Recently I read some literature about how people use the mean-field approximation to solve a particular physical problem. However, I saw people using it in a different way when they dealt with different Hamiltonians. A well-known example would be the BCS theory. People try to perform the Fourier transform on the Hamiltonian and use the mean-field approach to linearise the interaction term in k-space. Then they perform a Bogaliubove transform to obtain an effective single-particle Hamiltonian. However, when people try to deal with Bose Hubbard model, they often start by decoupling the hopping term and obtain an effective single site Hamiltonian in the real space and use the perturbation method to obtain the phase diagram of Mott-insulating and superfluid phase transition boundary, etc. My question is, how can we choose the proper term in our Hamiltonian for the mean-field treatment?
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$\begingroup$ I don't know if this fully answers the question, but this is possibly related to the Bose Einstein condensation. Since bosons can condense, they can form a superfluid, which is a coherent state and is such that $\langle b_i\rangle\neq 0$, where $b_i$ is the bosonic annihialtion operator on site i. On the other hand, fermions don't condense, and $\langle c_i\rangle=0$ ($c_i$ is the fermionic annihilation operator), thus preventing to use the same approach for bosons and fermions $\endgroup$– MatteoCommented Apr 6, 2023 at 11:28
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