Exterior integrability: Yang–Baxter form of non-equilibrium steady-state density operator

Abstract

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 is essentially different from the standard 4 × 4 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 infinite 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.