Exploring bulk-boundary correspondence in topological systems

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Topological systems are currently one of the most actively research areas of modern condensed matter physics. The key and most interesting physical manifestation of nontrivial topology is the existence of bound electron states. A multitude of bound-state structures is possible in various topological systems (such as topological insulators, superconductors, and semimetals).Bulk-boundary correspondence (BBC) is the theoretical claim that bulk topology determines the key properties of the bound states. This concept, central for topological systems, is widely believed to be universal and is routinely relied upon in both theoretical and experimental identification of topological states. However, despite its fundamental importance, the rigorous and general proof of BBC is still lacking and there are indications of its violation in some cases (for instance, in chiral-symmetric systems).In this research proposal, we plan to carry out a detailed theoretical analysis of BBC, which would include a precise formulation of BBC, deep understanding of the relation between bound states and bulk topology, and establishing precise bounds of validity and universality of BBC.As the main research method, we will use the symmetry-based formalism of constructing general analytical models that are free of microscopic assumptions. The main advantage of such models is that they are simultaneously simple enough that bound states (and their relation to bulk topology) can be calculated explicitly and general enough that their behavior is representative of a whole class of systems, providing an exhaustive characterization. As a result, BBC can be addressed in an explicit and physically illuminating fashion.