A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases (Gaussian field theory, 1D Ising model.) The great difficulty (e.g. when computing the partition function of the system) is the treatment of combinatorics generated by the interaction terms in the Hamiltonian when summing over all states. The goal of mean field theory (MFT, also known as self-consistent field theory) is to resolve these combinatorial problems
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