Quantum geometric bounds for observables: Linear responses, Drude weight, and orbital magnetization

The quantum geometric tensor (QGT) provides nontrivial bounds among physical quantities, as exemplified by the metric curvature inequality. How far can this idea be generalized, and what other observables obey such quantum-geometric constraints? The authors show here that generalized QGTs yield new inequalities among all linear responses, the Drude weight, and the orbital magnetization. They further find that this Drude orbital magnetization inequality approaches equality as bands become flatter, which is nearly satisfied in the experimentally observed orbital magnetization of twisted bilayer graphene.