Ising model

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.

1-dimensional Ising model

• 1-dimensional Ising model (exact solution)

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.

1. Lars Onsager "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition", Physical Review 65 pp. 117 - 149 (1944)
2. M. Kac and J. C. Ward "A Combinatorial Solution of the Two-Dimensional Ising Model", Physical Review 88 pp. 1332-1337 (1952)
3. Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 7 (freely available pdf)

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)

ANNNI model

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)

See also

• Critical exponents
• Potts model

References

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)