= Solution
<Mean-field theory> replaces fluctuating interactions with the average field produced by the other degrees of freedom. For an interacting <spin> system, write $\mathbf S_i=\mathbf S+\delta\mathbf S_i$ and neglect connected products $\delta\mathbf S_i\cdot\delta\mathbf S_j$. This reduces the many-body <partition function> to independent single-site factors coupled through a <self-consistency equation> for $\mathbf S$.
The approximation is most reliable when many effectively independent neighbors contribute, for sufficiently long-range interactions, or above the <upper critical dimension> away from the fluctuation-dominated critical region. It usually becomes least reliable near a continuous transition in low dimensions, where correlations extend over long distances. It can estimate ordering scales and reveal candidate broken symmetries while giving incorrect <critical exponents> or even a spurious transition. The two-dimensional isotropic model in the later parts is an especially important example of this limitation.
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