Generic transport formula for a system driven by Markovian reservoirs

We present a generic, compact formula for the current flowing in interacting and noninteracting systems which are driven out of equilibrium by biased reservoirs described by Lindblad jump operators. We show that, in the limit of high temperature and chemical potential, our formula is equivalent to the well-known Meir-Wingreen formula, which describes the current flowing through a system connected to fermionic baths, therefore bridging the gap between the two formalisms. Our formulation gives a systematic way to address the transport properties of correlated systems strongly driven out of equilibrium. As an illustration, we provide explicit calculations of the current in three cases : (i) a single-site impurity, (ii) a free-fermionic chain, and (iii) a fermionic chain with loss and gain terms along the chain. In this last case, we find that the current across the system has the same behavior for loss or gain terms and depends on the loss/gain rate in a nonmonotonic way.

Figure 1

The two types of systems considered in this study. (a) System coupled to the exterior via Lindblad-type creation and annihilation operators. There is no knowledge of the structure of the environment. All memory effects are discarded in this setting. (b) Explicit coupling to a fermionic bath described at equilibrium by the Fermi-Dirac distribution.

Figure 2

The different situations for which the current is computed with the transport formulas (19) and (20): (i) Single level coupled to reservoirs; (ii) free-fermionic chain; (iii) fermionic chain with loss terms along the chain.

Figure 3

Resonance spectrum (38) of the tight-binding chain as a function of energy and different system sizes $N$. We consider $\mathrm{\Delta }/J=0.1$. The plots are shifted vertically for different $N$, for readability.

Figure 4

Linear conductance $g$ [Eq. (29)], multiplied by the temperature $T$, as function of the chain size $N$, for $\mathrm{\Delta }/J=0.1$ and $\mu =0$ for fermionic bath boundaries. Different colors correspond to different temperatures $T$, listed in the legend.

Figure 5

Plot of the behavior of the current as a function of the loss of particles $\nu$ in the system for Lindblad boundary conditions. The different colors correspond to different system sizes $N$. Here $\mathrm{\Delta }/J=1$.

Figure 6

Plot of $J_{D,\mathcal{L}}$ for Lindblad boundary conditions for different system sizes. We see that as $\nu$ increases, the current leaking from the system to the environment also increases as expected.