Characterization for higher-order symmetry-protected topological phases by quantized Berry phases

We propose the 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. The integer Q is determined by the rotational symmetry the system has. As a concrete example, we show that the Berry phase is quantized in Z₄ 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 Z₄ Berry phase for the spin-model-analog of the BBH model. Furthermore, we demonstrate the Berry phase is quantized in Z₄ for the three-dimensional version of the BBH model. We also confirm the bulk-corner correspondence between the Z_Q Berry phase and the corner states in the HOSPT phases.