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

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.

Figure 1

(a) Lattice structure, determined by the primitive lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ of length $a$. Closed and open circles denote the triangular sublattice sites $A$ and $B$, respectively. We consider a strip geometry with open (periodic) boundary conditions in $x(y)$ direction. (b) The effective potential $V_{\mathrm{eff}}$ (red solid line) and the density profile of the ground state $U\overline{\rho }$ (black cross marks). These quantities are obtained numerically and are in excellent agreement with the Thomas-Fermi profile. Here $\lambda =w/2$, $M{\omega }^2{a}^2=0.02w$ and $U{N}_{\mathrm{str}}=800w$, where $N_{\mathrm{str}}$ is the total number of bosonic particles in a strip denoted by the (green) dashed lines in (a). The Thomas-Fermi radius is $x_{\mathrm{TF}}\approx 30a$.

Figure 2

Left: The energy spectrum corresponding to the spin-$1(-1)$ excitations. The solid lines correspond to numerical results, while the dashed line corresponds to the analytical expression Eq. (13) for a particular mode. Right: The effective potential in units of $w$ near the Thomas-Fermi radius $x_{\mathrm{TF}}\approx 30a$ (red) and the wave functions of the eigenmodes in arbitrary units at $k=2\pi$ (blue). All parameters are the same as in Fig. 1: $\lambda =w/2$, $M{\omega }^2{a}^2=0.02w$ and $U{N}_{\mathrm{str}}=800w$.