Probing chiral-symmetric higher-order topological insulators with multipole winding number

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The interplay between crystalline symmetry and band topology gives rise to unprecedented lower-dimensional boundary states in higher-order topological insulators (HOTIs). However, the measurement of the topological invariants of HOTIs remains a significant challenge. Here, we define a multipole winding number (MWN) for chiral-symmetric HOTIs by applying a corner twisted boundary condition. The MWN, arising from both bulk and boundary states, accurately captures the bulk-corner correspondence including boundary-obstructed topological phases. To address the measurement challenge, we leverage the perturbative nature of the corner twisted boundary condition and develop a real-space approach for determining the MWN in both two-dimensional and three-dimensional systems. The real-space formula provides an experimentally viable strategy for directly probing the topology of chiral-symmetric HOTIs through dynamical evolution. Our findings not only highlight the twisted boundary condition as a powerful tool for investigating HOTIs, but also establish a paradigm for exploring real-space formulas for the topological invariants of HOTIs.