Abstract We investigate the quantum metric and topological Euler number in a cyclically modulated Su-Schrieffer-Heeger (SSH) model with long-range hopping terms. By computing the quantum geometry tensor, we derive exact expressions for the quantum metric and Berry curvature of the energy band electrons, and we obtain the phase diagram of the model marked by the first Chern number. Furthermore, we also obtain the topological Euler number of the energy band based on the Gauss-Bonnet theorem on the topological characterization of the closed Bloch states manifold in the first Brillouin zone. However, some regions where the Berry curvature is identically zero in the first Brillouin zone result in the degeneracy of the quantum metric, which leads to ill-defined non-integer topological Euler numbers. Nevertheless, the non-integer "Euler number" provides valuable insights and an upper bound for the absolute values of the Chern numbers.
Fund: Project supported by the Beijing Natural Science Foundation (Grant No. 1232026), the Qinxin Talents Program of BISTU (Grant No. QXTCP C201711), the R&D Program of Beijing Municipal Education Commission (Grant No. KM202011232017), the National Natural Science Foundation of China (Grant No. 12304190), and the Research fund of BISTU (Grant No. 2022XJJ32).
Corresponding Authors:
Yu-Quan Ma
E-mail: abelish@163.com
Cite this article:
Xiang-Long Zeng(曾相龙), Wen-Xi Lai(赖文喜), Yi-Wen Wei(魏祎雯), and Yu-Quan Ma(马余全) Quantum geometric tensor and the topological characterization of the extended Su-Schrieffer-Heeger model 2024 Chin. Phys. B 33 030310
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