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We will discuss a general non-equilibrium setup by which one can approach the quantum transport problem in one dimension. One considers a strongly interacting quantum chain with fully coherent bulk dynamics and driven out of equilibrium in terms of Lindblad dissipators which only act on degrees of freedom near the boundary, i.e. at the ends of the chain. Non-equilibrium state carrying the physical currents is then approximated as the steady state of the corresponding Markovian master equation.
I will present several analytical solutions along these lines and discuss their remarkable algebraic properties. For example, in the Heisenberg XXZ chain under extreme boundary driving we will show that the steady-state density operator of a finite system of size n is -- apart from a normalization constant -- a polynomial of degree $2n-2$ in the coupling constant. In the isotropic case we find cosine spin profiles, $1/n^2$ scaling of the spin current, and long-range correlations in the steady state. Furthermore, the perturbative (weak coupling) version of our ansatz [Phys. Rev. Lett. 106, 217206 (2011)] is used to derive a novel pseudo-local conservation law of the anisotropic Heisenberg model, by means of which we rigorously estimate the spin Drude weight (the ballistic transport coefficient) in the easy-plane regime. This closes a long standing question in strongly correlated condensed matter physics.