Cyclic phases at an n-fold degeneracy

We study the sets of possible sign changes that can occur when n real quantum states are transported adiabatically around an n-fold degeneracy. The allowed sign changes include those predicted by the familiar spin j = (n - 1)/2 model as a subset, but we also find at least one other possible set of sign changes for each n > 2. Experimentally verifiable quadrilateral billiard examples are presented to illustrate these additional possibilities for 3- and 4-fold degeneracies.