# One-dimensional fermions with incommensuration

Sen, Diptiman and Lal, Siddhartha (2000) One-dimensional fermions with incommensuration. In: Physical Review B, 61 (13). pp. 9001-9013.

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## 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 $\pi$, 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 $q/\delta{\rightarrow}0$, the number of states lying inside the q=0 gap is nonzero and equal to $2\delta/{\pi^2}$. Thus the limit $q{\rightarrow}0$ 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 and a renormalization group analysis to argue that similar results hold; as $q{\rightarrow}0$, there is a nontrivial density of states near zero energy. However, the limit $q{\rightarrow}0$ and q=0 give the same results near commensurate wave numbers which are different from \pi; for both free and interacting fermions we find that a nonzero value of q is necessary to close the gap. Our results for free fermions are applied 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 free fermions with incommensurate hopping by going up to higher orders in \delta.

Item Type: Journal Article Copyright of this article belongs to American Physical Society. Division of Physical & Mathematical Sciences > Centre for Theoretical Studies 30 Jun 2006 19 Sep 2010 04:29 http://eprints.iisc.ernet.in/id/eprint/7713