Robust Topology and Tunable Geometry in Generalized BHZ Model with Fractional Dispersion
With recent developments in fractional quantum mechanics, we introduce a fractional generalization of the Bernevig-Hughes-Zhang (BHZ) model to investigate the effects of fractional dispersion on band topology and quantum geometry. In the low-energy limit, the model reduces to a fractional Dirac Hamiltonian while preserving momentum-space periodicity, thereby ensuring a compact Brillouin zone and a well-defined topological invariant. We further show that the corresponding real-space tight-binding model can be constructed directly through a Fourier-series transformation, providing a simpler and more general alternative to the methods typically employed for fractional lattice systems. This approach readily extends beyond the generalized BHZ model considered here. Using the Bloch eigenstates, we compute the quantum geometric tensor and analyze its components, namely the Berry curvature and quantum metric. We find that fractional tuning redistributes these quantities throughout the Brillouin zone as the dispersion exponent becomes fractional, leading to pronounced modifications of the local band geometry. In contrast, the Chern number remains invariant, demonstrating the robustness of the global topological phase against fractional deformation. We further argue that this invariance persists for a broad class of fractional models.