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arXiv:0806.3731 · Physical Review Letters

Twofold Advance in the Theoretical Understanding of Far-From-Equilibrium Properties of Interacting Nanostructures

E. Boulat, Hubert Saleur, Peter Schmitteckert

Published Sep 30, 2008• 175 citations
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Abstract

We calculate the full I-V characteristics at vanishing temperature in the self-dual interacting resonant level model in two ways. The first uses careful time dependent density matrix renormalization group with a large number of states per block and a representation of the reservoirs as leads subjected to a chemical potential. The other is based on integrability in the continuum limit, and generalizes early work by Fendley, Ludwig, and Saleur on the boundary sine-Gordon model. The two approaches are in excellent agreement, and uncover among other things a power law decay of the current at large voltages when U>0.

DOI
10.1103/physrevlett.101.140601

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Boulat, Saleur and Schmitteckert (BSS) reported results on the full $I-V$ characteristics of the interacting resonant level model (IRLM) exhibiting region with unexpected negative differential conductance (NDC). Using time-dependent density matrix renormalization group complemented with the exact solution performed at a special point (the self-dual point) in the parameter space BSS have shown that at nonzero Coulomb interaction $U$ the current flowing through the impurity level (IL) exhibits a power-law asymptotics as a function of large applied bias voltage. Similar conclusion was earlier reached by Doyon. Even though their results are solid and supported by both analytic and numeric arguments, BSS concluded that "the NDC at large voltage seems a truly nonperturbative behavior, with unclear physical origin". On the contrary, the remarkable physics of NDC can be explained by simple physical arguments and, as we shall show in this Comment, in certain circumstances can be calculated in the framework of perturbation theory.

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arXiv:1406.0254•Physical Review B•Oct 8, 2014•14 citations
We calculate the full I-V characteristics at vanishing temperature in the self-dual interacting resonant level model in two ways. The first uses careful time dependent density matrix renormalization group with a large number of states per block and a representation of the reservoirs as leads subjected to a chemical potential. The other is based on integrability in the continuum limit, and generalizes early work by Fendley, Ludwig, and Saleur on the boundary sine-Gordon model. The two approaches are in excellent agreement, and uncover among other things a power law decay of the current at large voltages when U>0.
Boulat, Saleur and Schmitteckert (BSS) reported results on the full I−VI-VI−V characteristics of the interacting resonant level model (IRLM) exhibiting region with unexpected negative differential conductance (NDC). Using time-dependent density matrix renormalization group complemented with the exact solution performed at a special point (the self-dual point) in the parameter space BSS have shown that at nonzero Coulomb interaction UUU the current flowing through the impurity level (IL) exhibits a power-law asymptotics as a function of large applied bias voltage. Similar conclusion was earlier reached by Doyon. Even though their results are solid and supported by both analytic and numeric arguments, BSS concluded that "the NDC at large voltage seems a truly nonperturbative behavior, with unclear physical origin". On the contrary, the remarkable physics of NDC can be explained by simple physical arguments and, as we shall show in this Comment, in certain circumstances can be calculated in the framework of perturbation theory.
We investigate equilibrium and steady-state nonequilibrium transport properties of a spinless resonant level locally coupled to two conduction bands of width $\ensuremath{\sim}\ensuremath{\Gamma}$ via a Coulomb interaction UUU and a hybridization ${t}^{\ensuremath{'}}$. In order to study the effects of finite bias voltages beyond linear response, a generalization of the functional renormalization group to Keldysh frequency space is employed. Being mostly unexplored in the context of quantum impurity systems out of equilibrium, we benchmark this method against recently published time-dependent density matrix renormalization group data. We thoroughly investigate the scaling limit $\ensuremath{\Gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$ characterized by the appearance of power laws. Most importantly, at the particle-hole symmetric point the steady-state current decays like $J\ensuremath{\sim}{V}^{\ensuremath{-}{\ensuremath{\alpha}}_{J}}$ as a function of the bias voltage $V⪢{t}^{\ensuremath{'}}$, with an exponent ${\ensuremath{\alpha}}_{J}(U)$ that we calculate to leading order in the Coulomb interaction strength. In contrast, we do not observe a pure power-law (but more complex) current-voltage-relation if the energy ϵϵϵ of the resonant level is pinned close to either one of the chemical potentials $\ifmmode\pm\else\textpm\fi{}V/2$.
Using time-dependent density matrix renormalization group calculations we study the transport properties ($I\ensuremath{-}V$ curves and shot noise) of the interacting resonant level model (IRLM) in a large range of the interaction parameter UUU, in the scaling limit. We find that these properties can be described remarkably well by those of the boundary sine-Gordon model (BSG), which are known analytically [Fendley et al., Phys. Rev. B 52, 8934 (1995).]. We argue that the two models are nevertheless in different universality classes out-of-equilibrium: this requires a delicate discussion of their infrared (IR) properties (i.e., at low bias), where we prove in particular that the effective tunneling charge is eee in the infrared regime of the IRLM (except at the self-dual point where it jumps to 2e2e2e), while it is known to be a continuously varying function of UUU in the BSG. This behavior is confirmed by careful analysis of the numerical data in the IR. The remarkable agreement of the transport properties, especially in the crossover region, remains however unexplained.
We present exact results for the thermodynamic properties of the interacting resonant level model in equilibrium. Detailed simulations using both the Numerical Renormalization Group and Density Matrix Renormalization Group were performed in order to compare with closed analytical expressions. By first bosonizing the model and then utilizing the integrability of the resulting boundary sine-Gordon model, one finds an analytic expression for the relevant energy scale TKT_KTK​ with excellent agreement to the numerical results. On the other hand, direct application of the Bethe ansatz of the interacting resonant level model does not correctly reproduce TKT_KTK​ - however if the bare parameters in the model are renormalised, then quantities obtained via the direct Bethe ansatz such as the occupation of the resonant level as a function of the local chemical potential do match the numerical results. The case of one lead is studied in the most detail, with many results also extending to multiple leads, although there still remain open questions in this case.
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 "resums" 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).
We develop a low-order conserving approximation for the interacting resonant-level model (IRLM), and apply it to (i) thermal equilibrium, (ii) nonequilibrium steady state, and (iii) nonequilibrium quench dynamics. Thermal equilibrium is first used to carefully gauge the quality of the approximation by comparing the results with other well-studied methods, and finding good agreement for small values of the interaction. We analytically show that the power-law exponent of the renormalized level width usually derived using renormalization group approaches can also be correctly obtained in our approach in the weak interaction limit. A closed expression for the nonequilibrium steady-state current is derived and analytically and numerically evaluated. We find a negative differential conductance at large voltages, and the exponent of the power-law suppression of the steady-state current is calculated analytically at zero temperature. The response of the system to quenches is investigated for a single lead as well as for two-lead setup at finite voltage bias at particle-hole symmetry using a self-consistent two-times Keldysh Green function approach, and results are presented for the time-dependent current for different bias and contact interaction strength.