We demonstrate that the exact nonequilibrium steady state of the one-dimensional Heisenberg $XXZ$ spin chain driven by boundary Lindblad operators can be constructed explicitly with a matrix product ansatz for the nonequilibrium density matrix where the matrices satisfy a quadratic algebra. This algebra turns out to be related to the quantum algebra $U_q[\mathrm{SU}(2)]$. Coherent state techniques are introduced for the exact solution of the isotropic Heisenberg chain with and without quantum boundary fields and Lindblad terms that correspond to two different completely polarized boundary states. We show that this boundary twist leads to nonvanishing stationary currents of all spin components. Our results suggest that the matrix product ansatz can be extended to more general quantum systems kept far from equilibrium by Lindblad boundary terms.

• Received 29 November 2012

DOI:https://doi.org/10.1103/PhysRevLett.110.047201