This thesis presents theoretical work devoted to the manifestation of the edge states of boson topological band insulators in optical lattice experiments with weakly-interacting spinor condensates. Although the investigation is presented mainly on a spin-one Kane-Mele model, many aspects of this thesis can be generalised to other lattice models. One major question this thesis addresses is the relevance of topological edge states in the dynamics of the interacting boson systems. This thesis shows how interactions in quenched spinor condensates can facilitate the manifestation of the edge states of two dimensional topological lattice models. Provided certain quench and symmetry-related conditions are fulfilled, the edge states are found to be populated exponentially fast right after the quench. A growing edge spin current is also described. A preliminary numerical computation for later times suggests a particle redistribution from the edge back into the bulk. The presence of a harmonic potential in optical lattice experiments often obscures the manifestation of the edge states. In relation to this problem, spinor condensates have been considered, and a sharpening of the boundaries is observed in the Thomas-Fermi regime. Moreover, for spin-
±1 states, despite the presence of the external potential, one can recover a band structure similar to that of a non-interacting model with hard-wall boundaries. Approximate analytical expressions for the edge-state energies in the presence of a harmonic potential are derived. The results presented in this thesis aim to widen the understanding of the manifestation of boson topological edge states, and are in line with the current experiments with ultracold atoms. The thesis suggests mechanisms of experimentally probing boson topological edge states and their quench dynamics with spinor condensates.
One of the most fascinating properties of topological phases of matter is their robustness to disorder and imperfections. Although several experimental techniques have been developed to probe the geometric properties of engineered topological Bloch bands with cold atoms, they almost exclusively rely on the translational invariance of the underlying lattice. This prevents direct studies of topology in the presence of disorder, further hindering an extension to disordered interacting topological phases. Here, we identify disorder-driven phase transitions between two distinct Floquet topological phases using the characteristic properties of topological edge modes with ultracold atoms in periodically-driven two-dimensional (2D) optical lattices. Our results constitute an important step towards studying the rich interplay between topology and disorder with cold atoms. Moreover, our measurements confirm that disorder indeed favors the anomalous Floquet topological regime over conventional Hall systems, indicating an enhanced robustness and paving the way towards observing exotic out-of-equilibrium phases such as the anomalous Floquet Anderson insulator.
We study two-particle states in a Su-Schrieffer-Heeger (SSH) chain with periodic boundary conditions and nearest-neighbor (NN) interactions. The system is mapped into a problem of a single particle in a two-dimensional (2D) SSH lattice with potential walls along specific edges. The 2D SSH model has a trivial Chern number but a non-trivial Zak's phase, the one-dimensional (1D) topological invariant, along specific directions of the lattice, which allow for the presence of topological edge states. Using center-of-mass and relative coordinates, we calculate the energy spectrum of these two-body states for strong interactions and find that, aside from the expected appearance of doublon bands, two extra in-gap bands are present. These are identified as bands of topological states localized at the edges of the internal coordinate, the relative distance between the two particles. As such, the topological states reported here are intrinsically many-body in what concerns their real space manifestation, having no counterpart in single-particle states derived from effective models. Finally, we compare the effect of Hubbard interactions with that of NN interactions to show how the presence of the topological bound states is specific to the latter case.