Ising model

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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, 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.

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

1-dimensional Ising model

Main article: 1-dimensional Ising model

The 1-dimensional Ising model has an exact solution.

2-dimensional Ising model

The 2-dimensional 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].

3-dimensional Ising model

Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice [10] [11]

ANNNI model

The axial next-nearest neighbour Ising (ANNNI) model [12] is used to study alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.

See also


  1. Ernst Ising "Beitrag zur Theorie des Ferromagnetismus", Zeitschrift für Physik A Hadrons and Nuclei 31 pp. 253-258 (1925)
  2. S. G. Brush "History of the Lenz-Ising Model", Reviews of Modern Physics 39 pp. 883-893 (1967)
  3. 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)
  4. 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. Lars Onsager "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition", Physical Review 65 pp. 117 - 149 (1944)
  6. M. Kac and J. C. Ward "A Combinatorial Solution of the Two-Dimensional Ising Model", Physical Review 88 pp. 1332-1337 (1952)
  7. Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 7 (freely available pdf)
  8. Rudolf Peierls "On Ising's model of ferromagnetism", Mathematical Proceedings of the Cambridge Philosophical Society 32 pp. 477-481 (1936)
  9. Robert B. Griffiths "Peierls Proof of Spontaneous Magnetization in a Two-Dimensional Ising Ferromagnet", Physical Review A 136 pp. 437-439 (1964)
  10. Three-dimensional proof for Ising model impossible, Sandia researcher claims to have shown
  11. 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)
  12. Walter Selke "The ANNNI model — Theoretical analysis and experimental application", Physics Reports 170 pp. 213-264 (1988)