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  • Letter
  • Open Access

Quantum geometric bounds in spinful systems with trivial band topology

Wojciech J. Jankowski1,*, Robert-Jan Slager1,2, and Gunnar F. Lange3,4,†

  • *Contact author: wjj25@cam.ac.uk
  • †Contact author: g.f.lange@fys.uio.no
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Phys. Rev. Research 7, L042011 – Published 9 October, 2025

DOI: https://doi.org/10.1103/zlxq-fxgc

Abstract

We derive quantum geometric bounds in spinful systems with spin topology characterized by a single 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 Z2 index and also bounds in systems with a trivial Z2 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.

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Phys. Rev. Research 7, L042011– Published 9 October, 2025
  • Received 4 February 2025
  • Accepted 25 August 2025
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