The quantum geometric tensor (QGT) provides nontrivial bounds among physical quantities, as exemplified by the metric-curvature inequality. In this paper, we investigate various bounds for different observables through certain generalizations of the QGT. First, we demonstrate that bounds hold for all linear responses, which are produced by a QGT extended to many-body states, finite temperature, and general parameter space. As an application, we show the thermodynamic inequality originating from the convexity of free energy can be further tightened. Second, we establish a bound between the Drude weight and the orbital magnetization. The equality is exactly satisfied in the Landau level system, and systems with nearly flat bands tend to approach equality as well. We apply the resulting inequality to two orbital ferromagnets and support that the twisted bilayer graphene system is close to the Landau level system. Moreover, we show that an analogous inequality also holds for a higher-order multipole, magnetic quadrupole. Finally, we discuss the analogy between the QGT and the uncertainty principle, emphasizing that the existence of nontrivial bounds necessarily reflects quantum effects.

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• Geometric & topological phases
• Magnetic order
• Orbital order
• Topological materials
• Twisted bilayer graphene
• Geometry
• Linear response theory