| Summary: | In this thesis, we study the di erent types of boundary modes found in Lieb-type tight-binding Hamiltonians (of chains, ribbons and square clusters) that are strongly dependent on the symmetries present in the lattice. A new topological description is developed, with the aim of predicting the behaviour of these edge states. Due to the unconventional symmetry features of the Lieb unit cell, a generalization of the Zak's phase invariant and the chiral pairing operation is realized and implemented to sustain our topological characterization. Our analysis reveals that, while a large set of boundary states have a common well de ned topological phase transition, other edge states re ect a topological non-trivial phase for any nite value of the hopping parameters and are responsible for the appearance of corner states in the two dimensional Lieb rotated square lattice when reaching a higher symmetry class. The introduction of symmetry preserving local onsite potentials in these Lieb-type systems lifts the "degeneracy" of the topological transition point, inducing a "cascade" of topological transitions. This feature is enhanced with increasing lattice spatial dimensions.
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