Mixed-state geometric phases and level-crossing correspondence

Investigate mixed-state geometric phases beyond the Uhlmann construction, including the interferometric phase, and determine the fate of the Uhlmann--Berry correspondence at genuine level crossings.

Background

The paper shows that, for the two-level system considered, the pure-state limit of the finite-temperature Uhlmann phase selects the osculating-plane regularization. It also notes that the Uhlmann--Berry correspondence is known to fail at genuine level degeneracies.

The authors explicitly leave unresolved the broader behavior of mixed-state geometric phases beyond the Uhlmann construction, specifically mentioning the interferometric phase and the status of the Uhlmann--Berry correspondence at genuine level crossings.

References

Several directions remain open. The analysis assumes a transversal crossing with nonvanishing transverse curvature, $a\neq\mathbf0$; inflectional crossings, where the osculating plane degenerates, and trajectories with multiple crossings, where the geodesics selected at successive degeneracies need not coincide, call for an extension of Theorem~\ref{thm:main}. On the physical side, the $\pi$ invariant and the reflection values $\pm\pi/2$ are directly testable in neutron and atom interferometry, and the interplay with mixed-state phases beyond the Uhlmann construction---the interferometric phase of, and the fate of the Uhlmann--Berry correspondence at genuine level crossings---deserves a systematic study.

— Geometric phase of open paths and a geodesic-selection rule at a level degeneracy  (2608.19679 - Yang et al., 20 Aug 2026) in Section 5, Conclusion

A related non-Hermitian problem is already present in the gapped rotating-frame generator $M$ of Eq.~eq:Mmatrix: its repeated-root locus is determined by the vanishing of the characteristic-polynomial discriminant derived in Appendix~\ref{app:eig}. Establishing where this locus is genuinely defective, and whether the resulting exceptional points leave distinct signatures in the lag or in the Uhlmann holonomy, is a complementary open problem.

— Geometric Phases of a Driven Qubit: Comparing Berry and Uhlmann Holonomies  (2608.22919 - Yang et al., 24 Aug 2026) in Section Conclusion