The quantum metric encodes the geometric structure of Bloch wave functions and governs a wide range of physical responses. Its Brillouin-zone integral, the quantum weight, appears in the structure factor and provides lower bounds on observables such as the optical gap and dielectric constant. In symmetry-protected topological (SPT) phases, the nontrivial band topology imposes a lower bound on the quantum weight and constraints on the observables. Here, we generalize the topological bound on quantum geometry to encompass systems beyond the SPT phases. We show that topological invariants defined via the projected spectrum lower-bound the quantum weight with a symmetry-breaking correction to the quantum metric. Our proposed bound holds even when the underlying symmetries are broken, and it would be amenable to experimental verification via the optical conductivity sum rule under external fields. We illustrate our theory by adding a nonzero spin-orbit coupling term to a spin Chern insulator model, where we show that our proposed bound applies even though the conventional topological bound does not hold.

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• Geometric & topological phases
• Optical conductivity
• Quantum geometry
• Sum rules
• Topological materials
• k dot p method