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arXiv:1008.4684 · Physical Review A

Flat bands, Dirac cones, and atom dynamics in an optical lattice

V. Apaja, M J Hyrkäs, M. Manninen

Published Oct 13, 2010• 163 citations
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Abstract

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.

DOI
10.1103/physreva.82.041402

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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.
Using ultracold atoms trapped in an optical lattice, we form a line-centered-square lattice in the condensed-matter physics, where a crossover from massive to massless Dirac fermion behavior can be easily achieved by tuning the laser intensities. The present Dirac fermions satisfy a three-component quantum equation for pseudospin-1 fermions, resulting in a single Dirac cone in the energy spectrum, a flat band touching at the Dirac point, and a vanishing Berry's phase. Interestingly, the massless Dirac fermions here may exhibit an all-angle Klein tunneling; i.e., the barrier is completely transparent for all incident angles.
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.
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.
Abstract We investigate Bloch–Zener oscillations and mean-field Bloch bands of a Bose–Einstein condensate (BEC) in a Lieb optical lattice. We find that the atomic interaction will break the point group symmetry of the system, leading to the destruction of the Dirac cone structure, while the flat band is preserved on the highly symmetric lines. Due to the nonlinear effect, a tubular band structure with a flat band will appear in the system. Furthermore, comparing with that the tight-binding (TB) model fails to describe the interacting bosonic systems in the honeycomb lattice, we show that the TB model is applicable to study the nonlinear energy band structures for the Lieb lattice. In addition, we show that the loop structure can be determined by the observation of the chaos of the state in the Bloch–Zener oscillations.
We consider the fate of the Dirac points in the spectrum of a honeycomb optical lattice in the presence of a harmonic confining potential. By numerically solving the tight binding model, we calculate the density of states and find that the energy dependence can be understood from analytical arguments. In addition, we show that the density of states of the harmonically trapped lattice system can be understood by application of a local density approximation based on the density of states in the homogeneous lattice. The Dirac points are found to survive locally in the trap as evidenced by the local density of states. Furthermore, they give rise to a distinct spatial profile of a noninteracting Fermi gas.
Cold atoms in optical lattices allow for accurate studies of many body dynamics. Rapid time-dependent modifications of optical lattice potentials may result in significant excitations in atomic systems. The dynamics in such a case is frequently quite incompletely described by standard applications of tight-binding models (such as, e.g., Bose-Hubbard model or its extensions) that typically neglect the effect of the dynamics on the transformation between the real space and the tight-binding basis. We illustrate the importance of a proper quantum mechanical description using a multiband extended Bose-Hubbard model with time-dependent Wannier functions. We apply it to situations directly related to experiments.