324 lines
15 KiB
TeX
324 lines
15 KiB
TeX
\documentclass[../ising_model.tex]{subfiles}
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\begin{document}
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\section{Methods}\label{sec:methods}
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\subsection{The Ising model}\label{subsec:ising_model}
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The Ising model consists of a lattice of spins, which can be thought of as atoms
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in a grid. In two dimensions, the length of a lattice is given by $L$, and the
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number of spins within a lattice is given by $N = L^{2}$. When we consider the
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entire lattice, the system spin configuration is represented as a matrix $L \times L$
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\begin{align*}
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\mathbf{s} &= \begin{pmatrix}
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s_{1,1} & s_{1,2} & \dots & s_{1,L} \\
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s_{2,1} & s_{2,2} & \dots & s_{2,L} \\
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\vdots & \vdots & \ddots & \vdots \\
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s_{L,1} & s_{L,2} & \dots & s_{L,L}
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\end{pmatrix} \ .
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\end{align*} % $\mathbf{s} = [s_{1}, s_{2}, \dots, s_{N}]$.
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The total number of possible spin configurations, also called microstates, is
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$|\mathbf{s}| = 2^{N}$.
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A given spin $i$ can take one of two possible discrete values $s_{i} \in \{-1, +1\}$.
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The spins interact with its nearest neighbors, and in a two-dimensional lattice
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each spin has up to four nearest neighbors. In our experiments we will use periodic
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boundary conditions, resulting in all spins having exactly four nearest neighbors.
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To find the analytical expressions necessary for validating our model implementation,
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we assume a $2 \times 2$ lattice.
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The hamiltonian of the Ising model is given by
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\begin{equation}
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E(\mathbf{s}) = -J \sum_{\langle k l \rangle}^{N} s_{k} s_{l} - B \sum_{k}^{N} s_{k}\ ,
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\label{eq:energy_hamiltonian}
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\end{equation}
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where $\langle k l \rangle$ denotes a spin pair. $J$ is the coupling constant, and
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$B$ is the external magnetic field. For simplicity, we consider the Ising model
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where $B = 0$, and find the total system energy given by
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\begin{equation}
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E(\mathbf{s}) = -J \sum_{\langle k l \rangle}^{N} s_{k} s_{l} \ .
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\label{eq:energy_total}
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\end{equation}
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We count the neighboring spins using the pattern visualized in Figure \ref{fig:tikz_boundary},
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to avoid counting a given spin pair several times.
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\begin{figure}
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\centering
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\begin{tikzpicture}
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\draw[help lines] (0, 0) grid (2, 2);
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% \node[inner] (s1) at (0.5, 0.5) {s};
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\node (s1) at (0.5, 1.5) {$s_1$};
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\node (s2) at (1.5, 1.5) {$s_2$};
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\node (s3) at (0.5, 0.5) {$s_3$};
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\node (s4) at (1.5, 0.5) {$s_4$};
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\node[gray] (s12) at (2.5, 1.5) {$s_1$};
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\node[gray] (s34) at (2.5, 0.5) {$s_3$};
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\node[gray] (s13) at (0.5, -0.5) {$s_1$};
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\node[gray] (s24) at (1.5, -0.5) {$s_2$};
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\draw[red, ->] (s1.east) -- (s2.west);
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\draw[red, ->] (s2.east) -- (s12.west);
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\draw[red, ->] (s1.south) -- (s3.north);
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\draw[red, ->] (s2.south) -- (s4.north);
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\draw[red, ->] (s3.east) -- (s4.west);
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\draw[red, ->] (s4.east) -- (s34.west);
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\draw[red, ->] (s3.south) -- (s13.north);
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\draw[red, ->] (s4.south) -- (s24.north);
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\end{tikzpicture}
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\caption{Visualization of counting rules for each pair of spins in a
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$2 \times 2$ lattice, where periodic boundary conditions are applied.}
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\label{fig:tikz_boundary}
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\end{figure}
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We also find the total magnetization of the system, which is given by
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\begin{equation}
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M(\mathbf{s}) = \sum_{i}^{N} s_{i} \ .
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\label{eq:magnetization_total}
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\end{equation}
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In addition, we have to consider the state degeneracy, the number of different
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microstates sharing the same value of total magnetization. In the case where we
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have two spins oriented up, the total energy have two possible values, as shown
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in Appendix \ref{sec:energy_special}.
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\begin{table}[H]
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\centering
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\begin{tabular}{cccc}
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\hline
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Spins up & $E(\mathbf{s})$ & $M(\mathbf{s})$ & Degeneracy \\
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\hline
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4 & -8J & 4 & 1 \\
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3 & 0 & 2 & 4 \\
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\multirow{2}*{2} & 0 & 0 & 4 \\
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& 8J & 0 & 2 \\
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1 & 0 & -2 & 4 \\
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0 & -8J & -4 & 1 \\
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\hline
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\end{tabular}
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\caption{Values of the total energy, total magnetization, and degeneracy for the
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possible states of a system for a $2 \times 2$ lattice, with periodic boundary
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conditions.}
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\label{tab:lattice_config}
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\end{table}
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We calculate the analytical values for a $2 \times 2$ lattice, found in Table \ref{tab:lattice_config}.
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However, we scale the total energy and total magnetization by the number of spins,
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to compare these quantities for lattices where $L > 2$. Energy per spin is given by
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\begin{equation}
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\epsilon (\mathbf{s}) = \frac{E(\mathbf{s})}{N} \ ,
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\label{eq:energy_spin}
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\end{equation}
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and magnetization per spin given by
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\begin{equation}
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m (\mathbf{s}) = \frac{M(\mathbf{s})}{N} \ .
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\label{eq:magnetization_spin}
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\end{equation}
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\subsection{Statistical mechanics}\label{subsec:statistical_mechanics}
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When we study the behavior of the Ising model, we have to consider the probability
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of a microstate $\mathbf{s}$ at a fixed temperature $T$. The probability distribution
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function (pdf) is given by
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\begin{equation}
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p(\mathbf{s} \ | \ T) = \frac{1}{Z} \exp^{-\beta E(\mathbf{s})},
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\label{eq:boltzmann_distribution}
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\end{equation}
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known as the Boltzmann distribution. This is an exponential distribution, where
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$\beta$ is given by
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\begin{equation}
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\beta = \frac{1}{k_{B}} \ ,
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\label{eq:beta}
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\end{equation}
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where and $k_{B}$ is the Boltzmann constant. $Z$ is a normalizing factor of the
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pdf, given by
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\begin{equation}
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Z = \sum_{\text{all possible } \mathbf{s}} e^{-\beta E(\mathbf{s})} \ ,
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\label{eq:partition}
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\end{equation}
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and is known as the partition function. We derive $Z$ in Appendix \ref{sec:partition_function},
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which gives us
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\begin{equation*}
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Z = 4 \cosh (8 \beta J) + 12 \ .
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\end{equation*}
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Using the partition function and Equation \eqref{eq:boltzmann_distribution}, the
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probability of a microstate at a fixed temperature is given by
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\begin{equation}
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p(\mathbf{s} \ | \ T) = \frac{1}{4 \cosh (8 \beta J) + 12} e^{-\beta E(\mathbf{s})} \ .
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\label{eq:pdf}
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\end{equation}
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% Add something about why we use the expectation values?
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We find the expected total energy
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\begin{equation}\label{eq:energy_total_result}
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\langle E \rangle = -\frac{8J \sinh(8 \beta J)}{\cosh(8 \beta J) + 3} \ ,
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\end{equation}
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and the expected energy per spin
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\begin{equation}\label{eq:energy_spin_result}
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\langle \epsilon \rangle = \frac{-2J \sinh(8 \beta J)}{ \cosh(8 \beta J) + 3} \ .
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\end{equation}
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We find the expected absolute total magnetization
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\begin{equation}\label{eq:magnetization_total_result}
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\langle |M| \rangle = \frac{2(e^{8 \beta J} + 2)}{\cosh(8 \beta J) + 3} \ ,
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\end{equation}
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and the expected magnetization per spin
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\begin{equation}\label{eq:magnetization_spin_result}
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\langle |m| \rangle = \frac{e^{8 \beta J} + 1}{2( \cosh(8 \beta J) + 3)} \ .
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\end{equation}
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We derive the analytical expressions for expectation values in Appendix
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\ref{sec:expectation_values}.
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We also need to determine the heat capacity
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\begin{equation}
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C_{V} = \frac{1}{k_{B} T^{2}} (\mathbb{E}(E^{2}) - [\mathbb{E}(E)]^{2}) \ ,
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\label{eq:heat_capacity}
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\end{equation}
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and the magnetic susceptibility
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\begin{equation}
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\chi = \frac{1}{k_{\text{B} T}} (\mathbb{E}(M^{2}) - [\mathbb{E}(M)]^{2}) \ .
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\label{eq:susceptibility}
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\end{equation}
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In Appendix \ref{sec:heat_susceptibility} we derive expressions for the heat
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capacity and the susceptibility, and find the heat capacity
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\begin{align*}
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\frac{C_{V}}{N} &= \frac{64J^{2} }{N k_{B} T^{2}} \bigg( \frac{3\cosh(8 \beta J) + 1}{(\cosh(8 \beta J) + 3)^{2}} \bigg) \ ,
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\end{align*}
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and magnetic susceptibility
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\begin{align*}
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\frac{\chi}{N} &= \frac{4}{N k_{B} T} \bigg( \frac{3e^{8 \beta J} + e^{-8 \beta J} + 3}{(\cosh(8 \beta J) + 3)^{2}} \bigg) \ .
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\end{align*}
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For scalability of our model, we want to work with unitless spins. To obtain this,
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we introduce the coupling constant $J$ as the base unit for energy. With the
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Boltzmann constant we derive the remaining units, which can be found in Table
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\ref{tab:units}.
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\begin{table}[H]
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\centering
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\begin{tabular}{ccc} % @{\extracolsep{\fill}}
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\hline
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Value & Unit & \\
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\hline
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$[E]$ & $J$ & \\
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$[M]$ & $1$ & (unitless) \\
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$[T]$ & $J / k_{B}$ & \\
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$[C_{V}]$ & $k_{B}$ & \\
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$[\chi]$ & $1 / J$ & \\
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\hline
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\end{tabular}
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\caption{Values of the total energy, total magnetization, and degeneracy for the possible states of a system for a $2 \times 2$ lattice, with periodic boundary conditions.}
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\label{tab:units}
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\end{table}
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\subsection{Phase transition and critical temperature}\label{subsec:phase_critical}
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We consider the Ising model in two dimensions, with no external external magnetic
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field. At temperatures below the critical temperature $T_{c}$, the Ising model will
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magnetize spontaneously. When increasing the temperature of the external field,
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the Ising model transitions from an ordered to an unordered phase. The spins become
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more correlated, and we can measure the discontinuous behavior as an increase in
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correlation length $\xi (T)$ \cite[p. 432]{hj:2015:comp_phys}. At $T_{c}$, the
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correlation length is proportional to the lattice size, resulting in the critical
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temperature scaling relation
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\begin{equation}
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T_{c}(L) - T_{c}(L = \infty) = aL^{-1} \ ,
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\label{eq:critical_infinite}
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\end{equation}
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where $a$ is a constant. For the Ising model in two dimensions, with a lattice of
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infinite size, the analytical solution of the critical temperature is
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\begin{equation}
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T_{c}(L = \infty) = \frac{2}{\ln (1 + \sqrt{2})} J/k_{B} \approx 2.269 J/k_{B}
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\label{eq:critical_solution}
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\end{equation}
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This result was found analytically by Lars Onsager in 1944 \cite{onsager:1944}. We can estimate the
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critical temperature of an infinite lattice, using the critical temperature of
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finite lattices of different sizes and linear regression.
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\subsection{The Markov chain Monte Carlo method}\label{subsec:mcmc_method}
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Markov chains consist of a sequence of samples, where the probability of the next
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sample depends on the probability of the current sample. Whereas the Monte Carlo
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method introduces randomness to the sampling, which allows us to approximate
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statistical quantities \cite{tds:2021:mcmc} without using the pdf directly. We
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generate samples of spin configuration, to approximate $\langle \epsilon \rangle$,
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$\langle |m| \rangle$, $C_{V}$, and $\chi$, using the Markov chain Monte Carlo
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(MCMC) method.
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The Ising model is initialized in a random state, and the Markov chain is evolved
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until the model reaches an equilibrium state. However, generating new random states
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requires ergodicity and detailed balance. A Markov chain is ergoditic when all system
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states can be reached at every current state, whereas detailed balance implies no
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net flux of probability. To satisfy these criteria we use the Metropolis-Hastings
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algorithm, found in Algorithm \ref{algo:metropolis}, to generate samples of microstates.
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One Monte Carlo cycle consists of $N$ attempted spin flips. When a spin is flipped,
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the change in energy is evaluated as
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\begin{align*}
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\Delta E &= E_{\text{after}} - E_{\text{before}} \ ,
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\end{align*}
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and accepted if $\Delta E \leq 0$. However, if $\Delta E > 0$ we have to compute
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the Boltzmann factor
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\begin{equation}
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p(\mathbf{s} | T) = e^{-\beta \Delta E}
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\label{eq:boltzmann_factor}
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\end{equation}
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Since the total system energy only takes three different values, the change in
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energy can take $3^{2}$ values. However, there are only five distinct values
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$\Delta E = \{-16J, -8J, 0, 8J, 16J\}$, we derive these values in Appendix \ref{sec:delta_energy}.
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We can avoid computing the Boltzmann factor, by using a look up table (LUT).
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We use $\Delta E$ as an index to access the resulting value of the exponential function,
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in an array.
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\begin{algorithm}[H]
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\caption{Metropolis-Hastings Algorithm}
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\label{algo:metropolis}
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\begin{algorithmic}
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\Procedure{Metropolis}{$lattice, energy, magnetization$}
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\For{ Each spin in the lattice }
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\State $ri \leftarrow \text{random index of the lattice}$
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\State $rj \leftarrow \text{random index of the lattice}$
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\State $dE \leftarrow 2 \cdot lattice_{ri,rj} \cdot (lattice_{ri,rj-1} + lattice_{ri,rj+1}
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+ lattice_{ri-1,rj} + lattice_{ri+1,rj} )$
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\State $rn \leftarrow \text{random number} \in [0,1)$
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\If{$rn \leq e^{- \frac{dE}{T}}$}
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\State $lattice_{ri,rj} = -1 \cdot lattice_{ri,rj}$
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\State $magnetization = magnetization + 2 \cdot lattice_{ri,rj}$
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\State $energy = energy + dE$
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\EndIf
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\EndFor
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\EndProcedure
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\end{algorithmic}
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\end{algorithm}
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The Markov process reaches an equilibrium after a certain number of Monte Carlo cycles,
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called burn-in time. After the burn-in time, the system state reflects the state of a real system,
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and we can start sampling microstates. The probability distribution of the samples
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will tend toward the actual probability distribution of the system.
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\subsection{Implementation and testing}\label{subsec:implementation_test}
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We implemented a test suite, and compared the numerical estimates to the analytical
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results from Section \ref{subsec:statistical_mechanics}. In addition, we set a
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tolerance to verify convergence, and a maximum number of Monte Carlo cycles to
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avoid potentially having the program run indefinitely.
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We used a pattern to access all neighboring spins, where all indices of the neighboring
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spins are put in an $L \times 2$ matrix. The indices are accessed using pre-defined
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constants, where the first column contain the indices for neighbors to the left
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and up, and the second column right and down. This method avoids the use of if-tests,
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so we can take advantage of the compiler optimization.
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We parallelized our code using both message passing interface (OpenMPI) and
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multi-threading (OpenMP). First, we divided the temperatures into smaller sub-ranges,
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and each MPI process received a sub-range of temperatures. Every MPI initialize a
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parallel region with a set of threads, which then initializes an Ising model and
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performs the Metropolis-Hastings algorithm. We limited the number of times threads
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are spawned and joined, by using single parallel regions, reducing parallel overhead.
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We used Fox \footnote{Technical specifications for Fox can be found at \url{https://www.uio.no/english/services/it/research/platforms/edu-research/help/fox/system-overview.md}},
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a high-performance computing cluster, to run our program for $1$ and $10$ million
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Monte Carlo cycles when analyzing phase transitions.
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\subsection{Tools}\label{subsec:tools}
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The Ising model and MCMC methods are implemented in C++, and parallelized using
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both \verb|OpenMPI| \cite{gabriel:2004:open_mpi} and \verb|OpenMP| \cite{openmp:2018}.
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We use the Python library \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce
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all the plots, \verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the
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figures. We also use a linear regression method and the normal distribution method
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from the Python library \verb|SciPy|. To optimize our implementation, we used a
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profiling tool \verb|Score-P| \cite{scorep}.
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\end{document}
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