\documentstyle[med_headings]{article} \oddsidemargin=0in\textwidth=6.4in\topmargin=-0.80in\textheight=10in \renewcommand{\textfraction}{0.01} \pagestyle{empty}\begin{document}\title{}\author{David J.C. MacKay }\date{} \noindent \input{psfig.tex} \input{/home/mackay/tex/newcommands1} \begin{center} {\large \bf Computational Methods in Physics \\ \medskip } \end{center} % \newcommand{\bb}{{\bf b}} \section*{Ising Models---Monte Carlo simulation} An Ising model' is an array of spins (\eg, atoms that can take states $\pm 1$) that are magnetically coupled to each other. In a ferromagnetic model it is energetically favourable for neighbours to be in the same state; in antiferromagnets, it is favourable for them to be in opposite states. Let the state $\bx$ of an Ising model with $N$ spins be a vector in which each component $x_n$ takes values $-1$ or $+1$. If two spins $m$ and $n$ are neighbours we write $(m,n) \in {\cal N}$. The energy of a state $\bx$ is \beq E(\bx;J,H) = - \left[ % \frac{1}{2} \sum_{(m,n) \in {\cal N}} J x_m x_n + \sum_{n} H x_n \right] , \eeq where $J$ is the coupling between spins, % $m$ and $n$, and $H$ is the applied field. If $J > 0$ then the model is ferromagnetic, and if $J < 0$ it is antiferromagnetic. (Note that each pair $(m,n)$ is counted just once in the first sum). % In Physics we may be interested in the properties of Ising models with % a large number $N$ of spins having regular geometric neighbourhood % relationships. In this problem we will study two-dimensional planar Ising models, using a simple Gibbs sampling' or heat bath Monte Carlo' method. Starting from some initial state, a spin $n$ is selected at random, and the probability that it should be $+1$ given the state of the other spins and the temperature is computed, \beq P(+1|h_n)= 1/(1+\exp(- 2 \beta b_n)), \eeq where $\beta = 1/k_{\rm B}T$ and $b_n$ is the local field \beq b_n = \sum_{m:(m,n) \in {\cal N}} J x_m + H. \eeq Spin $n$ is set to $+1$ with that probability, and then a new spin is selected at random. After sufficiently many iterations, this procedure converges to the equilibrium distribution, $P(\bx)=\frac{1}{Z}\exp(-\beta E(\bx;J,H))$. \medskip % \subsection*{Part a} (A) Write a program that simulates an Ising model with the square geometry (a) shown below, and with periodic boundary conditions. A line between two spins indicates that they are neighbours. % % To make a bite-sized example, we will set $b$ to 0 throughout, Set the external field $H=0$ and consider the two cases $J = \pm 1$. % which are a ferromagnet and antiferromagnet respectively. Start from a variety of initial states to check whether the initial state affects the results. \begin{center} \setlength{\unitlength}{2pt} (a) \begin{picture}(70,40)(0,-5) \newsavebox{\verticalfour} \savebox{\verticalfour}(0,0)[bl]{ \multiput(0,0)(0,10){4}{\circle{2}} % spins \multiput(0,5)(0,10){4}{\line(0,-1){3}} % lines down \multiput(0,-5)(0,10){4}{\line(0,1){3}} % lines up \multiput(2,0)(0,10){4}{\line(1,0){6}} % lines right } \multiput(0,0)(10,0){6}{\usebox{\verticalfour}} \end{picture} (b) \begin{picture}(70,45)(0,-5) \newsavebox{\verticalfourdiag} \savebox{\verticalfourdiag}(0,0)[bl]{ \multiput(0,0)(0,10){4}{\circle{2}} % spins \multiput(0,5)(0,10){4}{\line(0,-1){3}} % lines down \multiput(0,-5)(0,10){4}{\line(0,1){3}} % lines up \multiput(2,1)(0,10){4}{\line(2,1){6}} % lines rightup \multiput(2,-1)(0,10){4}{\line(2,-1){6}} % lines rightdown } \multiput(0,0)(20,0){3}{\usebox{\verticalfourdiag}} \multiput(10,5)(20,0){3}{\usebox{\verticalfourdiag}} \end{picture} \end{center} Track the average energy and the standard deviation of the energy of the system as a function of temperature. Repeat the simulation for a variety of values of $N$ and compare (start with $N=9,16,36$). Also track the mean square magnetization. Do the results appear to converge to some infinite $N$ limit? Examine both the cases $J = \pm 1$. Can you extract heat capacities of the Ising models from your computations? [Hints: $C = k_{\rm B} \beta^2 {\rm var}(E)$ might be a useful trick. One awkward decision that has to be made is how soon to start taking measurements of energy and magnetization after a new temperature has been established; it is virtually impossible to detect equilibrium'. A simple method is to pick a total number of iterations $I$ (which should obviously be significantly greater than the number of spins $N$), and to assume equilibrium had been reached after $I/3$ iterations; one can assess how well equilibrium has been reached by running through a sequence of temperatures first in one direction then in the other, looking to see if properties such as the mean energy show hysteresis. If they do, then $I$ needs to be bigger.] % \subsection*{Part b} (B) Repeat the above computations for a triangular Ising model. Do you expect the triangular Ising model with $J = \pm 1$ to show different physical properties from the square Ising model? Do you see evidence for differences? \subsection*{Optional extras} Switch on the applied field $H$ and find the energy and mean magnetization as a function of $\beta$. As $\b$ and $H$ vary do you find evidence for phase transitions? \medskip \hfill DJCM \today \end{document}