Order in metallic chains. I. the single chain

The nature of the order as T0 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 = -65F, 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 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 = -65F, 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|.