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We study quantum transport properties of an open Heisenberg XXZ spin 1/2chain driven by a pair of Lindblad jump operators satisfying a global`microcanonical' constraint, i.e. conserving the total magnetization. We willshow that this system has an additional discrete symmetry which is particularto the Liouvillean description of the problem. Such symmetry reduces thedynamics even more than what would be expected in the standard Hilbert spaceformalism and establishes existence of multiple steady states. Interestingly,numerical simulations of the XXZ model suggest that a pair of distinctnon-equilibrium steady states becomes indistinguishable in the thermodynamiclimit, and exhibit sub-diffusive spin transport in the easy-axis regime ofanisotropy Delta > 1.

A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains