The quantum metric measures how much Bloch wavefunctions change as momentum varies across the Brillouin zone. Its integral — the quantum weight — appears in observable quantities: optical conductivity, superfluid weight, orbital magnetization. In symmetry-protected topological phases, topological invariants impose lower bounds on the quantum weight: the topology guarantees a minimum geometric structure.
Hung et al. (arXiv:2603.13041) extend these bounds beyond symmetry-protected phases. When the protecting symmetry is broken (by spin-orbit coupling, disorder, or other perturbations), the conventional topological bound fails — the invariant that guaranteed the minimum quantum weight is no longer well-defined. But the authors show that the bound persists with a correction term proportional to the symmetry violation.
The correction is quantitative, not qualitative. The topological invariant doesn't suddenly become irrelevant when symmetry breaks — it contributes a bound that degrades continuously as the violation grows. For small symmetry breaking, the bound is nearly as tight as the exact topological bound. For large breaking, the correction dominates and the bound weakens. The transition is smooth.
They validate this on a spin Chern insulator with tunable spin-orbit coupling. In the spin-conserved limit, the spin Chern number exactly bounds the quantum weight. As spin-orbit coupling increases (breaking spin conservation), the conventional bound fails but the corrected bound continues to hold, tracking the actual quantum weight down to where the topological protection is almost fully eroded. The topology remembers its role even after the symmetry that defined it has been broken.