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arXiv:1307.7506 · Physical Review B

Transport through nanostructures: Finite time versus finite size

Peter Schmitteckert, Sam T. Carr, Hubert Saleur

Published Feb 5, 2014• 10 citations
arXivHTMLPDF

Abstract

Numerical simulations and experiments on nanostructures out of equilibrium usually exhibit strong finite size and finite measuring time ${t}_{\mathrm{m}}$ effects. We discuss how these affect the determination of the full counting statistics for a general quantum impurity problem. We find that, while there are many methods available to improve upon finite-size effects, any real-time simulation or experiment will still be subject to finite-time effects: In short size matters, but time is limiting. We show that the leading correction to the cumulant generating function (CGF) at zero temperature for single-channel quantum impurity problems is proportional to $ln{t}_{\mathrm{m}}$, where the constant of proportionality is universally related to the steady state CGF itself for non-interacting systems; universal in this context means independent of details of the quench procedure, i.e., independent of the switching on of both voltage and counting field. We give detailed numerical evidence for the case of the self-dual interacting resonant level model that this relation survives the addition of interactions. This allows the extrapolation of finite measuring time in our numerics to the long-time limit, in excellent agreement with Bethe-ansatz results.

DOI
10.1103/physrevb.89.081401

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arXiv:1801.05010•Physical review. B./Physical review. B•Mar 6, 2018•65 citations
Numerical simulations and experiments on nanostructures out of equilibrium usually exhibit strong finite size and finite measuring time tm{t}_{\mathrm{m}}tm​ effects. We discuss how these affect the determination of the full counting statistics for a general quantum impurity problem. We find that, while there are many methods available to improve upon finite-size effects, any real-time simulation or experiment will still be subject to finite-time effects: In short size matters, but time is limiting. We show that the leading correction to the cumulant generating function (CGF) at zero temperature for single-channel quantum impurity problems is proportional to lntmln{t}_{\mathrm{m}}lntm​, where the constant of proportionality is universally related to the steady state CGF itself for non-interacting systems; universal in this context means independent of details of the quench procedure, i.e., independent of the switching on of both voltage and counting field. We give detailed numerical evidence for the case of the self-dual interacting resonant level model that this relation survives the addition of interactions. This allows the extrapolation of finite measuring time in our numerics to the long-time limit, in excellent agreement with Bethe-ansatz results.
The full counting statistics of charge transport is the probability distribution that n electrons have flown through the system in measuring time t m . The cumulant generating function (CGF) of this distribution has been well studied in the long time limit , however there are relatively few results on the finite measuring time corrections to this. In this work, we study the leading finite time corrections to the CGF of interacting Fermi systems with a single transmission channel at zero temperature but driven out of equilibrium by a bias voltage. We conjecture that the leading finite time corrections are logarithmic in t m with a coefficient universally related to the long time limit. We provide detailed numerical evidence for this with reference to the self-dual interacting resonant level model. This model further contains a phase transition associated with the fractionalization of charge at a critical bias voltage. This transition manifests itself technically as branch points in the CGF. We provide numerical results of the dependence of the CGF on measuring time for model parameters in the vicinity of this transition, and thus identify features in the time evolution associated with the phase transition itself.
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.
Unless constrained by symmetry, measurement of an observable on an ensemble of identical quantum systems returns a distribution of values which are encoded in the full counting statistics. While the mean value of this distribution is important for determining certain properties of a system, the full distribution can also exhibit universal behavior. In this paper we study the full counting statistics of particle number in one-dimensional interacting Bose and Fermi gases which have been quenched far from equilibrium. In particular, we consider the time evolution of the Lieb-Liniger and Gaudin-Yang models quenched from a Bose-Einstein condensate initial state and calculate the full counting statistics of the particle number within a subsystem. We show that the scaled cumulants of the charge in the initial state and at long times are simply related and, in particular, the latter are independent of the model parameters. Using the quasiparticle picture we obtain the full time evolution of the cumulants and find that although their end points are fixed, the finite-time dynamics depends strongly on the model parameters. We go on to construct the scaled cumulant generating functions and from this determine the limiting charge probability distributions at long time which are shown to exhibit distinct, nontrivial, and non-Gaussian fluctuations and large deviations. Published by the American Physical Society 2024
In this thesis, certain aspects of time-dependent quantum transport properties in nanoscale structures are investigated using the non-equilibrium Green’s function(NEGF). For the transient behaviors of the electric current and heat current after a quantum quench, the exact analytical solution beyond the wide band limit (WBL) is obtained for the first time which paves a way to simulate realistic physical systems. The time-dependent thermoelectric effects are evaluated based on this general solution and it is shown that the time-dependent thermopower is significantly enhanced in the transient regime. By taking initial correlation into account, the combined Caroli and Cini scheme in quantum transport is proposed and its exact solution beyond WBL is derived for the first time as well. Furthermore, besides the average electric current, a current conserving and gauge invariant formalism for the total electric current operator is established at equilibrium state with the help of the electric displacement current operator formulated by taking the Coulomb interaction into account at the operator level. By means of the NEGF and the scattering matrix theory (SMT), a current conserving theory for equilibrium noise power at finite frequencies is obtained which solves one of the fundamental problems in quantum transport. A generalized equilibrium fluctuation-dissipation theorem is presented. Moreover, the entanglement entropy \nfluctuation and its distribution are investigated by introducing the entanglement entropy operator. The essential feature of the entanglement entropy generated by quantum transport is that it is a stochastic quantity whose fluctuation and distribution can be obtained from the generating function (GF). Combining the NEGF with the Grassmann algebras, it is found that the GF of entanglement \nentropy can be expressed in terms of the lesser Green’s function of the subsystem. The general relation connecting the full counting statistics (FCS) of charge transfer with the GF of entanglement entropy is found which indicates that, from the experimental point of view, the entanglement entropy fluctuation can be measured indirectly. As an illustration of the general theoretical framework of entanglement entropy fluctuation, the quantum point contact (QPC) system as well as the quantum dot (QD) system are studied. It is shown that for a QPC under dc bias, the entanglement entropy is maximized and fluctuationless when the transmission coefficient T is one-half. For the quantum dot system, at short times, a universal scaling relation among transient peak values of cumulants of entanglement entropy is shown.
We present here the details of a method [A. B. Culver and N. Andrei, Many-body wavefunctions for quantum impurities out of equilibrium, Phys. Rev. B 103, L201103 (2021)] for calculating the time-dependent many-body wavefunction that follows a local quench. We apply the method to the voltage-driven nonequilibrium Kondo model to find the exact time-evolving wavefunction following a quench where the dot is suddenly attached to the leads at t=0t=0t=0. The method, which does not use Bethe ansatz, also works in other quantum impurity models and may be of wider applicability. We show that the long-time limit (with the system size taken to infinity first) of the time-evolving wavefunction of the Kondo model is a current-carrying nonequilibrium steady state that satisfies the Lippmann-Schwinger equation. We show that the electric current in the time-evolving wavefunction is given by a series expression that can be expanded either in weak coupling or in strong coupling, converging to all orders in the steady-state limit in either case. The series agrees to leading order with known results in the well-studied regime of weak antiferromagnetic coupling and also reveals a universal regime of strong ferromagnetic coupling with Kondo temperature ${T}_{K}^{(F)}=D{e}^{\ensuremath{-}\frac{3{\ensuremath{\pi}}^{2}}{8}\ensuremath{\rho}|J|}$ ($J<0, \ensuremath{\rho}|J|\ensuremath{\rightarrow}\ensuremath{\infty}$). In this regime, the differential conductance dI/dVdI/dVdI/dV reaches the unitarity limit 2e2/h2{e}^{2}/h2e2/h asymptotically at large voltage or temperature.
The time-dependent full counting statistics of charge transport through an interacting quantum junction is evaluated from its generating function, controllably computed with the inchworm Monte Carlo method. Exact noninteracting results are reproduced; then, we continue to explore the effect of electron-electron interactions on the time-dependent charge cumulants, first-passage time distributions, and nnn-electron transfer distributions. We observe a crossover in the noise from Coulomb blockade to Kondo-dominated physics as the temperature is decreased. In addition, we uncover long-tailed spin distributions in the Kondo regime and analyze queuing behavior caused by correlations between single-electron transfer events.