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 $g_1 \mathrm{}(<0\mathrm{})\mathrm{}$ and $g_2$ is reexamined in the third-order renormalization-group approach. Our result indicates that the value of the invariant coupling $g_1^{\prime }$ varies, depending on whether the limit $\omega \to 0$, $q=0\mathrm{}$ or $\omega =0\mathrm{}$, $q\to 0$ is taken. For $\omega \to 0$, $q=0\mathrm{}$, the invariant coupling $g_1^{\prime }$ scales to $\frac{g_1^{\prime }}{\pi v}\simeq -1.24\mathrm{}$, and for $\omega =0\mathrm{}$, $q\to 0$, it scales to $\frac{g_1^{\prime }}{\pi v}\simeq -0.716\mathrm{}$. We also argue that the invariant coupling $g_1^{\prime }$ obtained from neither limit is appropriate for calculating the critical exponents of various response functions.