The bosonization procedure works equally well for fermions with spin, thus providing us with an essentially nonperturbative approach to one-dimensional metals. The foundations of this approach were laid in the late 1970s and early 1980s by various authors (see the review article by Brazovsky and Kirova (1984) and references therein).

The bosonization approach is based on the fact that coherent excitations in onedimensional interacting systems are not renormalized electrons, but collective excitations – bosons. For spinless fermions there is only one branch of collective excitations – charge density waves. For fermions with spin, another branch appears which represents spin density waves. Elementary excitations in the charge sector carry charge ±e and spin 0; excitations in the spin sector are neutral and carry spin 1/2. Since the different branches have different symmetry properties, one can expect them to have quite different spectra. This can even go to such extremes that one branch has a gap and the other does not. It is only natural under such circumstances that electrons, which carry quantum numbers from both the spin and charge sectors, cannot propagate coherently. Roughly speaking, the parts of the electron containing different degrees try to tear it in pieces; it is customary to call this phenomenon spin-charge separation. Empirically this loss of coherence is revealed as an absence of a quasi-particle pole in the single-electron Green's function, an effect whose existence is supported by experimental observations (see discussion at the end of the chapter).

All phenomenona described above (the spin-charge separation and existence of excitations with fractions of quantum numbers) can be described by the model considered below.