The Ising model is a prototypical model in statistical physics. 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 (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. A model of this type can be defined on lattices in any number of dimensions. Techniques that are applicable for small dimensions are not always useful for larger dimensions, and vice versa. In any dimension, the Ising model can be productively described by a locally varying mean field. The field is defined as the average spin value over a large region, but not so large so as to include the entire system. The field still has slow variations from point to point, as the averaging volume moves. These fluctuations in the field are described by a continuum field theory in the infinite system limit. The accuracy of this approximation improves as the dimension becomes larger. A deeper understanding of how the Ising model behaves, going beyond mean-field approximations, can be achieved using renormalization group methods.
Local field The field H is defined as the long wavelength Fourier components of the spin variable, in the limit that the wavelengths are long. There are many ways to take the long wavelength average, depending on the details of how high wavelengths are cut off. The details are not too important, since the goal is to find the statistics of H and not the spins. Once the correlations in H are known, the long-distance correlations between the spins will be proportional to the long-distance correlations in H. For any value of the slowly varying field H, the free energy (log-probability) is a local analytic function of H and its gradients. The free energy F(H) is defined to be the sum over all Ising configurations which are consistent with the long wavelength field. Since H is a coarse description, there are many Ising configurations consistent with each value of H, so long as not too much exactness is required for the match. Since the allowed range of values of the spin in any region only depends on the values of H within one averaging volume from that region, the free energy contribution from each region only depends on the value of H there and in the neighboring regions. So F is a sum over all regions of a local contribution, which only depends on H and its derivatives. By symmetry in H, only even powers contribute. By reflection symmetry on a square lattice, only even powers of gradients contribute. Writing out the first few terms in the free energy:
β F = ∫ d d x [ A H 2 + ∑ i = 1 d Z i ( ∂ i H ) 2 + λ H 4 + ⋯ ] . {\displaystyle \beta F=\int d^{d}x\left[AH^{2}+\sum _{i=1}^{d}Z_{i}(\partial _{i}H)^{2}+\lambda H^{4}+\cdots \right].}
On a square lattice, symmetries guarantee that the coefficients Zi of the derivative terms are all equal. But even for an anisotropic Ising model, where the Zi's in different directions are different, the fluctuations in H are isotropic in a coordinate system where the different directions of space are rescaled. On any lattice, the derivative term
Z i j ∂ i H ∂ j H {\displaystyle Z_{ij}\,\partial _{i}H\,\partial _{j}H}
is a positive definite quadratic form, and can be used to define the metric for space. So any translationally invariant Ising model is rotationally invariant at long distances, in coordinates that make Zij = δij. Rotational symmetry emerges spontaneously at large distances just because there aren't very many low order terms. At higher order multicritical points, this accidental symmetry is lost. Since βF is a function of a slowly spatially varying field, the probability of any field configuration is (omitting higher-order terms):
P ( H ) ∝ e − ∫ d d x [ A H 2 + Z | ∇ H | 2 + λ H 4 ] = e − β F [ H ] . {\displaystyle P(H)\propto e^{-\int d^{d}x\left[AH^{2}+Z|\nabla H|^{2}+\lambda H^{4}\right]}=e^{-\beta F[H]}.}
The statistical average of any product of H terms is equal to:
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