Generalized quantum geometric tensor and its properties in disordered lattices

Recent investigations into the role of the quantum geometric tensor (QGT) in exotic condensed matter phenomena have attracted considerable attention. However, the conventional QGT only captures the geometry of the Bloch states, leaving out contributions from non-trivial physical quantities such as magnetizations. We therefore generalize the conventional QGT to incorporate the orbital magnetization, energy magnetization, and Drude weight. To address the influence of disorder on these tensors, we further develop a set of real-space formulas for the generalized QGTs. Using the Haldane model as an example, we demonstrate that the semi-positive definiteness of these tensors still enforces meaningful constraints. Moreover, the real-space formulation provides a basis for studying more intriguing cases involving various types of disorder.