Topological edge states and Aharanov-Bohm caging with ultracold atoms carrying orbital angular momentum

We show that bosonic atoms loaded into orbital angular momentum $l=1$ states of a lattice in a diamond-chain geometry provide a flexible and simple platform for exploring a range of topological effects. This system exhibits robust edge states that persist across the gap-closing points, indicating the absence of a topological transition. We discuss how to perform the topological characterization of the model with a generalization of the Zak's phase and we show that this system constitutes a realization of a square-root topological insulator. Furthermore, the relative phases arising naturally in the tunneling amplitudes lead to the appearance of Aharanov-Bohm caging in the lattice. We discuss how these properties can be realized and observed in ongoing experiments.

Figure 1

(a) Schematic representation of the considered diamond chain. The inset shows, for harmonic oscillator traps, the dependence of the relative value of $J_2$ and $J_3$ on the intersite separation $d$, expressed in units of $\sigma =\sqrt{\hslash /\left(m\omega \right)}$. (b) Energy spectrum of the diamond chain with OAM $l=1$ states. The left plot shows the band structure computed for $d=3.5\sigma$, corresponding to $J_3/J_2=1.67$ (black solid line) and $d=6\sigma$, corresponding to $J_3/J_2=1.13$ (red dotted line). In the right plot, the corresponding exact diagonalization spectra of a diamond chain of $N_c=20$ unit cells are shown.

Figure 2

Schematic representation of the tight-binding models obtained through the different mappings. (a) $H^{+}$ diamond chain obtained after applying the basis rotation $\{|B_i,\pm \rangle ,|C_i,\pm \rangle \}\to \{|F_i,\pm \rangle ,|D_i,\pm \rangle \}$ to the original OAM $l=1$ diamond chain. (b) Modified SSH model obtained after applying the basis rotation $\{|D_i,+\rangle ,|F_i,-\rangle \}\to \left\{|G_i,\pm \rangle \right\}$ to the $H^{+}$ chain. $N_c$ is the total number of unit cells.

Figure 3

Density profiles of numerically obtained eigenstates for a diamond chain of $N_c=10$ units cells and intersite separation $d=6\sigma$, corresponding to $J_3/J_2=1.13$. (a) Two degenerate edge states. (b) Two states of the flat band. (c) The two degenerate ground states.

Figure 4

AB caging in a diamond chain of $N_c=5$ unit cells. (a) Snapshots at different times of the density profiles corresponding to the time evolution of a wave packet initially prepared in the state $|A_3,+\rangle$ in the perfect caging limit $J_2/J_3=1$. (b) Time evolution of the population of the same initial state as in (a) (black solid lines) and the total population of the cage (red dotted lines) for $J_3/J_2=1$ (left panel) and 1.1 (right panel).