# $\mathbb{Z}_Q$ Berry Phase for Higher-Order Symmetry-Protected Topological Phases

## TL;DR

This paper introduces the $Z_Q$ Berry phase as a new topological invariant for higher-order symmetry-protected topological phases in 2D and 3D systems, demonstrating its stability and quantization in specific models.

## Contribution

It proposes the $Z_Q$ Berry phase as a novel topological invariant for HOSPT phases and applies it to extended BBH models, including spin models, revealing new topological characterizations.

## Key findings

- The $Z_4$ Berry phase characterizes HOSPT phases in extended BBH models.
- The $Z_4$ Berry phase is quantized in 3D BBH models.
- Bulk-corner correspondence is confirmed via the $Z_4$ Berry phase.

## Abstract

We propose the $\mathbb{Z}_Q$ Berry phase as a topological invariant for higher-order symmetry-protected topological (HOSPT) phases for two- and three-dimensional systems. It is topologically stable for electron-electron interactions assuming the gap remains open. As a concrete example, we show that the Berry phase is quantized in $\mathbb{Z}_4$ and characterizes the HOSPT phase of the extended Benalcazar-Bernevig-Hughes (BBH) model, which contains the next-nearest neighbor hopping and the intersite Coulomb interactions. In addition, we introduce the $\mathbb{Z}_4$ Berry phase for the spin-model-analog of the BBH model, whose topological invariant has not been found so far. Furthermore, we demonstrate the Berry phase is quantized in $\mathbb{Z}_4$ for the three-dimensional version of the BBH model. We also confirm the bulk-corner correspondence between the $\mathbb{Z}_4$ Berry phase and the corner states in the HOSPT phases.