Quantum geometry, localization, and topological bounds of spin fluctuations

We study how topological crystalline defects--dislocations--reshape the real-space quantum geometric tensor and act as tunable sources of quantum geometry. We show that dislocations strongly enhance the quantum metric, establishing a direct link between lattice topology and the Hilbert-space geometry of states. We characterize the quantum geometry of topological magnons in ordered arrays of dislocations, demonstrating that defect-induced geometric enhancement controls their localization and topological protection. In disordered arrays, dislocation-driven geometry expands the accessible topological phase space and enables transitions to disorder-induced topological phases. Our results identify the quantum metric as a tunable bridge between crystalline topology, magnonic excitations, and emergent topological matter in aperiodic solid-state and synthetic systems.