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

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... 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-$\pm1$ 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.