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

Utilizing Bott index vectors formulated through a series of polynomials of position operators under open boundary conditions, we establish a universal, rigorous, and complete correspondence between the Bott index vector and topological zero-energy corner states in systems with chiral symmetry. Our framework covers systems of arbitrary shapes, including topological phases that are beyond the characterization by previously proposed invariants such as multipole moments or multipole chiral numbers. A key feature of our approach is its ability to capture the real-space patterns of zero-energy corner states, providing a deeper understanding of higher-order topological phases. We provide a rigorous analytical proof of its higher-order correspondence and sum rules for Bott index vectors under different boundary conditions. To demonstrate the effectiveness of our theory, we examine several model systems with representative patterns of zero-energy corner states that lie outside the scope of previous classification frameworks.