Abstract
The phase diagram for the interacting fermions in weak coupling is described by the perturbative renormalization group (RG) equations. However, these non-linear differential equations are not analytical, and the relevancy of couplings is difficult to tell in the numerical results. The scaling Ansatz we propose here is a breakthrough not only to predict the scale of the energy gap but also to classify the relevant couplings by building up a hierarchy of them. Applying the scheme to the two-leg ladder, we solve the long-standing phase problem and show that the mismatch of velocity leads no quantum phase transition. Further we investigate the influence of electron-phonon interactions in the fermionic ladders and find that the scaling Ansatz predicts a new type of superconductivity, Coulomb superconductors, emerging from repulsive interactions in the electron pair. In iron-pnictides superconductors, the inter-electron couplings are dominant and drive the electron-phonon couplings to become relevant, which is called the dressed Coulomb type. The scaling Ansatz also explains the anomalous isotope effect in experiments and predicts an extended s-wave pairing in spin gaps.