Geometry of degenerate quantum states, configurations of m-planes and invariants on complex Grassmannians

Understanding the geometric information contained in quantum states is valuable in various branches of physics, particularly in solid-state physics when Bloch states play a crucial role. While the Fubini-Study metric and Berry curvature form offer comprehensive descriptions of non-degenerate quantum states, a similar description for degenerate states did not exist. In this work, we fill this gap by showing how to reduce the geometry of degenerate states to the non-abelian (Wilczek-Zee) connection A and a previously unexplored matrix-valued metric tensor G. Mathematically, this problem is equivalent to finding the U (N) invariants of a configuration of subspaces in Cⁿ. For two subspaces, the configuration was known to be described by a set of m principal angles that generalize the notion of quantum distance. For more subspaces, we find 3 m² - 3 m + 1 additional independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces. We also develop a procedure for calculating these invariants as integrals of A and G over geodesics on the Grassmannain manifold. Finally, we briefly discuss possible application of these results to quantum state preparation and PT-symmetric band structures.