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Quantum effects in one-dimensional optical flat-band lattices

M J Hyrkäs

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

I numerically simulated one-dimensional lattice systems describable by the Hubbard-model and containing a flat-band. I studied the manner in which particles initially held in place by a parabolic confinement potential are dispersed across the lattice after the potential is removed. I also studied currents flowing within one-dimensional Hubbard-rings. In both of these cases I observed that the flat dispersion relation of the flat-band manifests itself as the immobility of the particles occupying it. Particles occupying flat-band states do not disperse even after the removal of any external confinement, neither do they contribute to a current within a ring.

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I numerically simulated one-dimensional lattice systems describable by the Hubbard-model and containing a flat-band. I studied the manner in which particles initially held in place by a parabolic confinement potential are dispersed across the lattice after the potential is removed. I also studied currents flowing within one-dimensional Hubbard-rings. In both of these cases I observed that the flat dispersion relation of the flat-band manifests itself as the immobility of the particles occupying it. Particles occupying flat-band states do not disperse even after the removal of any external confinement, neither do they contribute to a current within a ring.
We study atoms trapped with a harmonic confinement in an optical lattice characterized by a flat band and Dirac cones. We show that such an optical lattice can be constructed which can be accurately described with the tight-binding or Hubbard models. In the case of fermions the release of the harmonic confinement removes fast atoms occupying the Dirac cones while those occupying the flat band remain immobile. Using exact diagonalization and dynamics we demonstrate that a similar strong occupation of the flat band does not happen in the bosonic case and furthermore that the mean-field model is not capable of describing the dynamics of the boson cloud.
The difference between boson and fermion dynamics in quasi-one-dimensional lattices is studied by calculating the persistent current in small quantum rings and by exact simulations of the time evolution of the many-particle state in two cases: expansion of a localized cloud and collisions in a Newton's cradle. We consider three different lattices which in the tight-binding model exhibit flat bands. The physical realization is considered to be an optical lattice with bosonic or fermionic atoms. The atoms are assumed to interact with a repulsive short-range interaction. The different statistics of bosons and fermions lead to different dynamics. Spinless fermions are easily trapped in the flat-band states due to the Pauli exclusion principle, which prevents them from interacting, while bosons are able to push each other out from the flat-band states.
Symmetry-protected ideal flat bands in one-dimensional (1D) Hermitian lattices are populated by compact localized states (CLS) - a special class of localization with wavefunctions confined within a small region. In this work, we discover that the non-Hermitian skin effect (NHSE) can appear in a flat band. Unlike conventional NHSEs for dispersive bands that are protected by nontrivial point-gap topology, the flat band remains a point on the complex-energy plane and is therefore always topologically trivial. We found that, intriguingly, the flat-band skin effect (FBSE) is associated with the non-trivial spectral topology of the dispersive bands enclosing the flat band on the complex-energy plane, so it only emerges within a finite range of non-Hermitian parameters and can counterintuitively disappear at large non-Hermiticity. Moreover, the gaps between the flat and the dispersive bands can close at higher-order exceptional points under both periodic and open boundary conditions. The flat-band wavefunctions are discontinuous in quantum distance across these exceptional points, signifying that the gap-closing is singular. The FBSE was experimentally observed in a non-Hermitian mechanical lattice. Our work reveals flat-band phenomena unique to non-Hermitian systems and highlights new possibilities in quantum geometry and localization control.
We study properties of ultracold bosonic atoms in one-, two-, and three-dimensional optical lattices by large scale quantum Monte Carlo simulations of the Bose-Hubbard model in parabolic confining potentials. Our results indicate that local properties of the atoms can be accessed by probing the system's response to local potential perturbations. Furthermore, we show how the formation of Mott insulating regions is reflected in the momentum distribution of the atoms, amenable to experimental detection. We disprove previous claims concerning the relevance of fine structure in the momentum distribution function. Furthermore, we discuss limitations of local density approximations for confined systems, and demonstrate the absence of quantum criticality due to the inhomogenous potential. Instead, we show that quantum critical behavior can be observed in flat confining potentials. Our results indicate that the experimental detection of the Mott transition in moderately sized optical lattices would be significantly eased in flat confinement potentials.
We consider translationally invariant tight-binding all-bands-flat networks which lack dispersion. In a recent work [C. Danieli et al., Phys. Rev. B 104, 085131 (2021)], we identified the subset of these networks that shows nonlinear caging in the presence of Kerr-like local nonlinearities, i.e., it preserves nonexcited network sites and therefore keeps compact excitations compact. Here we replace nonlinear terms by Bose-Hubbard interactions and study quantum caging. We identify the quantum caging conditions that are related to the nonlinear caging conditions and that guarantee the existence of an extensive set of conserved quantities in any lattice dimension, as first revealed in Tovmasyan et al. [Phys. Rev. B 98, 134513 (2018)] for a set of specific networks. Consequently, transport is realized through moving pairs of interacting particles that break the single-particle caging. We further prove the existence of degenerate energy renormalized compact states for any finite number MMM of participating particles in one- and higher-dimensional lattice---results that explain and generalize previous observations for two particles on a diamond chain [Vidal et al., Phys. Rev. Lett. 85, 3906 (2000)].
Inspired by the growing interest in probing many-body phases in novel two-dimensional lattice geometries we investigate the properties of cold atoms as they could be observed in an optical Lieb lattice. We begin by computing Wannier functions localised at individual sites for a realistic experimental setup, and determining coefficients for a Hubbard-like model. Based on this, we show how experiments could probe the robustness of edge states in a Lieb lattice with diagonal boundary conditions to the effects of interactions and realise strongly correlated many-body phases in this geometry. We then generalise this to interacting particles in a half-filled 1D Lieb ladder, where excitations are dominated by flat band states. We show that for strong attractive interactions, pair correlations are enhanced even when there is strong mixing with the Dirac cone. These findings in 1D raise interesting questions about the phases in the full 2D Lieb lattice which we show can be explored in current experiments.