Exterior integrability: Yang-Baxter form of nonequilibrium steady state density operator

A new type of quantum transfer matrix, arising as a Cholesky factor for the steady state density matrix of a dissipative Markovian process associated with the boundary-driven Lindblad equation for the isotropic spin-1/2 Heisenberg (XXX) chain, is presented. The transfer matrix forms a commuting family of non-Hermitian operators depending on the spectral parameter which is essentially the strength of dissipative coupling at the boundaries. The intertwining of the corresponding Lax and monodromy matrices is performed by an infinitely dimensional Yang-Baxter R-matrix which we construct explicitly and which is essentially different from the standard XXX R-matrix. We also discuss a possibility to construct Bethe Ansatz for the spectrum and eigenstates of the non-equilibrium steady state density operator. Furthermore, we indicate the existence of a deformed R-matrix in the infinitely-dimensional auxiliary space for the anisotropic XXZ spin-1/2 chain which in general provides a sequence of new, possibly quasi-local, conserved quantities of the bulk XXZ dynamics.