# Mean Field Theory for Fermions/Bosons

1. Jun 26, 2015

### Xenosum

I'm not really sure if this counts as a homework problem (I was reluctant to post in that section since they evidently force you to ensure you've used the template, even though it's not very applicable here) so much as a general misunderstanding of mean field theory. So, in Michale Plischke and Birger Bergersen's Equilibrium Statistical Physics, they make the following approximation

$$\sum_{k,k^{'}}V_{k,k^{'}}b^{\dagger}_{k}b_{k^{'}} \approx \sum_{k,k^{'}}V_{k,k^{'}}\left( b^{\dagger}_k <b_{k^{'}}> + <b^{\dagger}_k > b_{k^{'}} - <b^{\dagger}_k > <b_{k^{'}}>\right)$$

where $b_k$ is a bosonic annihilation operator.

Can someone explain this? This is pretty much an invokation of mean field theory, but I'd like to generalize the approximation to other operators, e.g. $J\sum_{i} a^{\dagger}_i a_i \vec{S}_i \vec{S}_{i+1} a^{\dagger}_{i+1} a_{i+1}$. My guess is it has something to do with Wick's theorem, but I've only ever seen this in the context of QFT.

Thanks.

2. Jul 1, 2015