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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi mathvariant="double-struck">Z</a:mi></a:math> 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi mathvariant="double-struck">Z</c:mi><c:mn>2</c:mn></c:msub></c:math> index and also bounds in systems with a trivial <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msub><e:mi mathvariant="double-struck">Z</e:mi><e:mn>2</e:mn></e:msub></e:math> 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 Cramér-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.