\documentclass[../ising_model.tex]{subfiles}

\begin{document}
\appendix
\section{Ising model system states}\label{sec:system_states}
To avoid counting duplicates, we used 
\begin{figure}\label{fig:tikz_counting}
    \centering
    \begin{tikzpicture}
        \draw (0, 0) grid (2, 2);
        % \node[inner] (s1) at (0.5, 0.5) {s};
        \node (s1) at (0.5, 1.5) {$s_1$};
        \node (s2) at (1.5, 1.5) {$s_2$};
        \node (s3) at (0.5, 0.5) {$s_3$};
        \node (s4) at (1.5, 0.5) {$s_4$};
        \node[gray] (s12) at (2.5, 1.5) {$s_1$};
        \node[gray] (s34) at (2.5, 0.5) {$s_3$};
        \node[gray] (s13) at (0.5, -0.5) {$s_1$};
        \node[gray] (s24) at (1.5, -0.5) {$s_2$};
    
        \draw[red, ->] (s1.east) -- (s2.west);
        \draw[red, ->] (s2.east) -- (s12.west);
        \draw[red, ->] (s1.south) -- (s3.north);
        \draw[red, ->] (s2.south) -- (s4.north);
        \draw[red, ->] (s3.east) -- (s4.west);
        \draw[red, ->] (s4.east) -- (s34.west);
        \draw[red, ->] (s3.south) -- (s13.north);
        \draw[red, ->] (s4.south) -- (s24.north);
      \end{tikzpicture}
      \caption{Rules for multiplying spin pairs.}
\end{figure}


\section{Analytical expressions}\label{sec:analytical_expressions}
The Boltzmann distribution is normalized using a partition function $Z$ given by
\begin{align*}
	Z &= \sum_{\text{all possible } s_{i}}^{N} e^{-\beta E(\mathbf{s})} \\
	&= \dots \\
	&= 4 \cosh (8 \beta J) + 12 \\
\end{align*}


\end{document}