Nonabelian holonomy achieved by crossing point degeneracies

The holonomy, or, geometric phase, acquired after a cycle of parameter variation highlights a robust and controllable component in the temporal evolution of an oscillatory system. Nonabelian holonomies, where the order of cyclic parameter variations matter, are a promising avenue for fault-tolerant on-material computation. Current physical realizations are fine-tuned systems that manipulate design parameters to generate N-fold degenerate eigenstates, in essence exploiting the Wilczek-Zee phase. In this study, we investigate a new paradigm that leads to the acquisition of holonomies in generically nondegenerate systems by performing loops in parameter space that cross point degeneracies. These loops can exchange eigenstates and result in a nonabelian holonomy group of signed permutations, thereby generalizing Berry's abelian geometric phases. Numerical examples of mechanical resonators verify the holonomy and highlight its robust nature. The results extend to degenerate systems, where eigenspaces can be exchanged in addition to the usual Wilczek-Zee phase, and to complex-valued systems, where unitary-matrix-valued holonomies are acquired. This method of generating nonabelian holonomies can be used in practical realizations across acoustic, elastic, photonic, and quantum mechanical systems.