Interactions and Mobility Edges: Observing the Generalized Aubry-Andr\'e Model

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Interactions and Mobility Edges: Observing the Generalized Aubry-André Model

Fangzhao Alex An, Karmela Padavić, Eric J. Meier, Suraj Hegde, Sriram Ganeshan, J. H. Pixley, Smitha Vishveshwara, and Bryce Gadway

Phys. Rev. Lett. 126, 040603 – Published 29 January 2021

Abstract

Using synthetic lattices of laser-coupled atomic momentum modes, we experimentally realize a recently proposed family of nearest-neighbor tight-binding models having quasiperiodic site energy modulation that host an exact mobility edge protected by a duality symmetry. These one-dimensional tight-binding models can be viewed as a generalization of the well-known Aubry-André model, with an energy-dependent self-duality condition that constitutes an analytical mobility edge relation. By adiabatically preparing low and high energy eigenstates of this model system and performing microscopic measurements of their participation ratio, we track the evolution of the mobility edge as the energy-dependent density of states is modified by the model’s tuning parameter. Our results show strong deviations from single-particle predictions, consistent with attractive interactions causing both enhanced localization of the lowest energy state due to self-trapping and inhibited localization of high energy states due to screening. This study paves the way for quantitative studies of interaction effects on self-duality induced mobility edges.

Received 15 July 2020
Revised 4 December 2020
Accepted 8 January 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.040603

Physics Subject Headings (PhySH)

Anderson localization Atom optics Bose-Einstein condensates Localization

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied PhysicsGeneral Physics

Authors & Affiliations

Fangzhao Alex An^1,*,†, Karmela Padavić^1,*, Eric J. Meier¹, Suraj Hegde², Sriram Ganeshan^3,4,‡, J. H. Pixley^5,§, Smitha Vishveshwara^1,∥, and Bryce Gadway ^1,¶

¹Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA
²Max-Planck Institute for Physics of Complex Systems, 01187 Dresden, Germany
³Physics Department, City College of the CUNY, New York, New York 10031, USA
⁴CUNY Graduate Center, New York, New York 10031, USA
⁵Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA

^*These authors contributed equally to this work.
^†Present address: Honeywell Quantum Solutions, Golden Valley, Minnesota 55422, USA.
^‡sganeshan@ccny.cuny.edu
^§jed.pixley@physics.rutgers.edu
^∥smivish@illinois.edu
^¶bgadway@illinois.edu

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Vol. 126, Iss. 4 — 29 January 2021

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Figure 1
The generalized self-dual Aubry-André model. (a) The generalized Aubry-André potential and lattice site energies of Eq. (2) shown for $ϕ = 0$ and tuning parameter $α = - 0.5$ , 0, 0.5, with corresponding distributions of lattice site energies $ϵ_{n}$ . (b) Calculated eigenenergies and participation ratios (PRs, in color) vs $α$ for a noninteracting model just below the critical quasiperiodicity strength at $Δ / J = 1.8$ ( $N = 51$ sites). Away from $α = 0$ , eigenstates localize at different energies, forming a mobility edge. Dashed black lines show analytically predicted energy values of the ME [Eq. (3)].
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Figure 2
Probing localization by adiabatic Hamiltonian evolution. (a) Cartoon of the experimental sequence. Atoms initially localized for $Δ / J = \infty$ are slowly loaded into an eigenstate of the GAA model at a final quasiperiodicity-to-tunneling ratio $Δ / J$ . Bar color relates to the normalized participation ratio [see color bar inset in (b)], for $α = 0$ and no interactions. Bottom: atomic momentum distributions, corresponding to populations in the synthetic lattice, of the ES for $α = 0$ in the localized regime ( $Δ / J = 4.2$ ), near the delocalization transition ( $Δ / J = 2.1$ ), and in the delocalized regime ( $Δ / J = 0.9$ ). (b) Numerically calculated PRs overlaid on the eigenenergies of the GAA model for $α = - 0.5$ , $ϕ = π$ , and $N = 201$ sites. High-energy states localize at larger quasiperiodicity strengths than low-energy states, highlighting the presence of the mobility edge of Eq. (3) (dashed black line). The colors of the energy curves relate to states’ $PR / N$ values, according to the inset color bar. (c) $PR / N$ vs $Δ / J$ for GS (open blue circles) and ES (yellow diamonds) under $α = - 0.5$ , 0, 0.5. Numerical curves incorporate the exact experimental tunneling ramp and assume a mean-field energy $U / J = 0.48$ ( $U / h = 300 Hz$ ) for the dashed curves and zero interactions ( $U / J = 0$ ) for the solid curves. Error bars in (c) denote one standard error of the mean.
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Figure 3
Localization phase diagram of the GS and ES. Critical quasiperiodicity values for the onset of GS and ES delocalization (filled circles and open diamonds), overlaid on the difference in normalized participation ratio ( $PR / N$ , with difference shown according to the inset colorbar) of the numerically calculated extremal eigenstates for a mean-field interaction $U = 0.48 J$ . The GS and ES transition “lines” do not coincide, indicating a mobility edge, and they cross away from $α = 0$ , indicating a shift due to atomic interactions. Vertical error bars denote the 90% confidence intervals of the fit used to determine the critical $Δ / J$ values [43].
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Physical Review Letters

Interactions and Mobility Edges: Observing the Generalized Aubry-André Model

Fangzhao Alex An, Karmela Padavić, Eric J. Meier, Suraj Hegde, Sriram Ganeshan, J. H. Pixley, Smitha Vishveshwara, and Bryce Gadway

Phys. Rev. Lett. 126, 040603 – Published 29 January 2021

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