  We analyze a special separable and chiral-symmetric model with a quantized
quadrupole moment, extending the Benalcazar-Bernevig-Hughes model [Science 357,
61 (2017)]. Using nested-Wilson loop formalism, we give an exact expression for
Wannier centers, sector polarizations, and quadrupole moments. These are
connected to the winding numbers of the constitutive one-dimensional chains. We
prove that these winding numbers can characterize the model's Wannier topology
as a $\mathbb{Z}\times\mathbb{Z}$ set. These results clearly show that the
quantization of the quadrupole moment can arise without additional spatial
symmetry (except for translation symmetry) for the bulk. By switching from the
Wannier representation to the Bloch representation, we derive an alternative
expression for the bulk quadrupole moment and obtain its exact value. Combining
the bulk quadrupole and edge polarizations, we analytically calculate the
corner charge in a large square system and make the bulk-boundary
correspondence explicit in an edge-consistent gauge. Our work reveals the
relationship between zero-energy states at the boundary, charge localization,
and the bulk quadrupole of the extended model.

Strongly Correlated Electrons;Materials Science;Other Condensed Matter