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

Of the available classes of insulators which have been shown to contain topologically 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 encompasses non-trivial Z$_2$ topological insulators such as the quantum spin Hall (QSH) state and the 3D strong topological insulator. One of the defining properties of this system is the robustness of the state under the addition of disorder that preserves $T.$ In this letter, we explore the phase diagram of the disordered QSH state as a function of disorder strength and chemical potential by examining the entanglement spectrum for disordered class AII symplectic systems. As for the case of the $T$ breaking Chern insulator we show that there is a correspondence between the level-spacing statistics of the Hamiltonian and that of the level spacing statistics of the entanglement spectrum. We observe a feature in the statistics of the entanglement spectrum that aids the identification of delocalized states and consequently critical energies across which phase transitions occur.