Order in metallic chains. I. The single chain

The nature of the order as $T\to 0$ in the one-dimensional interacting electron gas is investigated. We consider both low-momentum-transfer ($g_2$ and $g_4$) and large-momentum-transfer ($g_1$ and $g_3$) electron-electron interactions, focusing upon the particular value of $g_1$ for which Luther and Emery have recently found an exact solution. For this value of $g_1=-\frac{6}{5\pi {\nu }_F}$, we calculate the low-frequency behavior of the zero-temperature response functions explicitly. For the singlet superconducting and charge-density-wave responses, our results are consistent with those of Luther and Emery, implying that these response functions are divergent as $\omega \to 0$ for certain values of $g_2$. For the triplet superconducting and spin-density-wave responses, however, our results differ from those of Luther and Emery, as we show explicitly that these responses have a gap for low frequencies, as was predicted by Lee. Furthermore, they do not diverge at the gap edge for any value of $g_2$. We also consider the effect of interactions between electrons on the same side of the Fermi "surface," and find that the low-momentum-transfer process of that type ($g_4$) does not change the response behavior qualitatively. For the large-momentum-transfer process between electrons on the same side of the Fermi "surface" ($g_3$), we may solve the problem exactly for $g_2=0$ and $g_1=-\frac{6}{5\pi {\nu }_F}$, and find that the ground state of the system exhibits only long-range charge-density-wave order. By a mapping onto the classical two-dimensional Coulomb gas problem, we may extend those results for $g_3\ne 0$ to the region $g_1<0$ and $g_1-2\mathrm{}{g}_2<\left|g_3\right|$.