Phase diagram and correlation exponents for interacting fermions in one dimension

A theory is given for the ground-state properties of short-ranged interacting spin-1/2 fermions moving in a periodic potential, based on a combination of renormalization-group ideas, the harmonic (Luttinger) liquid approach, and the symmetry properties of the Hubbard model, for a wide range of interactions and arbitrary fillings. For rational filling factors the outcome depends nontrivially on the parity of the order of the 2k_F umklapp scattering: even values fall into a universality class characterized by a separation of charge and spin degrees of freedom in the long-wavelength limit, while the action for the case of odd orders does not have this property, thus leading to distinct phase diagrams. It is argued that even when the long-wavelength action does exhibit a spin-charge separation it is not always an accurate approximation, particularly for the case of strong repulsion between electrons of the same spin. This leads to a classification of the possible ground states, associated correlation exponents, and phase transitions. The translational correlation exponents upon approaching insulator phases (existing at exact commensuration) by changing electron density can be related to the fractional charge of the solitons, using the Landauer rule.