Correspondence from Discrete to Continuum Models

Abstract

We have seen in the preceding chapter that the bosonization technique is a very powerful tool for studying one-dimensional electron systems. We have uncovered with it the properties of a continuous set of models showing critical behavior and governing the low-energy physics of a wide region of the parameter space of all the theories. The very fact of describing a number of critical properties is actually what justifies the study accomplished in the continuum limit. One may think that the assumption of an infinite linear dispersion relation for the electrons is a crucial step in the rigorous proof of the boson-fermion correspondence and wonder to what extent the results obtained by means of the bosonization program may apply to more realistic models. In particular, any model of fermions hopping on a lattice must have a compact dispersion relation of the kind shown in Fig. 3.1. There is empirical evidence, though, that the presence of the lattice does not change essentially the low-energy properties and that the Luttinger liquid paradigm continues applying to a wide class of discrete models. Some of these can be studied analytically by means of the Bethe ansatz technique, which provides exact results for the energy spectrum and some thermodynamic quantities. The case of the Hubbard model, for instance, is treated with detail in Chap. 10. From the exact resolution of the models one also gets information about the mapping to the Luttinger liquid universality class. This amounts to the knowledge of the parameters which characterize the model within the line of critical points obtained in Chap. 3.