Cyclic Evolution on Grassmann Manifold and Berry Phase

TL;DR

This paper explores the geometric evolution of subspaces in a Hilbert space, linking cyclic paths on the Grassmann manifold to unitary transformations, and analyzing their associated Berry phases.

Contribution

It introduces a method to construct paths in the Grassmann manifold with prescribed monodromy, connecting geometric evolution to unitary transformations.

Findings

Established a correspondence between cyclic evolutions and unitary monodromy

Provided a construction for paths with specific monodromy in the Grassmann manifold

Linked geometric phases to Berry phase in quantum systems

Abstract

For a given $k$-dimensional subspace $V_0$ in a Hilbert space $\hilb$ and a unitary transformation $g_0:V_0\To V_0$, we find a path in the Grassmann manifold the monodromy of which coincides with $g_0$.