Exact Universal Characterization of Chiral-Symmetric Higher-Order Topological Phases

 In recent years, higher-order topological phases (HOTPs) have attracted significant interest across various fields in physics. These phases support topologically protected (d-n)-dimensional boundary states in d-dimensional systems, where n>1 . Many efforts have recently been made to extend the framework of the first-order bulk-boundary correspondence—the heart of topological physics—to these higher-order phases. However, most studies often rely on implicit assumptions rather than rigorous demonstrations of correspondence between these invariants and HOTPs, and recent evidence suggests that previously defined invariants fail to accurately capture certain HOTPs. Furthermore, the patterns of HOTPs, such as the number of corner states at each corner—a characteristic unique compared to their first-order counterparts, cannot be captured by previous frameworks. Additionally, no existing theory accommodates arbitrary shapes and dimensions. In this manuscript, we introduce an exact analytical theory for chiral-symmetric systems that address these three critical issues above. Our approach, which employs polynomials of position operators and utilizes the Bott index vector from K-theory, provides exact correspondences and captures all distinct patterns of topological zero-energy corner states in lattice systems of arbitrary geometry, including those with concave boundaries or holes. We also derive a sum rule that decomposes the global Bott index into contributions from the bulk and various boundary features, bridging the gap between open and periodic boundary approaches. Our theory clarifies the existing ambiguities in the studies of HOTPs and establishes a robust foundation for future explorations into, for example, topological superconductors with Majorana corner modes. Moreover, our theory provides important guides to the search for new materials or classical systems exhibiting HOTPs.

Highlights:

- Rigorous Bott index vector and zero-energy corner state correspondence.

- Captures spatial corner state patterns.

- Universal theory for intrinsic and extrinsic HOTPs in arbitrary lattice geometry.

- Exact sum rules link Bott index vectors under open and periodic boundary conditions.

eTOC blurb

We establish a universal, rigorous correspondence between Bott index vectors and zero-energy corner states in chiral-symmetric higher-order topological phases. By utilizing position operator polynomials under open boundary conditions, our framework captures intricate corner state patterns across arbitrary lattice geometries and dimensions. We also introduce a general sum rule that decomposes global invariants into bulk and boundary contributions. This resolves long-standing inconsistencies and provides a robust tool for designing higher-order topological materials

Professor Meng Xiao from the School of Physics, Wuhan University, is the corresponding author of the paper.