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arXiv:1511.05685 · Physical review. B./Physical review. B

Dense gaps in the interacting Aubry-André model

Vieri Mastropietro

Published Jun 27, 2016• 12 citations
arXivHTMLPDF

Abstract

We consider the interacting Aubry-Andr\'e model describing fermions on a one-dimensional lattice with an incommensurate potential and a short-range many-body interaction. The single-particle spectrum has infinitely many gaps in the extended phase and at zero temperature is an insulator for almost all the chemical potentials. The many-body interaction has the effect that the gaps are strongly decreased or increased depending on the attractive or repulsive nature of the interaction, but even the smallest gaps remain open. The system is a band insulator for generic chemical potentials even in the presence of interaction, and a quantum phase transition is excluded at weak coupling.

DOI
10.1103/physrevb.93.245154

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We consider the interacting Aubry-Andr'e model describing fermions on a one-dimensional lattice with an incommensurate potential and a short-range many-body interaction. The single-particle spectrum has infinitely many gaps in the extended phase and at zero temperature is an insulator for almost all the chemical potentials. The many-body interaction has the effect that the gaps are strongly decreased or increased depending on the attractive or repulsive nature of the interaction, but even the smallest gaps remain open. The system is a band insulator for generic chemical potentials even in the presence of interaction, and a quantum phase transition is excluded at weak coupling.
We consider interacting electrons in a one-dimensional lattice with an incommensurate Aubry-André potential in the regime when the single-particle eigenstates are localized. We rigorously establish the persistence of ground state localization in the presence of weak many-body interaction, for almost all the chemical potentials. The proof uses a quantum many-body extension of methods adopted for the stability of tori of nearly integrable Hamiltonian systems and relies on number-theoretic properties of the potential incommensurate frequency.
The metal-insulator transition in one-dimensional fermionic systems with long-range interaction is investigated. We have focused on an excitation spectrum by the exact diagonalization technique in sectors with different momentum quantum numbers. At rational fillings, we have demonstrated gap opening transitions from the Tomonaga-Luttinger liquid to the Mott insulator associated with a discrete symmetry breaking by changing the interaction strength. Finite interaction range is crucial to have the Mott transition at a rational filling away from the half filling. It is consistent with the strong coupling picture where the Mott gap exists at any rational fillings with sufficiently strong interaction. The critical regions as a quantum phase transition are also investigated numerically. Nonanalytic behavior of the Mott gap is the characteristic in the weak coupling. It is of the order of the interaction in the strong coupling. It implies that the metal-insulator transition of the model is of the infinite order as a quantum phase transition at zero temperature. The fractal nature of the ground-state phase diagram is also revealed.
We study the influence of many-particle interaction in a system which, in the single particle case, exhibits a metal-insulator transition induced by a finite amount of onsite pontential fluctuations. Thereby, we consider the problem of interacting particles in the one-dimensional quasiperiodic Aubry-Andr\'e chain. We employ the density-matrix renormalization scheme to investigate the finite particle density situation. In the case of incommensurate densities, the expected transition from the single-particle analysis is reproduced. Generally speaking, interaction does not alter the incommensurate transition. For commensurate densities, we map out the entire phase diagram and find that the transition into a metallic state occurs for attractive interactions and infinite small fluctuations --- in contrast to the case of incommensurate densities. Our results for commensurate densities also show agreement with a recent analytic renormalization group approach.
We investigate the zero-temperature metal-insulator transition in a one-dimensional two-component Fermi gas in the presence of a quasiperiodic potential resulting from the superposition of two optical lattices of equal intensity but incommensurate periods. A mobility edge separating (low-energy) Anderson localized and (high-energy) extended single-particle states appears in this continuous-space model beyond a critical intensity of the quasiperiodic potential. To discern the metallic phase from the insulating phase in the interacting many-fermion system, we employ unbiased quantum Monte Carlo (QMC) simulations combined with the many-particle localization length familiar from the modern theory of the insulating state. In the noninteracting limit, the critical optical-lattice intensity for the metal-insulator transition predicted by the QMC simulations coincides with the Anderson localization transition of the single-particle eigenstates. We show that weak repulsive interactions induce a shift of this critical point towards larger intensities, meaning that repulsion favors metallic behavior. This shift appears to be linear in the interaction parameter, suggesting that even infinitesimal interactions can affect the position of the critical point.
Understanding the metal-insulator transition in disordered many-fermion systems, both with and without interactions, is one of the most challenging and consequential problems in condensed matter physics. In this paper, we address this issue from the perspective of the modern theory of the insulating state (MTIS), which has already proven to be effective for band and Mott insulators in clean systems. First, we consider noninteracting systems with different types of aperiodic external potentials: uncorrelated disorder (one-dimensional Anderson model), deterministic disorder (Aubry-Andr\'e Hamiltonian and its modification including next-nearest-neighbor hopping), and disorder with long-range correlations (self-affine potential). We show how the many-body localization tensor defined within the MTIS may be used as a powerful probe to discriminate the insulating and the metallic phases, and to locate the transition point. Then, we investigate the effect of weak repulsive interactions in the Aubry-Andr\'e Hamiltonian, a model which describes a recent cold-atoms experiment. By treating the weak interactions within a mean-field approximation we obtain a linear shift of the transition point towards stronger disorder, providing evidence for delocalization induced by interactions.
We study many-body localization (MBL) for interacting one-dimensional lattice fermions in random (Anderson) and quasiperiodic (Aubry-Andre) models, focusing on the role of interaction range. We obtain the MBL quantum phase diagrams by calculating the experimentally relevant inverse participation ratio (IPR) at half-filling using exact diagonalization methods and extrapolating to the infinite system size. For short-range interactions, our results produce in the phase diagram a qualitative symmetry between weak and strong interaction limits. For long-range interactions, no such symmetry exists as the strongly interacting system is always many-body localized, independent of the effective disorder strength, and the system is analogous to a pinned Wigner crystal. We obtain various scaling exponents for the IPR, suggesting conditions for different MBL regimes arising from interaction effects.