Order in metallic chains. I. The single chain

The nature of the order as T→0 in the one-dimensional interacting electron gas is investigated. We consider both low-momentum-transfer (g₂ and g₄) and large-momentum-transfer (g₁ and g₃) electron-electron interactions, focusing upon the particular value of g₁ for which Luther and Emery have recently found an exact solution. For this value of g₁=-65πνF, 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 ω→0 for certain values of g₂. 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 g₂. 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 (g₄) does not change the response behavior qualitatively. For the large-momentum-transfer process between electrons on the same side of the Fermi "surface" (g₃), we may solve the problem exactly for g₂=0 and g₁=-65πνF, 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 g₃≠0 to the region g₁<0 and g₁-2g₂<|g₃|.