Energy current and Casimir effect with phase transition in the nonequilibrium steady-state of a quantum anisotropic XY spin chain

The dynamics of quantum anisotropic XY spin chain driven out of equilibrium by a transverse magnetic field, that carry an energy current (JE) is investigated. Considering the spin exchange and the flow of energy trapped in magnetic field between two defects in the x y-plane separated by a distance d, we derive an analytical expression of the steady state Green's function  ij of the model using the Grassmann algebra in order to determine the Casimir force between two magnetic defects. As soon as done we find that, cooling the system as well as increasing the anisotropic parameter decrease the energy current and accordingly increase the range of the Casimir force. This symmetry is broken with a significant increase of distance between the defects. A critical temperature is then recorded exhibiting a phase transition according to the anisotropic parameter. An interesting feature of this model is that turning the anisotropic parameter as well as the temperature range will better manipulate the energy flux and control the range of the Casimir force.