rigorous definition of coherence length at mean field theory

so as far as I know, when we are doing mean field theory, in qft, we expand a action of a theory around a classical solution. so we find a classical solution, than we add quantum mechanical corrections to that solution.

also the coherence length in general defined as a inverse of a energy gap, so for gappless systems mean field theory does not has much meaning.

so, i think the mean field theory is valid, only at distances bigger than coherence legnth. I think the reason is this, when you expand the action around the classical solution at any order you want, you add corrections to that classical solution, and every higher order terms, you add decreases the length scale where is your theory is correct.

for example if I only add classical solution, than the theory will be valid only at very high length scales, if i make 2nd order correction, than it will be correct also at lower length scales and every term i add decreases the smallest valid length. however, however, the coherence length i think is the limit of that lowest possible length scale. for example if coherence length is infinite then your theory wont be valid no matter how much terms you add. I don't totally get this, but i think its the mean length of fluctuations, and if this length is infinite, it doesn't matter how many terms you add.

the thing, what is the actual formula for the coherence lengh for a general mean field theory.

• not sure I follow your statement that for gapless systems mean field theory does not have much meaning. Bose-Einstein condensates formed by atomic gases are gapless but their behaviors as weakly interacting superfluid is very well described by mean field theory. – wcc Mar 29 at 23:31