Nonquantized Bulk Multipole Moments with a Nonzero Chern Number

Topological multipole insulators have been widely explored due to their ability to exhibit quantized bulk multipole moments. The bulk-edge and bulk-corner correspondences are rigorously confirmed in terms of dipole and quadrupole moments, which are computed using the Wilson loop and nested Wilson loop methods, respectively. This work investigates Chern, dipole, and quadrupole topological phases in an extended two-dimensional Su–Schrieffer–Heeger model. Specifically, nonquantized bulk multipole moments are observed in regions with nonzero Chern number, contrasting with half-integer bulk multipole moments in regions with zero Chern number. Here, two invariants are introduced that capture the topology of nonquantized dipole and quadrupole topological phases. These invariants redefine the nonquantized dipole and quadrupole moments as quarter-integer values using the parity and topological indices, respectively. The main difference between half- and quarter-integer bulk multipole moments is that the former characterize topological states in the gap, while the latter characterize bound states in the continuum. Moreover, a scheme is proposed to construct a two-dimensional Su–Schrieffer–Heeger model in a topolectrical circuit. This work provides a more comprehensive characterization of dipole and quadrupole phases in Class-D Hermitian systems with (Formula presented.) rotational symmetry.