ZQ Berry phase for higher-order symmetry-protected topological phases

 Berry phase for higher-order symmetry-protected topological phases},
  author={Hiromu Araki and Tomonari Mizoguchi and Yasuhiro Hatsugai},
  journal={Physical Review Research},
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… 

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