We study the pattern of three-state topological phases that appears in systems with real Hamiltonians and wave functions. We give a simple geometric construction for representing these phases. We then apply our results to understand previous work on three-state phases. We show that the “mirror symmetry” of wave functions noticed in microwave experiments can be simply understood in our framework.

• Received 6 August 2001

DOI:https://doi.org/10.1103/PhysRevLett.87.260401