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Comments on boundary driven open XXZ chain: asymmetric driving and uniqueness of steady states

In this short note we provide two extensions on the recent explicit results on the matrix-product ansatz for the non-equilibrium steady state of a markovianly boundary-driven anisotropic Heisenberg XXZ spin 1/2 chain. We write a perturbative solution for the steady state density matrix in the system-batyh coupling for an arbitrary (asymmetric) set of four spin-flip rates at the two chain ends, generalizing the symmetric-driving ansatz of [Phys. Rev. Lett. 106, 217206 (2011)]. Furthermore, we generalize the exact (non-perturbative) form of the steady state for just two Lindblad channels (spin-up flipping on the left, and spin-down flipping on the right) to an arbitrary (asymmetric) ratio of the spin flipping rates [Phys. Rev. Lett. 107, 137201 (2011)]. In addition, we also indicate a simple proof of uniqueness of our steady states.