An explicit matrix product ansatz is presented, in the first two orders in the (weak) coupling parameter, for the nonequilibrium steady state of the homogeneous, nearest neighbor Heisenberg $XXZ$ spin $1/2$ chain driven by Lindblad operators which act only at the edges of the chain. The first order of the density operator becomes, in the thermodynamic limit, an exact pseudolocal conservation law and yields—via the Mazur inequality—a rigorous lower bound on the high-temperature spin Drude weight. Such a Mazur bound is a nonvanishing fractal function of the anisotropy parameter $\Delta$ for $|\Delta |<1$.

• Received 7 March 2011

DOI:https://doi.org/10.1103/PhysRevLett.106.217206