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Annalen der Physik

Conductance of inhomogeneous systems: Real‐time dynamics

Alexander Branschädel, G. Schneider, Peter Schmitteckert

Published Jul 6, 2010• 67 citations
Publisher

Abstract

Abstract Numerical time evolution of transport states using time dependent Density Matrix Renormalization Group (td‐DMRG) methods has turned out to be a powerful tool to calculate the linear and finite bias conductance of interacting impurity systems coupled to non‐interacting one‐dimensional leads. Several models, including the Interacting Resonant Level Model (IRLM), the Single Impurity Anderson Model (SIAM), as well as models with different multi site structures, have been subject of investigations in this context. In this work we give an overview of the different numerical approaches that have been successfully applied to the problem and go into considerable detail when we comment on the techniques that have been used to obtain the full I–V‐characteristics for the IRLM. Using a model of spinless fermions consisting of an extended interacting nanostructure attached to non‐interacting leads, we explain the method we use to obtain the current–voltage characteristics and discuss the finite size effects that have to be taken into account. We report results for the linear and finite bias conductance through a seven site structure with weak and strong nearest‐neighbor interactions. Comparison with exact diagonalisation results in the non‐interacting limit serve as a verification of the accuracy of our approach. Finally we discuss the possibility of effectively enlarging the finite system by applying damped boundaries and give an estimate of the effective system size and accuracy that can be expected in this case.

DOI
10.1002/andp.201000017

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Abstract Numerical time evolution of transport states using time dependent Density Matrix Renormalization Group (td‐DMRG) methods has turned out to be a powerful tool to calculate the linear and finite bias conductance of interacting impurity systems coupled to non‐interacting one‐dimensional leads. Several models, including the Interacting Resonant Level Model (IRLM), the Single Impurity Anderson Model (SIAM), as well as models with different multi site structures, have been subject of investigations in this context. In this work we give an overview of the different numerical approaches that have been successfully applied to the problem and go into considerable detail when we comment on the techniques that have been used to obtain the full I–V‐characteristics for the IRLM. Using a model of spinless fermions consisting of an extended interacting nanostructure attached to non‐interacting leads, we explain the method we use to obtain the current–voltage characteristics and discuss the finite size effects that have to be taken into account. We report results for the linear and finite bias conductance through a seven site structure with weak and strong nearest‐neighbor interactions. Comparison with exact diagonalisation results in the non‐interacting limit serve as a verification of the accuracy of our approach. Finally we discuss the possibility of effectively enlarging the finite system by applying damped boundaries and give an estimate of the effective system size and accuracy that can be expected in this case.
Numerical time evolution of transport states using time dependent Density Matrix Renormalization Group (td-DMRG) methods has turned out to be a powerful tool to calculate the linear and finite bias conductance of interacting impurity systems coupled to non-interacting one-dimensional leads. Several models, including the Interacting Resonant Level Model (IRLM), the Single Impurity Anderson Model (SIAM), as well as models with different multi site structures, have been subject of investigations in this context. In this work we give an overview of the different numerical approaches that have been successfully applied to the problem and go into considerable detail when we comment on the techniques that have been used to obtain the full I--V-characteristics for the IRLM. Using a model of spinless fermions consisting of an extended interacting nanostructure attached to non-interacting leads, we explain the method we use to obtain the current--voltage characteristics and discuss the finite size effects that have to be taken into account. We report results for the linear and finite bias conductance through a seven site structure with weak and strong nearest-neighbor interactions. Comparison with exact diagonalisation results in the non-interacting limit serve as a verification of the accuracy of our approach. Finally we discuss the possibility of effectively enlarging the finite system by applying damped boundaries and give an estimate of the effective system size and accuracy that can be expected in this case.
A new method to perform linear and finite bias conductance calculations in one dimensional systems based on the calculation of real time evolution within the Density Matrix Renormalization Group (DMRG) is presented. We consider a system of spinless fermions consisting of an extended interacting nanostructure attached to non-interacting leads. Results for the linear and finite bias conductance through a seven site structure with weak and strong nearest-neighbor interactions are presented. Comparison with exact diagonalization results in the non-interacting limit serve as verification of the accuracy of our approach. Our results show that interaction effects lead to an energy dependent self energy in the differential conductance.
We apply the adaptive time-dependent density-matrix renormalization-group method (tDMRG) to the study of transport properties of quantum-dot systems connected to metallic leads. Finite-size effects make the usual tDMRG description of the Kondo regime a numerically demanding task. We show that such effects can be attenuated by describing the leads by ``Wilson chains,'' in which the hopping matrix elements decay exponentially away from the impurity $({t}_{n}\ensuremath{\propto}{\ensuremath{\Lambda}}^{\ensuremath{-}n/2})$. For a given system size and in the linear-response regime, results for $\ensuremath{\Lambda}>1$ show several improvements over the undamped $\ensuremath{\Lambda}=1$ case: perfect conductance is obtained deeper in the strongly interacting regime and current plateaus remain well defined for longer time scales. Similar improvements were obtained in the finite-bias regime up to bias voltages of the order of the Kondo temperature. These results show that with the proposed modification, the tDMRG characterization of Kondo correlations in the transport properties can be substantially improved, while it turns out to be sufficient to work with much smaller system sizes. We discuss the numerical cost of this approach with respect to the necessary system sizes and the entanglement growth during the time evolution.
A procedure based on the recently developed ``adaptive'' time-dependent density-matrix-renormalization-group (DMRG) technique is presented to calculate the zero temperature conductance of nanostructures, such as quantum dots (QDs) or molecular conductors, when represented by a small number of active levels. The leads are modeled using noninteracting tight-binding Hamiltonians. The ground state at time zero is calculated at zero bias. Then, a small bias is applied between the two leads, the wave function is DMRG evolved in time, and currents are measured as a function of time. Typically, the current is expected to present periodicities over long times, involving intermediate well-defined plateaus that resemble steady states. The conductance can be obtained from those steady-state-like currents. To test this approach, several cases of interacting and noninteracting systems have been studied. Our results show excellent agreement with exact results in the noninteracting case. More importantly, the technique also reproduces quantitatively well-established results for the conductance and local density of states in both the cases of one and two coupled interacting QDs. The technique also works at finite bias voltages, and it can be extended to include interactions in the leads.
In this paper we present a novel approach combining linear response theory (Kubo) for the conductance and the Density Matrix Renormalization Group (DMRG). The system considered is one-dimensional and consists of non-interacting tight-binding leads coupled to an interacting nanostructure via weak links. Electrons are treated as spinless fermions and two different correlation functions are used to evaluate the conductance. Exact diagonalization calculations in the non-interacting limit serve as a benchmark for our combined Kubo and DMRG approach in this limit. Including both weak and strong interaction we present DMRG results for an extended nanostructure consisting of seven sites. For the strongly interacting structure a simple explanation of the position of the resonances is given in terms of hard-core particles moving freely on a lattice of reduced size.
Resonant tunneling through quantum dot under a finite bias voltage at zero temperature is investigated by using the adaptive time-dependent density matrix renormalization group(TdDMRG) method. Quantum dot is modeled by the Anderson Hamiltonian with the 1-D nearest-neighbor tight-binding leads. Initially the ground state wave function is calculated with the usual DMRG method. Then the time evolution of the wave function due to the slowly changing bias voltage between the two leads is calculated by using the TdDMRG technique. Even though the system size is finite, the expectation values of current operator show steady-like behavior for a finite time interval, in which the system is expected to resemble the real nonequilibrium steady state of the infinitely long system. We show that from the time intervals one can obtain quantitatively correct results for differential conductance in a wide range of bias voltage. Finally we observe an anomalous behavior in the expectation value of the double occupation operator at the dot $$ as a function of bias voltage.