Exploring bulk-boundary correspondence in topological systems

拓扑系统是现代凝聚态物理中最活跃的研究领域之一。非平凡拓扑的关键和最有趣的物理表现是束缚电子态的存在。在各种拓扑系统（如拓扑绝缘体、超导体和半金属）中可能存在多种束缚态结构。体边界对应（BBC）是体拓扑结构决定束缚态关键性质的理论主张。这个概念是拓扑系统的核心，被广泛认为是普遍的，并且在拓扑状态的理论和实验识别中经常依赖。然而，尽管它具有基本的重要性，严格和一般的BBC证明仍然缺乏，并且在某些情况下（例如，在手性对称系统中）有违反它的迹象。在本研究计划中，我们计划对BBC进行详细的理论分析，包括精确表述BBC，深入理解束缚态与体拓扑之间的关系，建立BBC的有效性和普适性的精确界限。作为主要的研究方法，我们将使用基于对称的形式主义来构建不含微观假设的一般分析模型。这种模型的主要优点是，它们同时足够简单，可以明确地计算束缚状态（以及它们与整体拓扑的关系），并且足够通用，它们的行为可以代表整个系统类，提供详尽的表征。因此，BBC可以用一种明确的、具有物理启发性的方式来称呼。

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.