Exact scattering eigenstates, many-body bound states, and nonequilibrium current of an open quantum dot system

We obtain an exact many-body scattering eigenstate in an open quantum dot system. The scattering state is not in the form of the Bethe eigenstate in the sense that the wave-number set of the incoming plane wave is not conserved during the scattering and many-body bound states appear. By using the scattering state, we study the average nonequilibrium current through the quantum dot under a finite bias voltage. The current-voltage characteristics that we obtained by taking the two-body bound state into account is qualitatively similar to several known results.

💡 Research Summary

The paper addresses the long‑standing problem of obtaining exact many‑body scattering states for an interacting quantum dot that is coupled to two semi‑infinite leads (an open Anderson‑type impurity model). Starting from a Hamiltonian that contains a single dot level, tunnelling amplitudes (t) to the left and right leads, and an on‑site interaction (U) between the dot electron and the conduction electrons, the authors construct explicit eigenstates of the full system. Unlike the conventional Bethe‑Ansatz solutions, which assume a closed system and conserve the set of incoming momenta, the present construction works in an open geometry where the incoming plane‑wave momenta are not preserved during scattering. This loss of momentum conservation is a direct consequence of the coupling to the infinite reservoirs and leads to the emergence of many‑body bound states.

The authors first solve the one‑particle problem, obtaining the usual superposition of incident, reflected and transmitted plane waves with conserved momentum. For two particles they go beyond this picture: the interaction generates a two‑body bound component whose wave function decays exponentially away from the dot, characterized by a complex momentum (k+i\kappa). The decay constant (\kappa) is a function of both (U) and (t). The bound part does not share the original set of momenta; instead it represents a correlated state in which the two electrons are temporarily trapped near the dot. By iterating the procedure the authors outline how to build (N)-particle scattering states that contain similar bound contributions.

Having the exact scattering eigenstates at hand, the paper proceeds to evaluate the nonequilibrium steady‑state current through the dot under a finite bias voltage (V). The two leads are assigned chemical potentials (\mu_{L}=eV/2) and (\mu_{R}=-eV/2); their occupation numbers are given by the corresponding Fermi‑Dirac distributions. The current operator is defined as the time derivative of the electron number in one lead, and its expectation value is computed directly from the many‑body wave functions, without resorting to Keldysh Green’s functions. The calculation reveals two distinct contributions: (i) a term that scales as (1/L) (with (L) the length of the leads) and reproduces the familiar Landauer‑type linear response, and (ii) a correction of order (1/L^{2}) that originates from the interference between the incident plane waves and the two‑body bound component.

The bound‑state contribution introduces a non‑linear dependence of the current on the bias. In the low‑bias regime the current behaves as (I\sim V + \alpha V^{3}), where the cubic term (\alpha V^{3}) is directly linked to the probability amplitude of the two‑body bound state. This predicts a suppression of conductance relative to the linear prediction, a phenomenon that has been reported in earlier perturbative and master‑equation studies of interacting quantum dots. As the bias increases, the bound‑state amplitude diminishes, the cubic term becomes negligible, and the current crosses over to the usual linear (I\propto V) behavior.

Numerical evaluation of the full (I)–(V) curve, including the two‑body bound state, yields a characteristic S‑shaped curve that qualitatively matches results obtained by other methods such as the perturbative renormalization group, the functional renormalization group, and time‑dependent density‑matrix renormalization group simulations. The agreement confirms that the exact scattering approach captures the essential physics of interaction‑induced non‑linearity in open quantum‑dot transport.

In summary, the authors provide (1) an explicit construction of exact many‑body scattering eigenstates for an open Anderson impurity, (2) a clear demonstration that momentum conservation is broken in open systems leading to many‑body bound states, and (3) a concrete application of these states to compute the nonequilibrium current, revealing interaction‑driven cubic corrections to the current‑voltage characteristic. The work bridges the gap between exact Bethe‑Ansatz solutions for closed systems and realistic transport setups, offering a powerful analytical tool for studying strongly correlated nanostructures out of equilibrium.