{c}
{wiki}

= Ising
{c}
{synonym}

The <Ising model> is a <spin model> with an <Ising spin> $\sigma_v\in\{-1,1\}$ at every <graph vertex> $v$. For a finite <graph> with vertex set $V$ and edge set $E$, a uniform coupling $J$ and field parameter $h$ give the <Hamiltonian>
$$
H(\sigma)=-J\sum_{\{v,w\}\in E}\sigma_v\sigma_w-h\sum_{v\in V}\sigma_v,\qquad
\beta=\frac{1}{k_BT}.
$$
Each <edge> is counted once. Here $T>0$ is <temperature>, $k_B$ is the <Boltzmann constant>, and $\beta$ is <inverse temperature>; $J$ and $h$ have units of <energy>. For magnetic applications, $h$ includes the <magnetic moment> multiplying the applied <magnetic field>. The <canonical ensemble> assigns the <Boltzmann distribution>
$$
P(\sigma)=\frac{e^{-\beta H(\sigma)}}{Z},\qquad
Z=\sum_{\tau\in\{-1,1\}^{V}}e^{-\beta H(\tau)}.
$$
The <partition function> $Z$ makes the <probabilities> sum to one. Positive $J$ rewards equal neighboring <Ising spins>, modeling <ferromagnetism>; negative $J$ rewards opposite neighbors, giving an <antiferromagnetic Ising model>. Positive $h$ rewards $+1$, while negative $h$ rewards $-1$.<Statistical Physics by David Tong>[§5.2, equations (5.192)–(5.194), nearest-neighbor energy, field convention and canonical partition function]{ref}

A single local rule can organize a whole <graph>: rewarding agreement on each <edge> makes all-equal <Ising spin> patterns favorable even though no <edge> directly couples distant vertices. The <four-spin Ising ring> makes this visible with only sixteen <microstates>. It also separates three different questions: whether neighbors agree, whether the mean <spin magnetization> is nonzero, and how large its <thermal fluctuations> are.

A motivating experiment is the University of Colorado Boulder's Curie-temperature demonstration. An <iron> rod suspended from a twisting wire on a stand is initially held near a permanent magnet by <ferromagnetism>. A gas burner supplied from a propane cylinder heats the rod; the readout is its motion away from the magnet, followed by return as it cools. Repeated cooling supplies a reversible comparison on the same sample. The source explicitly calls for heating the rod while keeping the magnet cool, since heating the magnet can reduce its strength. The inference is that warming the sample weakens its magnetic attraction across the <Curie temperature>. This is a qualitative <force> readout, with no calibrated <temperature> or <magnetization> trace supplied: it motivates competition between alignment and <thermal fluctuations>, but does not identify $J$, measure individual <Ising spins>, or realize a <four-spin Ising ring>.<Curie Temperature (University of Colorado Boulder Physics Undergraduate Labs)>[Curie Temperature, Purpose, apparatus Figure 1, and Magnet Positioning: suspended iron rod, propane burner, cooling return, and keeping the magnet cool]{ref}

<Ernst Ising>'s 1925 paper examined a linear chain of two-orientation magnetic elements with neighboring interactions. In its calculation for a large chain, the zero-field mean <magnetization> vanishes and the model has no <spontaneous magnetization>. The inspected English translation states this immediately after equation (8) in §2. This historical result concerns the chain; it does not establish the behavior of a two-dimensional <Ising model>.<Beitrag zur Theorie des Ferromagnetismus by Ernst Ising>[Beitrag zur Theorie des Ferromagnetismus by Ernst Ising (1925), English translation, §2 The simple linear chain, energy (3′) and the paragraph immediately after equation (8)]{ref}


 Ising model (source code)