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The interplay between crystalline symmetry and band topology gives rise tounprecedented lower-dimensional boundary states in higher-order topologicalinsulators (HOTIs). However, the measurement of the topological invariants ofHOTIs remains a significant challenge. Here, we propose the multipole windingnumber (MWN) for chiral-symmetric HOTIs, achieved by applying a corner twistedboundary condition. The MWN, arising from both bulk and boundary states, mayaccurately capture the bulk-corner correspondence in finite systems. To addressthe measurement challenge, we leverage the perturbative nature of the cornertwisted boundary condition and develop a real-space approach for determiningthe MWN in both two-dimensional and three-dimensional systems. The real-spaceformula provides an experimentally viable strategy for directly probing thetopology of chiral-symmetric HOTIs through dynamical evolution. Our findingsnot only highlight the twisted boundary condition as a powerful tool forinvestigating HOTIs, but also establish a paradigm for exploring real-spaceformulas for the topological invariants of HOTIs.

Probing Chiral-Symmetric Higher-Order Topological Insulators with Multipole Winding Number