One-dimensional fermions with incommensuration

We study the spectrum of fermions hopping on a chain with a weak incommensuration close to dimerization; both q, the deviation of the wave number from pi, and delta, the strength of the incommensuration, are small. For free fermions, we use a continuum Dirac theory to show that there are an infinite number of bands which meet at zero energy as q approaches zero. In the limit that the ratio q/ \delta --> 0, the number of states lying inside the q=0 gap is nonzero and equal to 2 \delta /\pi^2. Thus the limit q --> 0 differs from q=0; this can be seen clearly in the behavior of the specific heat at low temperature. For interacting fermions or the XXZ spin-1/2 chain close to dimerization, we use bosonization to argue that similar results hold; as q --> 0, we find a nontrivial density of states near zero energy. However, the limit q --> 0 and q=0 give the same results near commensurate wave numbers which are different from pi. We apply our results to the Azbel-Hofstadter problem of electrons hopping on a two-dimensional lattice in the presence of a magnetic field. Finally, we discuss the complete energy spectrum of noninteracting fermions with incommensurate hopping by going up to higher orders in delta.