# Ising model

The **Ising model** ^{[1]} (also known as the **Lenz-Ising** model) is commonly defined over an ordered lattice.
Each site of the lattice can adopt two states, . 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.

where is the Boltzmann constant, is the temperature, indicates that the sum is performed over nearest neighbors, and indicates the state of the i-th site, and is the coupling constant.

For a detailed and very readable history of the Lenz-Ising model see the following references:^{[2]}
^{[3]}
^{[4]}.

## 1-dimensional Ising model[edit]

*Main article: 1-dimensional Ising model*

The 1-dimensional Ising model has an exact solution.

## 2-dimensional Ising model[edit]

The 2-dimensional square lattice Ising model was solved by Lars Onsager in 1944
^{[5]}
^{[6]}
^{[7]}
after Rudolf Peierls had previously shown that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition
^{[8]} ^{[9]}.

#### Critical temperature[edit]

The critical temperature of the 2D Ising model is given by ^{[5]}

where is the interaction energy in the direction, and is the interaction energy in the direction. If these interaction energies are the same one has

#### Critical exponents[edit]

The critical exponents are as follows:

- Heat capacity exponent (Baxter Eq. 7.12.12)
- Magnetic order parameter exponent (Baxter Eq. 7.12.14)
- Susceptibility exponent (Baxter Eq. 7.12.15)

(see also: Ising universality class)

## 3-dimensional Ising model[edit]

Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice
^{[10]}
^{[11]}
In three dimensions, the critical exponents are not known exactly. However, Monte Carlo simulations, renormalisation group analysis and conformal bootstrap techniques provide accurate estimates ^{[12]}:

with a critical temperature of ^{[13]}

## ANNNI model[edit]

The **axial next-nearest neighbour Ising** (ANNNI) model ^{[14]} is used to study spatially modulated structures in alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.

## Cellular automata[edit]

The Ising model can be studied using cellular automata ^{[15]}^{[16]}^{[17]}^{[18]}.

## See also[edit]

## References[edit]

- ↑ Ernst Ising "Beitrag zur Theorie des Ferromagnetismus", Zeitschrift für Physik A Hadrons and Nuclei
**31**pp. 253-258 (1925) - ↑ S. G. Brush "History of the Lenz-Ising Model", Reviews of Modern Physics
**39**pp. 883-893 (1967) - ↑ 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) - ↑ Martin Niss "History of the Lenz–Ising Model 1950–1965: from irrelevance to relevance", Archive for History of Exact Sciences
**63**pp. 243-287 (2009) - ↑
^{5.0}^{5.1}Lars Onsager "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition", Physical Review**65**pp. 117-149 (1944) - ↑ M. Kac and J. C. Ward "A Combinatorial Solution of the Two-Dimensional Ising Model", Physical Review
**88**pp. 1332-1337 (1952) - ↑ Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 7 (freely available pdf)
- ↑ Rudolf Peierls "On Ising's model of ferromagnetism", Mathematical Proceedings of the Cambridge Philosophical Society
**32**pp. 477-481 (1936) - ↑ Robert B. Griffiths "Peierls Proof of Spontaneous Magnetization in a Two-Dimensional Ising Ferromagnet", Physical Review A
**136**pp. 437-439 (1964) - ↑ 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)
- ↑ Massimo Campostrini, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice", Physical Review E
**65**066127 (2002) - ↑ A. L. Talapov and H. W. J Blöte "The magnetization of the 3D Ising model", Journal of Physics A: Mathematical and General
**29**pp. 5727-5733 (1996) - ↑ Walter Selke "The ANNNI model — Theoretical analysis and experimental application", Physics Reports
**170**pp. 213-264 (1988) - ↑ Gérard Y. Vichniac "Simulating physics with cellular automata", Physica D: Nonlinear Phenomena
**10**pp. 96-116 (1984) - ↑ Y. Pomeau "Invariant in cellular automata", Journal of Physics A
**17**pp. L415-L418 (1984) - ↑ H. J. Herrmann "Fast algorithm for the simulation of Ising models", Journal of Statistical Physics
**45**pp. 145-151 (1986) - ↑ Michael Creutz "Deterministic ising dynamics", Annals of Physics
**167**pp. 62-72 (1986)

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