Topological Edge-State Manifestation of Interacting 2D Condensed Boson-Lattice Systems in a Harmonic Trap

Abstract

In this Letter, it is shown that interactions can facilitate the emergence of topological edge states of quantum-degenerate bosonic systems in the presence of a harmonic potential. This effect is demonstrated with the concrete model of a hexagonal lattice populated by spin-one bosons under a synthetic gauge field. In fermionic or noninteracting systems, the presence of a harmonic trap can obscure the observation of edge states. For our system with weakly interacting bosons in the Thomas-Fermi regime, we can clearly see a topological band structure with a band gap traversed by edge states. We also find that the number of edge states crossing the gap is increased in the presence of a harmonic trap, and the edge modes experience an energy shift while traversing the first Brillouin zone which is related to the topological properties of the system. We find an analytical expression for the edge-state energies and our comparison with numerical computation shows excellent agreement.PACS numbers: 37.10.Jk, 37.10.Gh, 34. Optical lattice experiments offer the possibility of simulating atomic crystal structures, creating a clean and well-controlled environment for probing many-body physics concepts and phenomena. In recent years, lattices reassembling the Hofstadter [1-3] and Haldane [4] models were created in optical lattice setups. A central focus in such systems lies in the manifestation of surface states which is a result of the nontrivial bulk topological properties. In topological insulators [5,6], for instance, electron excitations form a Fermi sea and provided the Fermi level is within the band gap the near-equilibrium dynamics is dominated by the states localized either at the sharp boundaries or at the interface with a system of a different topology. However, unlike solids, ultracold atoms are typically confined by a harmonic trapping potential. The influence of the harmonic trap on the band structure and its topology is therefore of significant importance.Theoretical studies suggest the presence of a confining potential can modify both the bulk and edge energy spectrum significantly, leading not only to a change in group velocity of edge modes but also to an emergence of additional edge states [7,8], their disappearance [9], or localization and a shrinking of the bulk region [8,10]. Possible ways of overcoming these difficulties include inducing topological interfaces [11,12] or creating so-called box traps [13][14][15]. Creation of such trapping potentials represents a separate challenge and cost for experimental setup. The role of mean field interactions in the Haldane boson model was also considered in Ref. [16] where it was shown that the bulk gap can close when the harmonic trap is taken into account. We, however, show that an interacting gas of spin-one bosons prepared in a polar ground state on a two-dimensional lattice can have a clear gap in the energy spectrum of the spin-±1 excitations. Furthermore the gap is crossed by edge states that reflect the topological structure of the lattice.Advances in t...

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