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 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.
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.