Thermal entanglement between π-electrons in silicene and photons; occurrence of phase transitions

In this article, the thermal entanglement between π-electronic states in a monolayer silicene sheet and a single mode quantized electromagnetic field is investigated. We assume that the system is in thermal equilibrium with the environment at a temperature T, so that the whole system is described by the Boltzmann distribution. Using the states of total Hamiltonian, the thermal density matrix and, consequently, its partially transposed one is computed, giving rise to the determination of negativity. Our analytical calculations, along with representative figures, show that the system is separable at zero temperature, exhibits a maximum, at a specific temperature, and asymptotically vanishes. Along these lines we also report the effects of electron-photon coupling, as well as the silicene buckling, on the entanglement. Specifically, we demonstrate that the maximal value of entanglement is larger for stronger electron-photon coupling, while it decreases for larger buckling effect. Moreover, we show that the gap in the total energy spectrum remains intact for any value of the buckling parameter. There is, however, one state whose energy changes sign, at a specific buckling, indicating a change of phase.