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. 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.

• Received 11 June 2019
• Revised 1 October 2019

DOI:https://doi.org/10.1103/PhysRevResearch.2.012009

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society