Quantum geometric tensor in systems with fractional band dispersion

We investigate the quantum geometric tensor, which is comprised of the Berry curvature and quantum metric, in a generalized Dirac two-band system with non-integer dispersion $E(\mathbf{k})\sim k^{\alpha}$. Our analysis reveals that this type of dispersion introduces significant and novel effects on quantum band geometry. We calculate the Berry curvature and observe its redistribution in momentum space as \(\alpha\) varies. Notably, despite this redistribution, the change in Chern number across topological transitions remains quantized as an integer, even for non-integer \(\alpha\). We illustrate the physical implications of this redistribution by computing the orbital magnetization. Furthermore, we demonstrate that the Berry curvature and quantum metric peak along the regions of momentum space where the energy band exhibits high curvature. While it is well-established that Berry curvature becomes highly concentrated at the band touching points, our findings indicate that they can also accumulate at sharp band corners, away from these band touching points. We further show that while the Berry curvature develops Dirac delta-like peaks away from $\mathbf{k}=0$, the Berry monopole, corresponding to the topological charge, remains located at the origin.