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

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.

• The paper derives an intrinsic monopole-connection formula for the solid angle of open Bloch-sphere paths ending at antipodal points, avoiding coordinate-dependent closure prescriptions.
• The paper shows that a regularization displaced by azimuth α changes the geometric phase by −α, while reversing the displacement preserves a universal π phase difference and produces deviations of 2α in solid angle.
• The paper identifies the osculating-plane great circle, determined by the control vector’s local velocity and acceleration, as the physically selected closure through the zero-temperature limit of the Uhlmann phase.

The problem of antipodal open paths

For a closed curve $C$ on the Bloch sphere, the geometric phase of a spin-$\tfrac12$ system is unambiguously $\gamma_g[C] = -\tfrac12\Omega[C]$, with $\Omega[C]$ the signed solid angle enclosed (2608.19679). The kinematic formulation of Aharonov–Anandan, Samuel–Bhandari, and Mukunda–Simon extends this to open paths by closing them with a geodesic arc, which contributes no phase — the geodesic rule. This construction fails when the endpoints are antipodal: infinitely many half-great-circle arcs join them, each enclosing a different solid angle, and the Pancharatnam relative phase between orthogonal endpoint states is undefined.

The paper by Yang and Noh identifies precisely when this situation arises physically: a two-level Hamiltonian $H(t) = -\mathbf{d}(t)\cdot\boldsymbol{\sigma}$ whose control vector crosses the origin transversally at an isolated time $t_0$. At that instant the levels are degenerate, and the direction field $$\hat{\mathbf{n}}(t) = \mathbf{d}(t)/|\mathbf{d}(t)|$$ jumps discontinuously between antipodal points $\pm\hat{\mathbf{v}}$, so its trajectory on $S^2$ is an open curve $C$ rather than a loop. Prior work — Rakhecha and Wagh's azimuthal analysis of $\tfrac12$0 jumps, Garza-Soto–Hagen's operational hierarchy of closing rules, and interferometric verifications by Wagh et al. and Zhou et al. — established prescriptions and observables but not a general coordinate-free derivation. This paper supplies one.

Intrinsic solid angle from the monopole connection

The central tool is the coordinate-free monopole one-form on $\tfrac12$1,

$\tfrac12$2

with curvature $\tfrac12$3; it equals minus twice the Berry connection in the gauge based at $\tfrac12$4. Two structural lemmas underpin everything that follows. First, along any great-circle arc through the base point, $\tfrac12$5 identically, since $\tfrac12$6 is normal to the plane containing both the base point and the arc. Second, if the curve terminates at the antipode $\tfrac12$7, every geodesic closure contributes zero to the line integral, because all great circles through antipodes pass through the base point's plane. Consequently the paper defines the open-path solid angle intrinsically as $\tfrac12$8, with the base point tied to the starting point of the curve (placing the Dirac string at the opposite endpoint). A triangulation argument via the van Oosterom–Strackee formula shows this line integral coincides with the signed solid angle seen from the sphere's centre.

The authors are careful to state what this does not mean: the intrinsic value $\tfrac12$9 is well defined, but the solid angle enclosed after closing depends on the chosen geodesic, two closures bounding a lune of area $\gamma_g[C] = -\tfrac12\Omega[C]$0. The identity "geometric phase = enclosed solid angle" therefore holds for exactly one closure, and identifying it is the paper's main result.

The main theorem: deviation $\gamma_g[C] = -\tfrac12\Omega[C]$1

Regularizing the degeneracy by displacing the control vector, $\gamma_g[C] = -\tfrac12\Omega[C]$2, produces a nonsingular closed loop with unambiguous solid angle. Writing the transverse part of $\gamma_g[C] = -\tfrac12\Omega[C]$3 in the Frenet frame $\gamma_g[C] = -\tfrac12\Omega[C]$4 at the crossing, with azimuth $\gamma_g[C] = -\tfrac12\Omega[C]$5 measured from the principal normal, the main theorem states:

$\gamma_g[C] = -\tfrac12\Omega[C]$6

equivalently $\gamma_g[C] = -\tfrac12\Omega[C]$7.

The proof rests on a clean separation of scales near the crossing: the polar angle relaxes to its pole over $\gamma_g[C] = -\tfrac12\Omega[C]$8, while the azimuth turns over $\gamma_g[C] = -\tfrac12\Omega[C]$9, so $\Omega[C]$0. The azimuthal turn imposed by the regularization is thus executed while the monopole weight $\Omega[C]$1 is pinned at $\Omega[C]$2 near the south pole (the Dirac string side) and undone where the weight vanishes at the north pole. This weight asymmetry — the turn counted once with weight 2 and once with weight $\Omega[C]$3 — is the entire mechanism behind the finite deviation $\Omega[C]$4. A supporting lemma shows the degeneracy itself contributes nothing concentrated to $\Omega[C]$5: the torsion-induced tilt of the bare transverse azimuth, $\Omega[C]$6, integrates to $\Omega[C]$7 within any window and vanishes as the window shrinks.

Three corollaries follow immediately:

• Berry's $\Omega[C]$8 invariant: reversing the regularization sends $\Omega[C]$9, so $H(t) = -\mathbf{d}(t)\cdot\boldsymbol{\sigma}$0, independent of direction.
• Reflection symmetry: for a planar control curve regularized along the mirror normal, $H(t) = -\mathbf{d}(t)\cdot\boldsymbol{\sigma}$1, $H(t) = -\mathbf{d}(t)\cdot\boldsymbol{\sigma}$2, and $H(t) = -\mathbf{d}(t)\cdot\boldsymbol{\sigma}$3.
• Non-universality of the average: $H(t) = -\mathbf{d}(t)\cdot\boldsymbol{\sigma}$4 retains dependence on $H(t) = -\mathbf{d}(t)\cdot\boldsymbol{\sigma}$5; only mirror symmetry forces agreement with $H(t) = -\mathbf{d}(t)\cdot\boldsymbol{\sigma}$6 modulo $H(t) = -\mathbf{d}(t)\cdot\boldsymbol{\sigma}$7.

The contrast between the first and third corollaries is instructive: universality holds for the difference because $H(t) = -\mathbf{d}(t)\cdot\boldsymbol{\sigma}$8 cancels, but fails for the average because it survives.

Uhlmann selection of the osculating-plane geodesic

The Frenet analysis singles out the canonical closure: the great circle in the osculating plane $H(t) = -\mathbf{d}(t)\cdot\boldsymbol{\sigma}$9, tangent to the curve at both antipodal endpoints ($t_0$0). The paper then shows this choice is not merely convenient but physically selected. For the Gibbs state $t_0$1 with $t_0$2, the Bloch vector passes straight through the maximally mixed point along $t_0$3 without discontinuity — unlike the pure-state path, which jumps. The Uhlmann connection's directional part squares out the sign, and $t_0$4 kills the central segment's contribution, so $t_0$5 is well defined without regularization for all $t_0$6. In the pure-state limit $t_0$7 and

$t_0$8

which combined with the main theorem yields $t_0$9 exactly when $$\hat{\mathbf{n}}(t) = \mathbf{d}(t)/|\mathbf{d}(t)|$$0: the mixed-state construction automatically realizes the osculating-plane closure. This converts the heuristic "close the path the way the physical path approaches the endpoint" into a computable prescription requiring only the local two-jet of $$\hat{\mathbf{n}}(t) = \mathbf{d}(t)/|\mathbf{d}(t)|$$1 at the degeneracy. The result is consistent with the known failure of the Uhlmann–Berry correspondence at genuine degeneracies, which here plays a structurally essential role.

Numerical verification

The appendix verifies the theorem on three fields (two generic, one reflection-symmetric). For Field A with $$\hat{\mathbf{n}}(t) = \mathbf{d}(t)/|\mathbf{d}(t)|$$2, predicted deviations $$\hat{\mathbf{n}}(t) = \mathbf{d}(t)/|\mathbf{d}(t)|$$3 across four regularization directions match measurements to within $$\hat{\mathbf{n}}(t) = \mathbf{d}(t)/|\mathbf{d}(t)|$$4 at $$\hat{\mathbf{n}}(t) = \mathbf{d}(t)/|\mathbf{d}(t)|$$5; halving $$\hat{\mathbf{n}}(t) = \mathbf{d}(t)/|\mathbf{d}(t)|$$6 reduces error by a factor consistent with the $$\hat{\mathbf{n}}(t) = \mathbf{d}(t)/|\mathbf{d}(t)|$$7 correction. The three-scale structure (weight pinned at $$\hat{\mathbf{n}}(t) = \mathbf{d}(t)/|\mathbf{d}(t)|$$8 for $$\hat{\mathbf{n}}(t) = \mathbf{d}(t)/|\mathbf{d}(t)|$$9, below $\pm\hat{\mathbf{v}}$0 for $\pm\hat{\mathbf{v}}$1) is confirmed directly. The $\pm\hat{\mathbf{v}}$2 invariant holds to $\pm\hat{\mathbf{v}}$3–$\pm\hat{\mathbf{v}}$4, Field C converges to $\pm\hat{\mathbf{v}}$5 with machine-precision sum cancellation, and $\pm\hat{\mathbf{v}}$6 agrees with $\pm\hat{\mathbf{v}}$7 to within $\pm\hat{\mathbf{v}}$8.

Limitations and open questions

The theorem assumes a transversal crossing with nonvanishing transverse curvature, $\pm\hat{\mathbf{v}}$9; inflectional crossings, where the osculating plane degenerates and the selection rule loses its anchor, are not covered. Trajectories with multiple crossings per period raise the question of whether geodesics selected at successive degeneracies can be composed consistently when they do not coincide. On the physical side, the interplay with mixed-state phases beyond Uhlmann's construction — notably the interferometric phase of Sjöqvist et al., and the fate of the Uhlmann–Berry correspondence at genuine level crossings — remains open. The proposed interferometric tests of the $S^2$0 invariant and $S^2$1 values in neutron and atom interferometry have not yet been performed for these specific predictions.

Conclusion

This paper resolves the geodesic ambiguity for open paths with antipodal endpoints in closed form. An intrinsic monopole line integral defines the open-path solid angle; every closure contributes zero to it, yet exactly one — the osculating-plane great circle fixed by the curvature of $S^2$2 at the degeneracy — realizes the identity between geometric phase and enclosed solid angle, with all other choices deviating by $S^2$3 due to the asymmetry of the monopole weight across the Dirac string. Berry's $S^2$4 invariant and the reflection values $S^2$5 follow as corollaries, and the pure-state limit of the Uhlmann phase selects the canonical closure automatically. The result replaces heuristic closing rules with a prescription computable from the local velocity and acceleration of the control curve, subject to extension beyond single transversal crossings with nonzero curvature.