arXiv:1004.4811 · Physical Review Letters

Shot Noise in the Self-Dual Interacting Resonant Level Model

Alexander Branschädel, E. Boulat, Hubert Saleur, Peter Schmitteckert

Published Oct 1, 2010• 38 citations

Abstract

By using two independent and complementary approaches, we compute exactly the shot noise in an out-of-equilibrium interacting impurity model, the interacting resonant level model at its self-dual point. An analytical approach based on the thermodynamical Bethe ansatz allows us to obtain the density matrix in the presence of a bias voltage, which in turn allows for the computation of any observable. A time-dependent density matrix renormalization group technique that has proven to yield the correct result for a free model (the resonant level model) is shown to be in perfect agreement with the former method.

DOI: 10.1103/physrevlett.105.146805

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arXiv:1004.4811 · Physical Review Letters

Shot Noise in the Self-Dual Interacting Resonant Level Model

Alexander Branschädel, E. Boulat, Hubert Saleur, Peter Schmitteckert

Published Oct 1, 2010• 38 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.105.146805

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We present a general technique to obtain the zero temperature cumulant generating function of the full counting statistics of charge transfer in interacting impurity models out of equilibrium from time-dependent simulations on a lattice. We demonstrate the technique with application to the self-dual interacting resonant level model, where very good agreement between numerical simulations using the density matrix renormalization group and those obtained analytically from the thermodynamic Bethe ansatz is found. We show from the exact form of counting statistics that the quasiparticles involved in transport carry charge 2e in the low bias regime and e/2 in the high bias regime.We present a general technique to obtain the zero temperature cumulant generating function of the full counting statistics of charge transfer in interacting impurity models out of equilibrium from time-dependent simulations on a lattice. We demonstrate the technique with application to the self-dual interacting resonant level model, where very good agreement between numerical simulations using the density matrix renormalization group and those obtained analytically from the thermodynamic Bethe ansatz is found. We show from the exact form of counting statistics that the quasiparticles involved in transport carry charge 2e in the low bias regime and e/2 in the high bias regime.

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arXiv:1810.08246Physical review. B./Physical review. BFeb 12, 201914 citations

We study the transport properties of the interacting resonant level model (IRLM) for arbitrary values of the interaction parameter

U

, and compare them with those of the Boundary sine-Gordon model (BSG), which are known analytically (Fendley, Ludwig and Saleur, 1995). While the IRLM and BSG models are expected not to be equivalent (except at two particular points: the free-fermion model and the self-dual point), we find that their

I

V

curves are extraordinarily close, and can be distinguished only by a delicate analysis in the Infra Red (IR) regime. We also find that the shot noise in the two models is very similar for small enough values of

U

, but that marked differences appear for larger values of

U\gtrsim 5

. The numerical calculations of the IRLM are done using time-dependent Density Matrix renormalization Group (tDMRG) simulations. The shot noise is obtained by analyzing the second cumulant of the transferred charge, whose expectation value grows linearly with time, and extracting the corresponding rate from the simulations.

U

, 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

e

in the infrared regime of the IRLM (except at the self-dual point where it jumps to

2e

), while it is known to be a continuously varying function of

U

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.

U

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

ϵ

of the resonant level is pinned close to either one of the chemical potentials $\ifmmode\pm\else\textpm\fi{}V/2$.

T_K

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

T_K

- 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.

{T}_{K}

with excellent agreement to the numerical results. On the other hand, direct application of the Bethe ansatz of the interacting resonant level mode does not correctly reproduce

{T}_{K}

---however, if the bare parameters in the model are renormalized, 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.