note_ising.tex

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\begin{document}


\title{Susceptibility of the Ising model}
\author{Matthijs Hogervorst}
\date\today
\maketitle

Consider the Ising model on a finite lattice of size $V = L^d$. There
are $2^V$ configurations $\{\sigma_x\}_x$, $\sigma_x = \pm 1$. The
Hamiltonian is
\beq
H = - \sum_{\expec{x,y}} \sigma_x \sigma_y,
\quad
Z = \sum_{\sigma} e^{-\beta H}.
\eeq
The susceptibility is
\beq
\chi \ldef \expec{\mca{X}} = \frac{1}{Z} \sum_\sigma \mca{X} e^{-\beta H},
\quad
\mca{X} \ldef \frac{1}{V^2} \sum_{x,y} \sigma_x \sigma_y.
\eeq
Note that
\beq
\mca{X} = \left(\frac{1}{V} \sum_x \sigma_x\right)^2 \geq 0
\eeq
meaning that $\mca{X} \geq 0$ for \emph{every} configuration
$\{\sigma_x\}_x$. Moreover clearly $\mca{X} \leq 1$. We can measure
$\chi$ directly for small $V$, by summing over all $2^V$
configurations. The results are shown in Fig.~\ref{fig:susc}.
\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=0.5]{suscepPlot.png}
    \caption{\label{fig:susc}Susceptibility for a range of different lattices,
      scanning over $\tanh(\beta)$.}
  \end{center}
\end{figure}

Moreover notice that for any observable $\Oo$
\beq
\frac{\pd \expec{\Oo}}{\pd \beta} = \expec{\Oo} E -  \expec{\Oo H}
\quad
\Rightarrow
\quad
\frac{\pd\chi}{\pd\beta} = \chi \cdot E - \expec{\mca{X} H}.
\eeq
Using this, we can compute $\pd_\beta \chi$ directly too; see Fig.~\ref{fig:ds}.
\begin{figure}[htb]
  \begin{center}
    \includegraphics[scale=.5]{der_suscepPlot.png}
    \caption{\label{fig:ds}Derivative of the
      susceptibility (right) for a range of different lattices,
      scanning over $\tanh(\beta)$.}
  \end{center}
\end{figure}
Finally we can plot the energy, see Fig.~\ref{fig:en}.
\begin{figure}[htb]
  \begin{center}
    \hspace{0mm}
    \includegraphics[scale=.5]{energyPlot.png}
    \caption{\label{fig:en}Energy $\expec{H}$ for a range of different lattices,
      scanning over $\tanh(\beta)$.}
  \end{center}
\end{figure}

As for analytics, we can show that
\beq
\chi \, \limu{\beta \to 0} \,  \frac{1}{V} + \frac{2d}{V}\, \tanh(\beta) + \ldots
\qaq
\chi \limu{\beta \to \infty} 1.
\eeq
For the energy we have
\beq
E \limu{\beta \to 0} 0 \qaq
E \limu{\beta \to \infty} -dV.
\eeq
This seems to agree with the data.

   
\end{document}