Mean Field Theory for Fermions/Bosons

[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.

 

[eljose79]

Hello,

Thank you for your post. The approximation you mentioned is indeed related to mean field theory, specifically the Hartree-Fock approximation. This is a common technique used in statistical physics to simplify calculations and make them more tractable.

In the case of the first equation you mentioned, the mean field approximation is used to approximate the interaction term in the Hamiltonian, which is the sum over all possible pairs of bosonic annihilation and creation operators. This approximation assumes that each boson interacts only with the average value of the other boson, rather than with the specific values of the other boson. This simplifies the calculation and makes it easier to solve.

To generalize this approximation to other operators, as in the second equation you mentioned, the same principle applies. The interaction term is approximated by the average value of the operators involved, rather than the specific values. This can be extended to other operators, as long as the interaction term can be written in terms of the average values of the operators.

You are correct in thinking that this approximation is related to Wick's theorem, which is a useful tool in calculating averages of products of operators. However, in mean field theory, the approximation is applied to the interaction term, while Wick's theorem is used to calculate the averages themselves.

I hope this helps to clarify the concept of mean field theory and its application in approximating interaction terms in statistical physics. If you have any further questions, please feel free to ask.