The non-equilibrium dynamics of an open quantum systems is often described in terms of a quantum Master equation in the Lindblad form. We consider an anisotropic Heisenberg XXZ spin chain coupled at the edges with baths of a fixed, and different polarizations. Such a coupling introduces a dissipation which brings the quantum system with time in a non-equilibrium steady state with gradients and currents. We demonstrate that the exact non-equilibrium steady state of the XXZ spin chain driven by boundary Lindblad operators targeting two different completely polarized boundary states, can be constructed explicitly with a matrix product ansatz for the non-equilibrium density matrix where the matrices satisfy a quadratic algebra, related to the quantum algebra \(U_{q}\)[SU(2)] [1]. Our results suggest that the matrix product ansatz can be extended to more general quantum systems kept far from equilibrium by Lindblad boundary terms. The method allows to investigate analytically quantum systems of large sizes and in the thermodynamic limit. Our Matrix Product ansatz solution generalizes and simplifies the results of an earlier remarkable paper of T. Prosen [2].
[1] D. Karevski, V. Popkov, and G. M. Schütz, Phys. Rev. Lett. 110, 047201 (2013)
[2] T. Prosen, Phys. Rev. Lett. 107, 137201 (2011).