Mean field models

A mean field model, or a mean field solution of a model, is an approximation to the actual solution of a model in statistical physics. The model is made exactly solvable by treating the effect of all other particles on a given one as a mean field (hence its name). It appear in different forms and different contexts, but all mean field models have this feature in common.

Mean field solution of the Ising model

A well-known mean field solution of the Ising model, known as the Bragg-Williams approximation goes as follows. From the original Hamiltonian, \[ U = - J \sum_i^N S_i \sum_{<j>} S_j , \] suppose we may approximate \[ \sum_{<j>} S_j \approx n \bar{s}, \] where \(n\) is the number of neighbors of site \(i\) (e.g. 4 in a 2-D square lattice), and \(\bar{s}\) is the (unknown) magnetization: \[ \bar{s}=\frac{1}{N} \sum_i S_i . \]

Therefore, the Hamiltonian turns to \[ U = - J n \sum_i^N S_i \bar{s} , \] as in the regular Langevin theory of magnetism (see Curie's_law): the spins are independent, but coupled to a constant field of strength \[H= J n \bar{s}.\] The magnetization of the Langevin theory is \[ \bar{s} = \tanh( H/k_B T ). \] Therefore: \[ \bar{s} = \tanh(J n\bar{s}/k_B T). \]

This is a self-consistent expression for \(\bar{s}\). There exists a critical temperature, defined by \[k_B T_c= J n .\] At temperatures higher than this value the only solution is \(\bar{s}=0\). Below it, however, this solution becomes unstable (it corresponds to a maximum in energy), whereas two others are stable. Slightly below \(T_c\), \[\bar{s} =\pm\sqrt{3\left(1 - \frac{T}{T_c}\right)}. \]

General discussion

The solution obtained shares a number of features with any other mean field approximation:

• It largely ignores geometry, which may be important in some cases. In particular, it reduces the lattice details to just the number of neighbours.
• As a consequence, it may predict phase transitions where none are found: the 1-D ising model \(n=2\) is known to lack any phase transition (at finite temperature)
• In general, the theory underestimates fluctuations
• It also leads to classical critical exponents, like the \(\left(1 - \frac{T}{T_c}\right)^{1/2}\) decay above. In 3-D, the magnetization follows a power law with a different exponent.
• Nevertheless, above a certain space dimension the critical exponents are correct. This dimension is 4 for the Ising model, as predicted by a self-consistency requirement due to E.M. Lipfshitz (similar ones are due to Peierls and L.D. Landau)