Tuning corner states in proximitized second-order topological insulators with bulk-boundary obstruction

Second-order topological insulators (SOTIs) support topological states beyond the usual bulk-boundary correspondence and provide important connections between quantum chemistry and topology. A hallmark of the two-dimensional (2D) SOTIs is the emergence of corner states, which usually arise from the topologically nontrivial obstructed states in the bulk. In contrast, we reveal a very different scenario where even trivial obstructed bulk states can induce corner states due to their open boundaries. Remarkably, we show that these two types of corner states can coexist in a single system and predict, from first-principles calculations, that the monolayer ${\text{C}}_{2}\text{N}$ is a promising candidate for their observation. To overcome the limitation in manipulating corner states, we demonstrate it can be accomplished using a magnetic exchange field, where the corner states can be fully spin polarized and moved into the bulk states. Focusing on the example of the ${\text{C}}_{2}\text{N}/\text{Cr}{\text{I}}_{3}$ van der Waals heterostructure, we put forth a class of proximitized materials which enable the versatile control of corner states through strain-controlled magnetic proximity effects. Our work reveals another type of topological state, and provides a universal proposal for topological corner state modulations and applications.