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

TL;DR

This paper develops a geometric framework for understanding degenerate quantum states using invariants on complex Grassmannians, extending non-abelian connections and metrics to characterize subspace configurations.

Contribution

It introduces a set of invariants for degenerate quantum states, generalizing quantum distance and Berry phase, and provides methods to compute them via integrals over Grassmannian geodesics.

Findings

Defined matrix-valued metric tensor G for degenerate states

Identified 3m^2 - 3m + 1 invariants for multiple subspaces

Connected invariants to quantum state geometry and potential applications

Abstract

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 $\mathbb{C}^n$. 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 $3 m^2 - 3 m + 1$ 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.