# Ising Model

The Ising model (/ˈaɪsɪŋ/; ), named after the physicist Ernst Ising, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic spins that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually, a lattice, allowing each spin to interact with its neighbors. The model allows the identification of phase transitions, as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition.

The Ising model was invented by the physicist Wilhelm Lenz (1920), who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model has no phase transition and was solved by Ising (1925) himself in his 1924 thesis. The two-dimensional square lattice Ising model is much harder, and was given an analytic description much later, by Lars Onsager (1944). It is usually solved by a transfer-matrix method, although there exist different approaches, more related to quantum field theory.

In dimensions greater than four, the phase transition of the Ising model is described by mean field theory.

### Other articles related to "model, ising model, models, ising":

Spherical Model - Formulation
... The model describes a set of particles on a lattice which contains N sites ... It differs from the Ising model in that the are no longer restricted to, but can take all real values, subject to the constraint that which in a homogenous system ensures that the average of the ... The partition function generalizes from that of the Ising model to where is the Dirac delta function, are the edges of the lattice, and and, where T is the temperature of the system, k is ...
Barry M. Mc Coy
... known for his contributions to classical statistical mechanics, integrable models and conformal field theories ... from Harvard University (1967), the thesis entitled Spin Correlations of the Two Dimensional Ising Model advised by Tai Tsun Wu ... The two of them also wrote the book The Two Dimensional Ising Model (Harvard University Press, 1973) ...
Eight-vertex Model - Equivalence With An Ising Model
... There is a natural correspondence between the eight-vertex model, and the Ising model with 2-spin and 4-spin nearest neighbour interactions ... The states of this model are spins on faces of a square lattice ... The analogue of 'edges' in the eight-vertex model are products of spins on adjacent faces The most general form of the energy for this model is where, describe the horizontal, vertical and ...
Ising Model - Applications - Spin Glasses
... With the Ising model the so-called spin glasses can also be described, by the usual Hamiltonian where the S-variables describe the Ising spins, while the Ji,k are taken from a ... When p=0 we have the original Ising model ... Much attention has been also attracted by the related bond and site dilute Ising model, especially in two dimensions, leading to intriguing critical behavior ...
Landau Theory
... For example, the Ising model free energy may be written as the following where the parameter for physical reasons ... Note that the Ising model exhibits the following discrete symmetry If every spin in the model is flipped, such that, where is the value of the spin, the Hamiltonian (and consequently the free energy ... For the Ising model case, the equilibrium magnetization assumes the following value below the critical temperature At the time, it was known experimentally that the liquid-gas ...

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