Ising model

Also known as the Lenz-Ising model.

Ising Model

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.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{U}{k_B T} = - K \sum_{\langle ij \rangle} S_i S_j }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle ij \rangle } indicates that the sum is done over nearest neighbors, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_i } indicates the state of the i-th site.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K } is called the Coupling constant.

1-dimensional Ising model

• 1-dimensional Ising model (exact solution)

2-dimensional Ising model

Solved by Lars Onsager in 1944.

• 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)

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

• History of the Ising model