Diagrammatic Formulation of Transport in Liouville Space

The Nakajima-Zwanzig formalism naturally leads to a diagrammatic formulation for the propagator and current kernels of an interacting nanojunction in Liouville space. This allows one to obtain exact formal expressions for the average current as well as to devise suitable approximation schemes. In the weak tunneling regime, the diagrammatics soon recovers the second order (sequential) tunneling dynamics and facilitates a classification of the various virtual processes contributing in fourth order. As the tunneling coupling is increased, appropriate selections of diagrams to all orders in the coupling are needed to properly capture the interplay of tunneling and interactions. Finally, the connection to the non-equilibrium Green’s function formulation of the current is elucidated.