Non-Abelian Berry Phase — Wilczek-Zee

Holonomy of a degenerate quantum system under cyclic adiabatic evolution

Loop Radius Loop Tilt Speed

U = [ ]

For a degenerate subspace, adiabatic cyclic transport yields a non-Abelian (matrix-valued) Berry phase — the Wilczek-Zee holonomy. The state vector traces a path on the Bloch sphere while the SU(2) holonomy matrix U accumulates. Colors show the two degenerate eigenstates.