New Method for Studying Steady States in Quantum Impurity Problems: The Interacting Resonant Level Model

Abstract

We develop a new perturbative method for studying any steady states of quantum impurities, in or out of equilibrium. We show that steady-state averages are completely fixed by basic properties of the steady-state (Hershfield's) density matrix along with dynamical "impurity conditions". This gives the full perturbative expansion without Feynman diagrams (matrix products instead are used), and "re-sums" into an equilibrium average that may lend itself to numerical procedures. We calculate the universal current in the interacting resonant level model (IRLM) at finite bias V to first order in Coulomb repulsion U for all V and temperatures. We find that the bias, like the temperature, cuts off low-energy processes. In the IRLM, this implies a power-law decay of the current at large V (also recently observed by Boulat and Saleur at some finite value of U ).PACS numbers: 73.63. Kv, 72.15.Qm, 72.10.Fk Impurity models describe mesoscopic quantum objects in contact with large conducting leads. Quantum dots are much-studied examples, and experiments in a bias voltage [1] lead to accurate descriptions of their nonequilibrium steady-state properties. In impurity models, such states are of high interest as they pose the theoretical challenge of capturing the effect of non-equilibrium in a truly quantum system, yet they constitute the simplest non-equilibrium situation, where properties are time independent. At low energies, interactions between the leads' Landau quasi-particles and the impurity occur in the s-wave channel, and the spectrum may be linearised around the two Fermi points. Then, the universal behaviour is described by free massless fermions on the half line (bulk conformal field theory) with a non-conformal boundary interaction at the end-point. Many methods are known for studying equilibrium behaviors in such models, but understanding and accessing properties of non-equilibrium steady states is still a much harder task. Wilson's picture does not apply, and, for instance, the effect of a bias on low-energy processes (which determines the large-bias current) is still under study [2,3].Two calculational schemes exist:the real-time Schwinger-Keldysh formulation and the scattering-state Lippman-Schwinger formulation. The former relies on the infinite extent of the half line in order to absorb the energy released by relaxation from an appropriate "uninteracting" density matrix to the steady state. It requires relaxation mechanisms (see, e.g. [3]), whose absence may lead to pathologies in perturbative expansions [4]. The latter describes directly the expected end result just from "how the state looks" asymptotically far from the impurity. Hershfield's Y operator [5] (see also the studies [3,6,7]) gives a "steady-state density matrix" that encodes these scattering states. This is interesting, since a non-equilibrium steady state is not described by the usual density matrix, but it is still hard to apply to interacting systems. Exact results for integable models occur when the exact quasi-particles do not mix the bias...

Out-of-Equilibrium Properties and Nonlinear Effects for Interacting Quantum Impurity Systems in the Strong-Coupling Regime

“…This model [39] has recently earned the status of benchmark in out-of-equilibrium physics in QIM [1,14,[40][41][42] as one of the simplest QIM allowing for non equilibrium in the presence of interactions. The IRLM consists of two baths of free spinless electrons coupled via tunnel hopping to a single level, that interacts capacitively with the electrodes via a short range potential with strength U .…”

Universal electric current of interacting resonant-level models with asymmetric interactions: An extension of the Landauer formula

“…It is clear that the way of realizing the nonequilibrium steady states in our extension of the Landauer formula is different from the Keldysh formalism [12][13][14][15][16]. By employing a renormalization-group technique with the Callan-Symanzik equation [24,27], we arrived at the universal electric current in the first order of the interaction parameter U . We found that the negative differential conductance of the universal electric current is characterized by the same scaling parameter T K as that obtained by other approaches [24,27,31].…”

“…Clearly, this is a phenomenon out of the linear-response regime. To see the feature and to compare the results obtained by different approaches, the universal electric current characterized by a single scaling parameter T K is useful [24,26,27]. Indeed, it is found that, for large bias voltages V , the universal electric current shows a power-law decay I ∝ (V /T K ) −U/π with the parameter U of the Coulomb interaction [24,25].…”

“…The original IRLM, which consists of a single impurity coupled to a conduction band, was introduced for studying the Kondo problem in equilibrium systems [21]. Recently, the IRLM with two external leads has been employed as a minimal model of open QD systems with Coulomb interactions; it plays an important role in verifying the theoretical approaches such as the nonequilibrium Bethe-ansatz approach [22], the perturbation theory with the numerical renormalization group [23], a new method called * nishino@kanagawa-u.ac.jp impurity conditions [24], and the time-dependent densitymatrix renormalization-group method [25].…”

Real-time simulations of nonequilibrium transport in the single-impurity Anderson model

“…However, the understanding of open systems out of equilibrium remains an extremely challenging area of research. While recent studies have succeeded in calculating the currentvoltage (I-V) characteristics of the interacting resonant level model, 4,5,6,7,8 the focus of current research has been devoted toward the more difficult single-impurity Anderson model incorporating Kondo correlations. For example, experimental results for the finite-bias transport properties of a quantum dot still await a complete theoretical explanation.…”