After playing with Green functions for some time, we found that the perturbation method using Feynman diagrams is quite difficult to calculate, especially for high order terms. It seems that one can use mean field theory to get some qualitative results. However, if the fluctuations locally are important, mean filed theory will fail to work. I'm not sure how it works on our problem. But my supervisor noticed that the calculation can not converge properly. Another way to solve this problem may be renormalization group method, which is claimed quite powerful. However, I have no experience on this subject, so I'm reading some papers on that which date back to 1970s.
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