Hierarchy of relevant couplings in perturbative renormalization group transformations

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 nonlinear 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 nontrivial 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.

Figure 1

(Color online) RG flows of the eight couplings in the doped two-leg ladder. The initial couplings are generated from the on-site interaction $U/t={10}^{-5}$ and the velocity ratio $v_2/v_1=7.8$. We only plot the flows within the perturbative regime where all couplings $g_i<1$.

Figure 2

(Color online) Exponents extracted from the numerical solutions for the RG equations at different velocity ratios $v_2/v_1$ in the scaling regime. For small velocity ratios, all couplings except $f_{12}^{\sigma }$ are equally dominant with exponents ${\gamma }_i=1$. As the velocity ratio increases, the relevance of the couplings can be classified by the exponents accordingly.

Figure 3

(Color online) RG flows up to two-loop order for the charge sector in the one-dimensional Hubbard model.