The Lyapunov equation and Green’s functions for non-equilibrium stationary states

Extended reservoir approaches provide a versatile framework for simulating many--body, open quantum systems, including quantum transport. These are frequently benchmarked on non-interacting systems (i.e., ones with quadratic Hamiltonians), necessitating the need for robust, scalable computational tools to provide the exact solution in this scenario. We study two such tools here. The first is the use of the Lyapunov equation that was recently provided as the solution to the accumulative reservoir construction (a bridge unifying two distinct extended reservoir approaches). The second is the use of Green's functions to calculate the correlation matrix of the junction system and the extended reservoir modes. We demonstrate that the Keldysh equation for non-interacting Green's functions is related to the formal solution of the Lyapunov equation. Diagonalization of the (retarded) Green's function, thus, not only provides a route to analytic results for the non-interacting correlation matrix, but also provides a new perspective on the Lyapunov equation. We formulate a third approach by integrating out subsystems of modes in the standard way to yield a system of integrals.