Quantum geometric bounds in spinful systems with trivial band topology

We derive quantum geometric bounds in spinful systems with spin topology characterized by a single $\mathbb{Z}$ index protected by a spin gap. Our bounds provide geometric conditions on the spin topology, distinct from the known quantum geometric bounds associated with Wilson loops and nontrivial band topologies. As a result, we obtain broader bounds in time-reversal symmetric systems with a nontrivial $\mathbb{Z}_2$ index and also bounds in systems with a trivial $\mathbb{Z}_2$ index, where the quantum metric should be otherwise unbounded. We benchmark these findings with first-principles calculations in elemental bismuth realizing a nontrivial even spin-Chern number. Moreover, we connect these bounds to optical responses and show their robustness in the presence of disorder within a real space marker formulation, demonstrating that spin-resolved quantum geometry is observable in realistic experimental settings of impure materials. Finally, we connect spin bounds to quantum Cram\'er-Rao bounds that are central to quantum metrology, showing that elemental Bi and other spin-topological phases hold promises for topological free fermion quantum sensors.