Order in metallic chains. I. The single chain

Abstract

The nature of the order as $T\ensuremath{\rightarrow}0$ 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 ${g}_{1}=\ensuremath{-}\frac{6}{5\ensuremath{\pi}{\ensuremath{\nu}}_{F}}$, 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 $\ensuremath{\omega}\ensuremath{\rightarrow}0$ 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 ${g}_{1}=\ensuremath{-}\frac{6}{5\ensuremath{\pi}{\ensuremath{\nu}}_{F}}$, 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 ${g}_{3}\ensuremath{\ne}0$ to the region ${g}_{1}<0$ and ${g}_{1}\ensuremath{-}2{g}_{2}<|{g}_{3}|$.