We develop an exact non-perturbative framework to compute steady-state properties of quantumimpurities subject to a finite bias. We show that the steady-state physics of these systems is captured by nonequilibrium scattering eigenstates which satisfy an appropriate Lippman-Schwinger equation. Introducing a generalization of the equilibrium Bethe-Ansatz -the Nonequilibrium Bethe-Ansatz (NEBA), we explicitly construct the scattering eigenstates for the Interacting Resonance Level model and derive exact, nonperturbative results for the steady-state properties of the system. PACS numbers: 72.63. Kv, 72.15.Qm, 72.10.Fk The recent spectacular progress in nanotechnology has made it possible to study quantum impurities out-ofequilibrium [1]. The impurity is typically realized experimentally as a quantum dot, a tiny island of electron liquid attached via tunnel junctions to two leads (baths or reservoirs) held at different chemical potentials. As a result of the potential difference, an electric current flows from one lead to another across the quantum impurity. The description of such an out-of-equilibrium situation in a strongly correlated system is a long standing problem and has not been given even in the simplest case of when the system is in a steady state.In a steady state the system properties do not change with time even when out of equilibrium. Such a state is reached only under special conditions: each lead needs to be a good thermal bath and infinite in size (equivalently, the bath level spacing tends to zero.) It then follows that particles transferred from one lead to another dissipate their extra energy in the lead and equilibrate [2].There are two equivalent ways, time-dependent and time-independent, to describe the establishment of a steady state in the system. In the time dependent picture the quantum impurity is coupled to the two baths in the far past, t 0 , and is allowed to evolve adiabatically under the conditions described above. After a sufficiently long time, at t = 0 say, a steady state is reached. Two elements are required to fully determine the system: a hamiltonian to describe the time evolution and an initial condition, ρ 0 , describing the system in the far past. The hamiltonian is chosen to be of the form, H(t) = H 0 +e ηt H 1 , where H 0 describes the two free leads (thermal baths), H 1 is the interaction term between the leads and the quantum impurity, and η an infinitesimal parameter, small enough to ensure adiabaticity yet large compared to the level spacing in the leads. The initial condition is typically given by,with µ i and N i the chemical potential and number operator for particles in lead i. Subsequently, at times t ≥ t 0 , the system is described by a density matrix ρ(t) = T {e}, and the properties of the system are calculated in the usual manner, Ô (t) = T r{ρ(t)Ô}. The establishment of a steady state follows, in this language, from the existence of the limit t 0 → −∞ with the expectation value becoming time-independent, Ô = T r{ρ sÔ } where ρ s = ρ(0). At T = 0 the descri...
