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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 solversin practical nonequilibrium calculations. We review the main characteristics ofthe methods and discuss the scaling of the associated numerical effort. Wesubstantiate our discussion with explicit numerical results for thenonequilibrium transport properties of a single-site Anderson impurity. Thenumerical effort of the HQME scheme scales linearly with the simulation timebut increases (at worst exponentially) with decreasing temperature. Incontrast, CT-QMC is less restricted by temperature at short times, but ingeneral the cost of going to longer times is also exponential. Afterestablishing the numerical exactness of the HQME scheme, we use it to elucidatethe influence of different ways to induce transport through the impurity on theinitial dynamics, discuss the phenomenon of coherent current oscillations,known as current ringing, and explain the non-monotonic temperature dependenceof the steady-state magnetization as a result of competing broadening effects.We also elucidate the pronounced non-linear magnetization dynamics, whichappears on intermediate time scales in the presence of an asymmetric couplingto the electrodes.

physical review. b, condensed matter and materials physics

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