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.
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.
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 improve the density-matrix renormalization group (DMRG) evaluation of the Kubo formula for the zero-temperature linear conductance of one-dimensional correlated systems. The dynamical DMRG is used to compute the linear response of a finite system to an applied ac source-drain voltage; then the low-frequency finite-system response is extrapolated to the thermodynamic limit to obtain the dc conductance of an infinite system. The method is demonstrated on the one-dimensional spinless fermion model at half filling. Our method is able to replicate several predictions of the Luttinger liquid theory such as the renormalization of the conductance in a homogeneous conductor, the universal effects of a single barrier, and the resonant tunneling through a double barrier.
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.
Interesting discoveries have recently been made in the area of solid state physics and nano technology regarding electronic systems that are confined to reduced spatial dimensions. Under these circumstances the quantum nature of the electrons begins to surface. In the one-dimensional case this class of systems is therefore called quantum- or nano wires. The conductance in such quantum wires becomes quantized and is dominated by electronic correlations. The goal of this thesis was to develop a fast and efficient procedure to describe a realistic transport measurement for such systems. The method presented here is in- spired by the work of Dan Bohr et al. who used the Density Matrix Renormalization Group algorithm(DMRG) to evaluate the Kubo formula for the linear response to an applied potential. In a similar way the procedure of this thesis also computes the Kubo formula using the dynamical DMRG in order to calculate the low frequency response to an AC source-drain voltage of finite systems. Making use of a special finite size scaling allows to extrapolate the results into the thermodynamic limit and to finally obtain the linear DC-conductance. First, the theoretical framework for the new method was developed. This was fol- lowed by testing the new approach in different model systems. The first model com- prised of a tight-binding chain that was filled with spinless, noninteracting fermions. In subsequent steps the model was expanded by introducing a homogeneous in- teraction in the Luttinger liquid regime followed by inhomogeneous interaction for different chain sections. The potential impact of impurities was assessed in the above models. Lastly, the two cases of the Hubbard chain of spin- 2 1 electrons and of the Holstein model in one dimension, where Einstein phonons interact with spinless fermions, were studied. The results of the proposed method are consistent with Luttinger liquid and Lan- dauer scattering theory. In the Luttinger liquid regime the influence of phonons to the charge structure factor was confirmed. A dependency on the length of the quantum wire in the form of dissipation effects for the conductance was not observed.
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.
This thesis develops a rigorous large-scale response theory for gapless lattice Fermi systems in low dimensions, focusing on the mathematical validity of linear response -- the Kubo formula -- for transport in gapless systems, where the absence of a spectral gap prevents the direct use of standard adiabatic theorems. After reviewing fundamental results on many-body lattice fermions, we first analyse the transport properties of non-interacting, gapless, one-dimensional quantum systems and of the edge modes of two-dimensional topological insulators. We prove the validity of the Kubo formula, in the zero-temperature and infinite-volume limit, for perturbations that are weak and slowly varying in space and time. The proof relies on expressing the real-time Duhamel series in imaginary time, which ensures its convergence uniformly in the scaling parameter and system size, at low temperatures. This representation also reveals a key cancellation in the scaling limit, linked to the emergent anomalous chiral symmetry of relativistic one-dimensional fermions. As a consequence, in the combined low-temperature and scaling limit, the linear response provides the full physical response. In particular, the method yields a dynamical proof of the quantisation of edge conductance in two-dimensional quantum Hall systems. The analysis is then extended to weakly interacting, gapless one-dimensional fermionic systems, for which we obtain explicit expressions for the response functions in terms of renormalised Fermi velocities. Here, sharp estimates for interacting Euclidean correlation functions are established via rigorous renormalisation-group methods. The asymptotic exactness of linear response persists due to a cancellation in the scaling limit of the correlations -- reminiscent of bosonisation -- derived rigorously from emergent chiral Ward identities.