Ising Models

From SklogWiki

The Ising model is also known as the Lenz-Ising model. For a history of the Lenz-Ising model see Refs. 1 and 2. The Ising model is commonly defined over an ordered lattice. Each site of the lattice can adopt two states, $S \in \{-1, +1 \}$ . Note that sometimes these states are referred to as spins and the values are referred to as down and up respectively.

The energy of the system is the sum of pair interactions between nearest neighbors.

$\frac{U}{k_B T} = - K \sum_{\langle ij \rangle} S_i S_j$

where $k B$ is the Boltzmann constant, $T$ is the temperature, $\langle ij \rangle$ indicates that the sum is performed over nearest neighbors, and $S i$ indicates the state of the i-th site, and $K$ is the coupling constant.

[edit] 1-dimensional Ising model

1-dimensional Ising model (exact solution)

[edit] 2-dimensional Ising model

Solved by Lars Onsager in 1944. Rudolf Peierls had previously shown (1935) that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition.