% miko:type: atomic
% miko:title: Ising Model
% miko:summary: A minimal lattice model for ferromagnetism and phase transitions.
% miko:tags: statmech
% miko:difficulty: undergrad
% miko:related: phase-transition, fluctuation-response, complete-statistical-mechanics
% miko:keywords: spin, criticality, magnetization

\documentclass{article}
\usepackage{amsmath}
\title{Ising Model}
\begin{document}
\maketitle

\section{Hamiltonian}

The Ising model places spins $s_i=\pm1$ on a lattice with Hamiltonian

$$
H=-J\sum_{\langle i,j\rangle}s_i s_j-h\sum_i s_i.
$$

For $J>0$, neighboring spins prefer alignment.

\section{Physics}

The model captures competition between energy and entropy. In dimensions two
and higher it can exhibit a finite-temperature phase transition with
spontaneous magnetization.

\end{document}
