Correlated Fermions in a One-Dimensional Quasiperiodic Potential

Abstract

We study analytically one-dimensional interacting spinless fermions in a Fibonacci potential. We show that the effects of the quasiperiodic modulation are intermediate between those of a commensurate potential and a disordered one. The system exhibits a metal-insulator transition whose position depends both on the strength of the correlations and on the position of the Fermi level. Consequently, the conductivity displays a power-law-like size and frequency behavior characterized by a nontrivial exponent. PACS numbers: 71.30. + h, 61.44.Br, Since the discovery of quasicrystals in 1984 by Shechtman et al. [1], the electronic properties of quasiperiodic systems have been intensively studied. These metallic alloys are notably characterized by a low electrical conductivity s which increases when either temperature or disorder increases [2]. The very low temperature behavior of s is still an open question and depends on the materials. For example, in AlCuFe [3] and AlCuRu [4], a finite conductivity at zero temperature is expected, whereas recent results [5] seem to confirm a Mott's variable range hopping mechanism for i-AlPdRe down to 20 mK.Many theoretical works have attempted to understand how the quasiperiodic order could induce such exotic behaviors. In particular, the case of independent electrons in one-dimensional (1D) systems has been deeply investigated for different structures (Harper model, Fibonacci chain, . . . [6,7]), giving rise to singular continuous spectra with an infinite number of gaps. Moreover, the corresponding eigenstates are neither extended nor localized but critical, and are known to be responsible of anomalous diffusion [8,9]. For higher dimensional systems, similar studies had also displayed complex and intricated spectra, with analogous characteristics of the electronic states [10-13]. However, given the complexity of these problems due to the geometry alone, the interactions between electrons have often been neglected. Even in 1D incommensurate structures, few results have been obtained [14][15][16][17][18].In this Letter, we investigate the effect of the interactions considering a Hubbard-like model for spinless fermions embedded in a Fibonacci potential. We take the correlations into account using a bosonization technique whereas the quasiperiodicity is treated perturbatively. Using a renormalization group approach, we show that the quasicrystalline system displays a metal-insulator transition (MIT) induced by the interactions. The corresponding critical MIT point is found to be strongly dependent on the Fermi level. In marked contrast with the simple cases of disordered or commensurate potentials, the quasiperi-odicity leads to a power-law-like dependence of the conductivity either in size or frequency with an exponent depending both on the interactions and on the position of the Fermi level. Though our analysis is performed for a Fibonacci potential, we stress that these results can be extended to any potential having a nonflat dense Fourier spectrum.Let us consider a model of interac...

Stability of the superfluid state in a disordered one-dimensional ultracold fermionic gas

“…(1), correlations of the potential studied in [18] are very strong lim n→∞ V (n)V (0) = 0. However different reasons prevent from a direct comparison between these models: a) in [18] only spinless fermions are considered, b) for λ ≪ 1, the limit in which the formalism of [18] is applicable, our system is in the metallic region with properties almost identical as those of a periodic potential, c) for no interactions the spectrum of the Fibonacci chain is singular continuous for all λ. This leads to eigenstates that are power-law localized and quantum superdiffusion [20].…”

“…The perturbative treatment of [18] showed that the critical disorder depends on both the strength of the interactions and the position of the Fermi level. We note that, as in Eq.…”

“…Bosonization techniques combined with a renormalization group analysis were employed in [18] to investigate the effect of interactions in another 1D quasiperiodic system, the Fibonacci chain [20]. The perturbative treatment of [18] showed that the critical disorder depends on both the strength of the interactions and the position of the Fermi level.…”

Interacting particles at a metal-insulator transition

“…Whereas for repulsive interactions the ground state remains localized, we find a region of extended states for attractive interaction due to the interplay between interaction and quasiperiodic potential. The behavior may be described by a weak-coupling renormalization group (RG) treatment [31]. Thus, the physics of the model is dominated by whether the density is commensurate or incommensurate and only then by interaction effects.…”

Quasiperiodic spin chains in a magnetic field

“…Also, recent investigations of quasi-periodicity involving either the couplings [11] or the magnetic field [12], have been addressed using Abelian bosonization along with RG and numerical techniques. Here we focus on the combined effect of a quasi-periodic exchange modulation under a uniform magnetic field.…”