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Of the available classes of insulators which have been shown to containtopologically non-trivial properties one of the most important is class AII,which contains systems that possess time-reversal symmetry $T$ with $T^2=-1.$This class has been the subject of significant attention as it encompassesnon-trivial Z$_2$ topological insulators such as the quantum spin Hall (QSH)state and the 3D strong topological insulator. One of the defining propertiesof this system is the robustness of the state under the addition of disorderthat preserves $T.$ In this letter, we explore the phase diagram of thedisordered QSH state as a function of disorder strength and chemical potentialby examining the entanglement spectrum for disordered class AII symplecticsystems. As for the case of the $T$ breaking Chern insulator we show that thereis a correspondence between the level-spacing statistics of the Hamiltonian andthat of the level spacing statistics of the entanglement spectrum. We observe afeature in the statistics of the entanglement spectrum that aids theidentification of delocalized states and consequently critical energies acrosswhich phase transitions occur.

Signature of Phase Transitions in the Disordered Quantum Spin Hall State From the Entanglement Spectrum