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.
1-dimensional Ising model
- 1-dimensional Ising model (exact solution)
2-dimensional Ising model
- Lars Onsager "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition", Physical Review 65 pp. 117 - 149 (1944)
3-dimensional Ising model
Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice:
- Three-dimensional proof for Ising model impossible, Sandia researcher claims to have shown
- Sorin Istrail "Statistical mechanics, three-dimensionality and NP-completeness: I. Universality of intracatability for the partition function of the Ising model across non-planar surfaces", Proceedings of the thirty-second annual ACM symposium on Theory of computing pp. 87 - 96 (2000)
The axial next-nearest neighbour Ising (ANNNI) model is used to study alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.
- Walter Selke "The ANNNI model — Theoretical analysis and experimental application", Physics Reports 170 pp. 213-264 (1988)