Symmetry-protected topological phases host gapless modes at their boundary with a featureless environment of the same dimension or a trivial vacuum. In this study, we explore their behavior in a higher-dimensional environment, which itself is nontrivial—a topological vacuum. In particular, we embed a one-dimensional topological wire within a two-dimensional Chern insulator, allowing the zero-dimensional edge modes of the wire to interplay with the surrounding chiral boundary states created by the environment. In contrast to a trivial vacuum, we show, depending on the nature of low-energy modes, the topology of the environment selectively influences the topological phase transitions of the wire. Interestingly, such selectivity leads to scenarios where the environment trivializes the wire and even induces topological character in an otherwise trivial phase—an example of “proximity-induced topology.” Using both numerical and analytical approaches, we establish the general framework of such embedding and uncover the role of symmetries in shaping the fate of low-energy theories. Our findings provide a deeper understanding of heterostructural topological systems, paving the way for their experimental exploration.

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• Chern insulators
• Defects
• Topological insulators
• Lattice models in condensed matter
• Su-Schrieffer-Heeger model
• Tight-binding model