Quantum geometric bounds in spinful systems with trivial band topology

TL;DR

This paper establishes new quantum geometric bounds specific to spinful systems with trivial and nontrivial band topology, linking geometry, optical responses, disorder robustness, and quantum metrology.

Contribution

It introduces geometric bounds on spin topology that extend beyond known bounds related to Wilson loops, applicable to systems with various topological indices.

Findings

Derived bounds for spin topology in trivial and nontrivial systems.

Benchmarking with first-principles calculations on bismuth.

Connected quantum bounds to optical responses and quantum metrology.

Abstract

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.