Single-particle Entanglement of the Entanglement Hamiltonian Eigen-modes

Single-particle entanglement entropy (SPEE) is calculated for entanglement Hamiltonian eigen-mode in a one-dimensional free fermion model that undergoes a delocalized-localized phase transition. In this numerical study, we show that SPEE of entanglement Hamiltonian eigen-mode has the same behavior as EPEE of Hamiltonian eigen-mode at the Fermi level: as we go from delocalized phase toward localized phase, SPEE of both modes decreases in the same manner. Furthermore, fluctuations of SPEE of entanglement Hamiltonian eigen-mode -- which can be obtained through the calculation of moments of SPEE -- signature very sharply the phase transition point. These two modes are also compared by calculation of single particle R\'{e}yni entropy (SPRE). We show that SPEE and SPRE of entanglement Hamiltonian eigen-mode can be used as a phase detection parameter.