Ring states in topological materials

TL;DR

This paper introduces ring states in topological materials, which are impurity-induced states pinned by band topology, serving as fundamental units for boundary modes and elucidating bulk-boundary correspondence.

Contribution

It reveals that band topology causes zeros in the impurity Green's function, leading to topologically protected ring states that underpin boundary modes.

Findings

Ring states are pinned by band topology within topological gaps.

These states are orthogonal to impurity eigenstates and weakly depend on impurity strength.

Ring states serve as building blocks for boundary modes, linking bulk and boundary properties.

Abstract

Ingap states are commonly observed in semiconductors and are often well characterized by a hydrogenic model within the effective mass approximation. However, when impurities are strong, they significantly perturb all momentum eigenstates, leading to deep-level bound states that reveal the global properties of the unperturbed band structure. In this work, we discover that the topology of band wavefunctions can impose zeros in the impurity-projected Green's function within topological gaps. These zeros can be interpreted as spectral attractors, defining the energy at which ingap states are pinned in the presence of infinitely strong local impurities. Their pinning energy is found by minimizing the level repulsion of band eigenstates onto the ingap state. We refer to these states as ring states, marked by a mixed band character and a node at the impurity site, guaranteeing their orthogonality to the bare impurity eigenstates and a weak energy dependence on the impurity strength. We show that the inability to construct symmetric and exponentially localized Wannier functions ensures topological protection of ring states. Linking ring states together, the edge or surface modes can be recovered for any topologically protected phase. Therefore, ring states can also be viewed as building blocks of boundary modes, offering a framework to understand bulk-boundary correspondence.