  Working in analogy with topological insulators in condensed matter, photonic
topological insulators (PTI) have been experimentally realized, and protected
electromagnetic edge-modes have been demonstrated in such systems. Moreover,
PTI technology also emulates a synthetic spin-1/2 degree of freedom (DOF) in
the reflectionless topological modes. The spin-1/2 DOF is carried by Quantum
Valley Hall (QVH) / Quantum Spin Hall (QSH) interface modes created from the
bianisotropic meta waveguide (BMW) platform, and realized both in simulation
and experiment. We employ the PTI setting to build an ensemble of wave chaotic
1D metric graphs that display statistical properties consistent with Gaussian
Symplectic Ensemble (GSE) statistics. The two critical ingredients required to
create a physical system in the GSE universality class, the half-integer-spin
DOF and preserved time-reversal invariance, are clearly realized in the QVH/QSH
interface modes. We identify the anti-unitary T-operator for the PTI
Hamiltonian underlying our experimental realization. An ensemble of
PTI-edgemode metric graphs are proposed and experimentally demonstrated. We
then demonstrate the Kramers degeneracy of eigenmodes of the PTI-graph systems
with both numerical and experimental studies. We further conduct spectral
statistical studies of the edgemode graphs and find good agreement with the GSE
theoretical predictions. The PTI chaotic graph structures present an innovative
and easily extendable platform for continued future investigation of GSE
systems.
