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arXiv:2111.01120 · Physical review. B./Physical review. B

Conductance of correlated many-fermion systems from charge fluctuations

Yuchi He, Dante Marvin Kennes, V. Meden

Published Apr 13, 2022• 3 citations
arXivHTMLPDF

Abstract

We put forward a relation between the static charge fluctuations and the conductance of correlated many-fermion systems at zero temperature, avoiding the use of time-dependent fluctuations as in the fluctuation-dissipation theorem. Static charge fluctuations can efficiently be computed for low-dimensional systems using tensor network approaches, while the conductance is often significantly more difficult to obtain, requiring a challenging low-frequency linear response computation or an explicit time evolution. We put this relation to the test for quantum dot and quantum point contact setups, where in limiting cases exact results are known. Our study includes systems in which the one-dimensional reservoirs are interacting.

DOI
10.1103/physrevb.105.165120

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We put forward a relation between the static charge fluctuations and the conductance of correlated many-fermion systems at zero temperature, avoiding the use of time-dependent fluctuations as in the fluctuation-dissipation theorem. Static charge fluctuations can efficiently be computed for low-dimensional systems using tensor network approaches, while the conductance is often significantly more difficult to obtain, requiring a challenging low-frequency linear response computation or an explicit time evolution. We put this relation to the test for quantum dot and quantum point contact setups, where in limiting cases exact results are known. Our study includes systems in which the one-dimensional reservoirs are interacting.
The low-temperature AC conductance of a one-dimensional electron system with a strong interaction of finite range is calculated by using linear response theory. The conductance factorizes into parts which depend on the internal properties of the system, and the external probe. For short-range interaction, the result resembles that for non-interacting electrons, but with the zero-frequency limit and the Fermi velocity renormalized by the interaction strength. For strong and long-range interaction, the conductance shows a peak that is related to charge-wave excitations. In this limit, the AC conductance can be simulated by a quantum capacitance and a quantum inductance .
We study the dynamics in low dimensional correlated quantum systems. Advances in this thesis are three-fold. First, as a methodical point we extend the real-time formulation of the functional renormalization group approach introduced in [1]. Describing the transient time dependence of a given microscopic model prepared in an initial non-equilibrium state poses a formidable challenge in the presence of interaction. We successfully extend the functional renormalization group approach of [1] to finite temperature as well as extended interacting geometries. Furthermore, for both extensions we test the validity of the perturbatively motivated treatment by explicitly benchmarking to complementary methods and find that for not too large interactions the functional renormalization group proves to be quantitatively accurate. Second, we examine the non-equilibrium dynamics of open as well as closed quantum systems in the presence of correlations. We study a prominent prototype model for coherence in small quantum systems coupled to dissipative environments, namely the spin-boson model. We reveal that prevailing literature does not cover the full complexity of the dynamics encountered. The relaxation dynamics formerly classified as coherent and incoherent should be refined according to the results presented in this thesis: we propose a subdivision of the coherent regime into partially and asymptotically coherent. For the former the coherence signature of the dynamics is restricted only to intermediate times, while in the latter it extends in the asymptotic regime. Interestingly, we explicitly show that elevating temperature might be conducive to coherence, which is opposite to the expected behavior. For those closed quantum systems known to fall into the so called Luttinger Liquid universality class in equilibrium, we provide evidence that the equilibrium concept of universality seemingly carries over to the non-equilibrium steady-state. This is rather counter-intuitive, as universality is usually attributed to the low energy degrees of freedom only. Third, we study the thermoelectric transport properties of quantum dots. Quantum dots are promising candidates for highly efficient thermoelectric devices as they are not bounded by the Wiedemann-Franz law which strongly limits the efficiency of ordinary bulk systems. For this we study a charge fluctuating model and a model which additionally allows for spin fluctuations and phononic degrees of freedom. We find that in certain regimes quantum dots show very high efficiency when operated as power engines. [1] D. M. Kennes, ‘A Renormalization Group Approach to the Time Evolution of Correlated Quantum Dots’ Master’s thesis, RWTH Aachen (2011).
The low-temperature AC conductance of a one-dimensional electron system with a strong interaction of finite range is calculated by using linear response theory. The conductance factorizes into parts which depend on the internal properties of the system, and the external probe. For short-range interaction, the result resembles that for non-interacting electrons, but with the zero-frequency limit and the Fermi velocity renormalized by the interaction strength. For strong and long-range interaction, the conductance shows a peak that is related to charge-wave excitations. In this limit, the AC conductance can be simulated by a quantum capacitance and a quantum inductance.
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.
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.
Abstract The understanding of out-of-equilibrium fluctuation relations in small open quantum systems has been a focal point of research in recent years. In particular, for systems with adiabatic time-dependent driving, it was shown that the fluctuation relations known from stationary systems do no longer apply due the geometric nature of the pumping current response. However, the precise physical interpretation of the corrected pumping fluctuation relations as well as the role of many-body interactions remained unexplored. Here, we study quantum systems with many-body interactions subject to slow time-dependent driving, and show that fluctuation relations of the charge current can in general not be formulated without taking into account the total energy current put into the system through the pumping process. Moreover, we show that this correction due to the input energy is nonzero only when Coulomb-interactions are present. Thus, fluctuation response relations offer an until now unrevealed opportunity to probe many-body correlations in quantum systems. We demonstrate our general findings at the concrete example of a single-level quantum dot model, and propose a scheme to measure the interaction-induced discrepancies from the stationary case.