Reexamination of the third-order renormalization-group calculation for a one-dimensional interacting Fermi system

A one-dimensional interacting Fermi system with weak-coupling parameters g₁ (<0) and g₂ is reexamined in the third-order renormalization-group approach. Our result indicates that the value of the invariant coupling g₁^' varies, depending on whether the limit ω→0, $q=0$ or ω=0, q→0 is taken. For ω→0, $q=0$, the invariant coupling g₁^' scales to g₁^'πv-1.24, and for ω=0, q→0, it scales to g₁^'πv-0.716. We also argue that the invariant coupling g₁^' obtained from neither limit is appropriate for calculating the critical exponents of various response functions.