Ising model

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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: either UP (S=+1) or DOWN (S=-1).

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.

1-dimensional Ising model

2-dimensional Ising model

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

3-dimensional Ising model

Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice:

ANNNI model

The axial next-nearest neighbour Ising (ANNNI) model is used to study alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.


  1. S. G. Brush "History of the Lenz-Ising Model", Reviews of Modern Physics 39 pp. 883-893 (1967)
  2. Martin Niss "History of the Lenz-Ising Model 1920-1950: From Ferromagnetic to Cooperative Phenomena", Archive for History of Exact Sciences 59 pp. 267-318 (2005)