GSE statistics without spin

Energy levels 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 based on geometric symmetries of a quantum system which alleviates the need for spin. As an example, we construct a quantum graph with a particular discrete symmetry given by the quaternion group Q8. GSE statistics is then observed within one of its subspectra.