Ising model: Difference between revisions

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The '''Ising model''' is also known as the '''Lenz-Ising''' model. For a history of the Lenz-Ising model see <ref>[http://dx.doi.org/10.1103/RevModPhys.39.883  S. G. Brush "History of the Lenz-Ising Model", Reviews of Modern Physics '''39''' pp. 883-893 (1967)]</ref>
The '''Ising model''' is also known as the '''Lenz-Ising''' model. For a history of the Lenz-Ising model see <ref>[http://dx.doi.org/10.1103/RevModPhys.39.883  S. G. Brush "History of the Lenz-Ising Model", Reviews of Modern Physics '''39''' pp. 883-893 (1967)]</ref>
<ref>[http://dx.doi.org/10.1007/s00407-004-0088-3 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)]</ref>
<ref>[http://dx.doi.org/10.1007/s00407-004-0088-3 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)]</ref>
<ref>[http://dx.doi.org/10.1007/s00407-008-0039-5 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)]</ref>.
The Ising model is commonly defined over an ordered lattice.  
The Ising model is commonly defined over an ordered lattice.  
Each site of the lattice can adopt two states, <math>S \in \{-1, +1 \}</math>. Note that sometimes these states are referred to as ''spins'' and the values are referred to as ''down'' and ''up'' respectively.  
Each site of the lattice can adopt two states, <math>S \in \{-1, +1 \}</math>. Note that sometimes these states are referred to as ''spins'' and the values are referred to as ''down'' and ''up'' respectively.  

Revision as of 15:28, 19 April 2010

The Ising model is also known as the Lenz-Ising model. For a history of the Lenz-Ising model see [1] [2] [3]. The 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.

1-dimensional Ising model

2-dimensional Ising model

Solved by Lars Onsager in 1944 [4] [5] [6]. Rudolf Peierls had previously shown (1935) that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition.

3-dimensional Ising model

Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice [7] [8]

ANNNI model

The axial next-nearest neighbour Ising (ANNNI) model [9] is used to study alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.

See also

References