The critical behavior of the BCS order parameter: a straightforward derivation

where the prefactor B is a non-universal coefficient and the exponent has the classical value α = 1/21. Then one may read that this is a standard result for any mean-field theory, although the student may wonder, why it does not follow straightforwardly from the model? Yet in the literature this outstandingly simple statement is obtained in a rather roundabout manner. Furthermore α and B are computed only in the weakcoupling limit ~ωD kBTc, where ωD is the Debye frequency. This is certainly an aesthetically not very pleasing situation and I doubt the student really wants to grind through the approximations just to get this simple result. The following lines show a little trick straightening out this situation. It will hopefully find its way to the textbooks. In the BCS theory the order-parameter ∆(T ) satisfies the non-linear integral equation2