An exact solution of the one-dimensional electron gas on a lattice is given for the special case in which two of the four coupling constants have particular values. It is shown that umklapp scattering has the same effect on the charge-density waves as backward scattering across the Fermi "surface" has on the spin-density waves and that the method of Luther and Emery can be used to solve this more general problem. For repulsive electron-electron interactions the umklapp scattering produces a gap in the charge-density wave spectrum and this appears for low-lying excitations when there is a half-filled band. The use of renormalization-group scaling to solve for general values of the coupling constants is discussed.
The energy spectrum of the one-dimensional electron gas, solved by Luther and Emery for a particular attractive interaction strength
U of the backward scattering across the Fermi surface, has been calculated for small deviations from this particular limit. The energy spectrum shows a gap at the Fermi energy, whose magnitude is very sensitive to the interaction
U. The density of states shows a gap at zero temperature, which is smeared out by the quasiparticle lifetime for $T\ensuremath{\ne}0$.
Phase transitions of a quasi-one-dimensional Fermi gas are studied by the use of the renormalization group method for the fermion system without the bosonization. Intrachain backward and forward scatterings, pair electron hopping between different chains and exchange scattering of electrons in different chains are taken account of. Pair electron hopping gives rise to the singlet and triplet superconductor phase transitions. Electron exchange scattering gives rise to the spin density wave and charge density wave transitions. Calculation is made by the use of the one-dimensional dispersion. Following Menyhárd's method (Ref. 1) for the model with interchain backward scattering only, the change of the fluctuation from the one-dimensional region to the three-dimensional region is described. The transition temperatures are given by the bare couplings at high temperatures.