system with nearest neighbour interactions. It consists of $N$ particles with spin up or down. The nature of the interaction between these spins is described by the coupling constant $J$. If this constant is positive, parallel neighbour spins have a lower energy than anti-parallel neighbours, and the system is ferromagnetic. If $J$ is negative, this is the other way round, and the system is antiferromagnetic. At $J = 0$ a system of non-interacting spins is described. In addition to the force between the particles, an external magnetic field $h$ can be applied.

The energy of a certain state of the system is given by equation \ref{eq:energy}, where $\mu$ is the magnetic moment, and $\sigma_i$ is the sign (1 or -1) of spin $i$. The first sum in this equation is over all four nearest neighbours of $i$.
\begin{equation}\label{eq:energy}
H(\sigma) = - \sum_{nn} J \sigma_i \sigma_j - \mu \sum_j h \sigma_j
\end{equation}

\section{Metropolis algorithm}
In the simplest Monte Carlo simulation of the Ising model, the \emph{Metropolis Monte Carlo}, the subsequent states of the systems are links in a \emph{Markov Chain}. This means that the system is updated by flipping individual spins, so that each state of the system only depends on the previous state. One spin is picked at random, and flipped. If this lowers the energy of the system, given by equation \ref{eq:energy}, the spin stays in its new position. If the energy of the system increases, the new position is kept with a probability $P$, given in equation \ref{eq:flip_prob}.
\begin{equation}\label{eq:flip_prob}
P = e^{-\left(E_{new} - E_{old}\right)/k_B T} = e^{-\beta \Delta E}
\end{equation}
By repeating this process indefinitely, the state of the system is updated. If the temperature is lower than the critical temperature $T_C$, all of the spins will move to either the up or the down position. Since at most one spin is flipped in every step however, it will take more than $N$ steps to update the entire system, and often several times longer to reach an equilibrium state.

\section{Wolff algorithm}
A different method to simulate the Ising model makes use of clusters. This method is called the Wolff algorithm\cite{Wolff89}. In this method a recursive loop is used, where new spins are added to the cluster and then the neighbours of those new spins that are not yet in the cluster are added to the cluster. The process takes place in x steps:
\begin{enumerate}
\item A random spin is picked, flipped, and added to the cluster, which now contains one spin.
\item If the neighbours of this spin are not yet in the cluster, they are added with a probability of $P = 1 - e^{-2J/k_B T}$
\item If a spin is added to the cluster, we try to add the neighbours of this spin too (going back to step 2)
\item When no new spins are added, the whole cluster is flipped.
\end{enumerate}

\section{System parameters}
From a simple model like the Ising model, certain physical system parameters can be extracted. They change with temperature and magnetization.

\subsection{Critical temperature}
In the Ising model, a phase transition takes place once the temperature of the system reaches a certain critical temperature $T_C$. Above this temperature, the total magnetization stays zero on average. Below this temperature, the magnetization converges sharply towards either -1 or +1, where all of the spins are pointing in the down or up direction respectively. This critical temperature can be calculated analytically, where the temperature is a function of the coupling constant\cite{Onsager44}:
\begin{equation}\label{eq:crittemp}
T_C = \frac{J}{0,44 k_b}
\end{equation}
If we have a coupling constant of 1, equation \ref{eq:crittemp} tells us that the critical temperature should have a value of 2,3K. The critical temperature can also be extracted from Ising model simulations. We start the system at a temperature well below the critical temperature. We allow it to relax, en then we increase the temperature  by an amount $\Delta T$. We keep doing this until the magnetization suddenly flips from a value of $\pm 1$ to something close to zero. Our \emph{Metropolis Monte Carlo} simulations of the magnetization for a system of 256 particles tell us that the critical temperature is ... , as can be seen in figure \ref{fig:magnetiz} below.

\subsection{Physical properties}
\begin{equation}
\chi_m = \left(\frac{\partial m}{\partial H}\right)_T
\end{equation}
Magnetic susceptibility: from the exact solution, $\gamma = 7/4$. \cite{CompPhys}
\begin{equation}
\chi_m(T) \propto |T - T_c|^{-\gamma}
\end{equation}
Specific heat: from the exact solution, $\alpha = 0$. \cite{CompPhys}
\begin{equation}
c_h(T) \propto |T - T_c|^{-\alpha} \propto \text{ln}|T - T_c|
\end{equation}
Correlation length: from the exact solution, $\nu = 1$. \cite{CompPhys}
\begin{equation}
\xi(T) \propto |T - T_c|^{-\nu}
\end{equation}
Magnetization: from the exact solution, $\beta = 1/8$. \cite{CompPhys}
\begin{equation}
m(T) \propto \left(-T + T_c\right)^{\beta} ; T < T_c
\end{equation}
Magnetization with varying magnetic field at $T_c$: from the exact solution, $\delta = 15$. \cite{CompPhys}
\begin{equation}
m(H)|_{T_c} \propto H^{1/\delta}
\end{equation}

\section{Conclusions and discussion}

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