  The phase diagram for the interacting fermions in weak coupling is described
by the perturbative renormalization group equations. Due to the lack of
analytic solutions for these coupled non-linear differential equations, it is
rather subtle to tell which couplings are relevant or irrelevant. We propose a
powerful classification scheme to build up the hierarchy of the relevant
couplings by a scaling Ansatz found numerically. To demonstrate its superiority
over the conventional classification for the relevant couplings, we apply this
scheme to a controversial phase transition in the two-leg ladder and show that
it should be a non-trivial crossover instead. The scaling Ansatz we propose
here can classify the relevant couplings in hierarchical order without any
ambiguity and can improve significantly how we interpret the numerical outcomes
in general renormalization group methods.
