  Very recently, increasing attention has been focused on non-Abelian
topological charges, e.g. the quaternion group Q8. Different from Abelian
topological band insulators, these systems involve multiple tangled bulk
bandgaps and support non-trivial edge states that manifest the non-Abelian
topological features. Furthermore, a system with even or odd number of bands
will exhibit significant difference in non-Abelian topological classifications.
Up to now, there is scant research investigating the even-band non-Abelian
topological insulators. Here, we both theoretically explored and experimentally
realized a four-band PT (inversion and time-reversal) symmetric system, where
two new classes of topological charges as well as edge states are
comprehensively studied. We illustrate their difference from four-dimensional
rotation senses on the stereographically projected Clifford tori. We show the
evolution of bulk topology by extending the 1D Hamiltonian onto a 2D plane and
provide the accompanying edge state distributions following an analytical
method. Our work presents an exhaustive study of four-band non-Abelian
topological insulators and paves the way to other even band systems.
