Geometric phase of open paths and a geodesic-selection rule at a level degeneracy

• URL: http://arxiv.org/abs/2608.19679v1
• Date: Thu, 20 Aug 2026 06:14:15 GMT
• Title: Geometric phase of open paths and a geodesic-selection rule at a level degeneracy
• Abstract summary: We show that the geodesic rule for the open-path geometric phase becomes ambiguous when many geodesics close the path.<n>A coordinate-free monopole connection defines the open-path solid angle $[C]$ intrinsically.<n>We show that the pure-state limit of the finite-temperature Uhlmann phase selects the osculating-plane closure automatically.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: When the control field of a qubit, a polarization state, or a spin-$\tfrac12$ system is swept through a level degeneracy, its direction traces an open curve on the Bloch sphere whose endpoints are antipodal, and the geodesic rule for the open-path geometric phase becomes ambiguous: infinitely many geodesics close the path, and different closures enclose different solid angles. We resolve this ambiguity in closed form. A coordinate-free monopole connection defines the open-path solid angle $Ω[C]$ intrinsically, and displacing the degeneracy by $ε\uhat$ closes the path with enclosed solid angle $Ω(ε\uhat)=Ω[C]+2α+O(ε)$, where $α$ is the azimuth of the transverse part of $\uhat$ measured from the principal normal of the control curve at the crossing. The identity between geometric phase and enclosed solid angle therefore holds for exactly one closing geodesic---the great circle in the osculating plane ($α=0$)---supplied by the curvature at the degeneracy. Berry's $π$ invariant under reversal of the displacement and the values $\pmπ/2$ under a reflection symmetry follow as corollaries, and the pure-state limit of the finite-temperature Uhlmann phase selects the osculating-plane closure automatically, turning the heuristic closing rules of the open-path literature into a computable prescription.

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