GSE statistics without spin

Energy level statistics following the Gaussian Symplectic Ensemble (GSE) of Random
Matrix Theory have been predicted theoretically and observed numerically in numerous quantum chaotic systems. However, in all these systems there has been one unifying feature: the combination of half-integer spin and time-reversal invariance. Here we provide an alternative mechanism for obtaining GSE statistics that is derived from geometric symmetries of a quantum system which alleviates the need for spin. As an example, we construct a quantum graph with a discrete symmetry given by the quaternion group Q8 and observe GSE statistics within one of its subspectra. We then show how to isolate this subspectrum and construct a quantum graph with a scalar valued wave function and a pure GSE spectrum.