One-dimensional fermions with incommensuration

Abstract

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 π, and δ, 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/δ→0, the number of states lying inside the q=0 gap is nonzero and equal to 2δ/π². 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-½ chain close to dimerization, we use bosonization and a renormalization group analysis 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 π. 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 δ.