Properties of a one-dimensional metallic system in the third-order renormalization-group approximation

A one-dimensional system of electrons with interaction parameters $g_1$ and $g_2$ is studied by applying the third-order renormalization-group technique. For $g_1>0\mathrm{}$ our result is essentially the same as that obtained previously in the first-order scaling. For $g_1<0\mathrm{}$ the invariant coupling $\frac{{g^{\prime }}_1}{\pi v}\simeq -0.87\mathrm{}$ is considerably weaker than that $\frac{{g^{\prime }}_1}{\pi v}=-2\mathrm{}$ in the second-order renormalization obtained by Menyhárd and Sólyom in the low-frequency limit. The phases of the ground state in the present approximation are the same as those predicted in the second-order renormalization. The critical exponents for response functions which describe various critical fluctuations have also been calculated.