Some properties of the one-dimensional Fermi model

A model of one-dimensional fermions characterized by a large momentum-transfer interaction $g_1$ and a small momentum-transfer interaction $g_2$ is considered. Results obtained previously by the renormalization-group method are extended by studying the triplet Cooper-pair response function and uniform charge and spin response functions. The sign of $g_1$ dominates the behavior of the system. For $g_1>0\mathrm{}$, $g_1-2\mathrm{}{g}_2>0\mathrm{}$ the triplet Cooper response diverges as $T\to 0$ and is seen to dominate the singlet Cooper response which is also divergent. The uniform static spin susceptibility is nondivergent for $g_1>0\mathrm{}$ and diverges to zero for $g_1<0\mathrm{}$. The specific heat is found to be linear in temperature in all regions. These results, obtained by the second-order renormalization group, yield the correct qualitative results in regions where exact solutions are available.