One-dimensional fermions with incommensuration close to dimerization

We study a model of fermions hopping on a chain with a weak incommensuratio n close to dimerization; both q, the deviation of the wave number from pi, and delta, the strength of the incommensuration, are assumed to be small. F or free fermions, we show that there are an infinite number of energy bands which meet at zero energy as q approaches zero. The number of states lying inside the q = 0 gap remains nonzero as q/delta --> 0. Thus the limit q -- > 0 differs from q = 0, as can be seen clearly in the low-temperature speci fic heat. For interacting fermions or the XXZ spin-(1/2) chain, we use boso nization to argue that similar results hold. Finally, our results can be ap plied to the Azbel-Hofstadter problem of particles hopping on a two-dimensi onal lattice in the presence of a magnetic field.