Many-body quadrupolar sum rule for higher-order topological insulator

The modern theory of polarization establishes the bulk-boundary correspondence for the bulk polarization. We attempt to extend it to a sum rule of the bulk quadrupole moment by employing a many-body operator introduced in Ref.s 1 and 2. The sum rule consists of the alternating sum of four observables, which are the phase factors of the many-body operator in different boundary conditions. We demonstrate the validity of the sum rule through extensive numerical computations on various non-interacting tight-binding models. We also observe that individual terms in the sum rule correspond to the bulk quadrupole moment, the edge-localized polarizations, and the corner charge in the thermodynamic limit on some models.

[1] Byungmin Kang, Ken Shiozaki, and Gil Young Cho, “Many-body order parameters for multipoles in solids,” Phys. Rev. B 100, 245134 (2019).

[2] William A. Wheeler, Lucas K. Wagner, and Taylor L. Hughes, “Many-body electric multipole operators in extended systems,” Phys. Rev. B 100, 245135 (2019).