Implications of setting ds = 0 to remove Berry-phase ambiguity

Ascertain the mathematical and physical implications of the heuristic replacement ds = 0 that eliminates the Berry-phase–induced non-uniqueness in the spinor definition under coordinate rotation, specifically in the diagonalization of H(\mathrm{d}\boldsymbol{x}) and the construction of covariant spinor bases using V(\mathrm{d}\boldsymbol{x}) and (\mathrm{d}s^2)^{1/4}.

Background

When diagonalizing the Weyl Hamiltonian H(\mathrm{d}\boldsymbol{x}) with differential-form parameters, coordinate rotations induce a Berry phase, leading to non-uniqueness in the spinor basis up to a phase. The authors show that by formally setting \mathrm{d}s → 0, the Berry phase tends to unity, thereby removing the ambiguity.

However, the authors explicitly note that the implication of this heuristic procedure is unclear, motivating a careful investigation into its justification and consequences for the proposed spinor framework based on differential forms.