A way to do mean field theory for the Ising model is as follows.
- First take the Ising Hamiltonian: $$H=-J \sum_{\left} \sigma_i\sigma_j$$
Let $\sigma_i=\sigma_i-M+M$ and likewise for $\sigma_j$ to get: $$H=-J \sum_{\left} (M^2 +(\sigma_i-M)M+(\sigma_j-M)M+\underbrace{(\sigma_i-M)(\sigma_j-M)}_{\bigstar})$$
Ignore the stared ($\star$) term.
Write down the partition function, apply a self-consistency condition etc.
Given that in the Ising model $\sigma_i=\pm1$ thus for any given $i$ and $j$, the ($\star$) term is not going to be small. What is the standard justificiation for then ignoring it?
Answer
Even though $\sigma_i-M$ is not small, expectation value of its square is small as that corresponds to the variance, hence fluctuations, which are assumed to be next order terms in the mean-field-theory. That's why summation $\sum (\sigma_i-M)(\sigma_j-M)$, which is basically autocovariance function along lattice sites, should be small. By the way, here we should have $M=<\sigma_i>$.
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