Transport through an Anderson impurity: Current ringing, non-linear magnetization and a direct comparison of continuous-time quantum Monte Carlo and hierarchical quantum master equations

TL;DR

This paper compares the hierarchical quantum master equation and continuous-time quantum Monte Carlo methods for nonequilibrium transport in an Anderson impurity, highlighting their efficiencies, limitations, and physical insights into current ringing and magnetization dynamics.

Contribution

It provides a detailed comparison of HQME and CT-QMC methods, demonstrating the applicability of HQME for nonequilibrium impurity problems and analyzing complex dynamical phenomena.

Findings

HQME scales linearly with simulation time but exponentially with decreasing temperature.

CT-QMC is efficient at short times and higher temperatures but becomes costly at longer times.

The study reveals non-linear magnetization dynamics and current ringing phenomena.

Abstract

We give a detailed comparison of the hierarchical quantum master equation (HQME) method to a continuous-time quantum Monte Carlo (CT-QMC) approach, assessing the usability of these numerically exact schemes as impurity solvers in practical nonequilibrium calculations. We review the main characteristics of the methods and discuss the scaling of the associated numerical effort. We substantiate our discussion with explicit numerical results for the nonequilibrium transport properties of a single-site Anderson impurity. The numerical effort of the HQME scheme scales linearly with the simulation time but increases (at worst exponentially) with decreasing temperature. In contrast, CT-QMC is less restricted by temperature at short times, but in general the cost of going to longer times is also exponential. After establishing the numerical exactness of the HQME scheme, we use it to elucidate the influence of different ways to induce transport through the impurity on the initial dynamics, discuss the phenomenon of coherent current oscillations, known as current ringing, and explain the non-monotonic temperature dependence of the steady-state magnetization as a result of competing broadening effects. We also elucidate the pronounced non-linear magnetization dynamics, which appears on intermediate time scales in the presence of an asymmetric coupling to the electrodes.