Basic Properties and History
The most studied case of the Ising model is the translation-invariant ferromagnetic zero-field model on a d-dimensional lattice, namely, Λ = Zd, Jij = 1, h = 0.
In his 1924 PhD thesis, Ising solved the model for the 1D case, which can be thought of as a linear horizontal lattice where each site only interacts with its left and right neighbor. In one dimension, the solution admits no phase transition. Namely, for any positive β, the correlations <σiσj> decay exponentially in |i−j|:
and the system is disordered. On the basis of this result, he incorrectly concluded that this model does not exhibit phase behaviour in any dimension.
The Ising model undergoes a phase transition between an ordered and a disordered phase in 2 dimensions or more. Namely, the system is disordered for small β, whereas for large β the system exhibits ferromagnetic order:
This was first proved by Rudolph Peierls in 1933, using what is now called a Peierls argument.
The Ising model on a two dimensional square lattice with no magnetic field was analytically solved by Lars Onsager (1944). Onsager showed that the correlation functions and free energy of the Ising model are determined by a noninteracting lattice fermion. Onsager announced the formula for the spontaneous magnetization for the 2-dimensional model in 1949 but did not give a derivation. Yang (1952) gave the first published proof of this formula, using a limit formula for Fredholm determinants, proved in 1951 by Szegő in direct response to Onsager's work.
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