Quantum-State Projectors on Grassmannian: Geometry, Holonomy, and Topology
An isolated group of $k$ bands in an $N$-level quantum system defines a rank-$k$ spectral projector and hence a map into the complex Grassmannian $\mathrm{Gr}(k,N)$. For a smooth gapped Hamiltonian, this projector is globally smooth and periodic even when band topology obstructs a globally smooth periodic, or symmetry-compatible, Bloch frame. We take the globally defined differential $\dd P$ as the central object: it is the tangent field of the Grassmannian map, removes unphysical rotations within the selected subspace, and retains the physical interband transition. The interband block of this tangent data simultaneously determines the quantum metric and Berry curvature; the associated horizontal generator produces finite Grassmannian motion, while wedge products of $\dd P$ enter topological forms. We construct a shortest path between two projectors in the ambient Grassmannian and show that the singular values of its horizontal generator block are the principal angles between the endpoint subspaces. This construction provides a piecewise-geodesic interpretation of discrete geometric phases. In a separate development, we derive a basis-independent expression for the determinant of a multiband Wilson loop from traces of powers of an ordered projector product, without decomposing a degenerate band multiplet into individual bands. Finally, the same tangent-vector calculus organizes Chern characters, chiral winding numbers, and the time-reversal $\mathbb Z_2$ index. The resulting framework unifies local quantum geometry, finite subspace distance, holonomy, and topology while avoiding global gauge fixing.