Many-body index for quadrupole insulators

A quadrupole insulator hosts bound states at corners of the system under the open boundary condition and is expected to possess a “bulk quadrupole moment” under the periodic boundary condition. However, there have been conflicting discussions on the latter and characterization of a quadrupole insulator in terms of the bulk quadrupole moment is still an open problem especially for interacting systems. In this study, we propose a new index for characterization of an interacting quadrupole insulator which is closely related to the bulk quadrupole moment. It can naturally lead to a bulk-boundary correspondence and also are well compatible with the periodicity of the system. We demonstrate these properties for representative models.