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Out-of-equilibrium transport in the interacting resonant level model and comparison with the boundary sine-Gordon model

Kemal Bidzhiev, Grégoire Misguich, Hubert Saleur

Published Oct 25, 2018• 1 citation
OpenAlex

Abstract

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.

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We study the transport properties of the interacting resonant level model (IRLM) for arbitrary values of the interaction parameter UUU, 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 III-VVV 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 UUU, but that marked differences appear for larger values of U≳5U\gtrsim 5U≳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.
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.
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.
Here, the authors investigate the out-of-equilibrium properties of a simple quantum impurity model: the interacting resonant level model. They focus on the scaling regime, where the bandwidth of the fermions in the leads is larger than all the other energies, so that the lattice and the continuum versions of the model become equivalent. Using time-dependent density matrix renormalization group simulations, the new results cover a range of parameters, complementing previous analytical results that exist for only a few points in parameter space. The chief results are a demonstration of scaling collapse for III-VVV curves and for entanglement in terms of the Kondo temperature.
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.
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 present exact results for the susceptibility 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 TK{T}_{K}TK​ 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 TK{T}_{K}TK​---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.