Wannier Topology and Quadrupole Moments for a generalized Benalcazar-Bernevig-Hughes Model

TL;DR

This paper extends the Benalcazar-Bernevig-Hughes model to analyze quadrupole moments and Wannier topology using nested-Wilson loops, providing exact formulas and revealing bulk-boundary relationships without additional symmetries.

Contribution

It introduces an exact analytical framework for quadrupole moments and Wannier topology in a generalized model, expanding understanding of topological quadrupole insulators.

Findings

Exact expressions for Wannier centers, sector polarizations, and quadrupole moments.

Winding numbers characterize the Wannier topology as a old set.

Quantized quadrupole moments can occur without additional spatial symmetries.

Abstract

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.