Dynamic correlations of the spinless Coulomb Luttinger liquid

The dynamic density response function, structure factor, and spectral function of a Luttinger liquid with Coulomb interaction are studied with the emphasis on the short-range electron correlations. The Coulomb interaction changes dramatically the density response function as compared to the case of the short-ranged interaction. The coordinate dependence of the density response is smoothing with time, and the oscillatory structure appears. However, the spectral functions remain qualitatively the same. The dynamic structure factor contains the delta peak in the long-wave region, corresponding to one-boson excitations. In addition, the multi-boson-excitations band exists in the wave-number region near to 2k(F), where k(F) is the Fermi wave number. We develop a method to analyze the asymptotic behavior of the spectral functions near to the edges of the multi-boson-excitations band at zero temperature. The dynamic structure factor diverges at the edges of this band, while the dielectric function goes to zero there. Hence, a new collective charge mode appears in the one-dimensional electron liquid as a consequence of short-range dynamic correlations. The mode energy goes to zero at 2k(F), which means that the 2k(F) mode is soft.