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arXiv:1304.2216 · New Journal of Physics

Generalized projected dynamics for non-system observables of non-equilibrium quantum impurity models

Guy Cohen, Eli Y. Wilner, Eran Rabani

Published Jul 8, 2013• 67 citations
arXivHTMLPDF

Abstract

The reduced dynamics formalism has recently emerged as a powerful tool to study the dynamics of non-equilibrium quantum impurity models in strongly correlated regimes. Examples include the non-equilibrium Anderson impurity model near the Kondo crossover temperature and the non-equilibrium Holstein model, for which the formalism provides an accurate description of the reduced density matrix of the system for a wide range of timescales. In this work, we generalize the formalism to allow for non-system observables such as the current between the impurity and leads. We show that the equation of motion for the reduced observable of interest can be closed with the equation of motion for the reduced density matrix and demonstrate the new formalism for a generic resonant level model.

DOI
10.1088/1367-2630/15/7/073018

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The reduced dynamics formalism has recently emerged as a powerful tool to study the dynamics of non-equilibrium quantum impurity models in strongly correlated regimes. Examples include the non-equilibrium Anderson impurity model near the Kondo crossover temperature and the non-equilibrium Holstein model, for which the formalism provides an accurate description of the reduced density matrix of the system for a wide range of timescales. In this work, we generalize the formalism to allow for non-system observables such as the current between the impurity and leads. We show that the equation of motion for the reduced observable of interest can be closed with the equation of motion for the reduced density matrix and demonstrate the new formalism for a generic resonant level model.
A hierarchical equations of motion based numerical approach is developed for accurate and efficient evaluation of dynamical observables of strongly correlated quantum impurity systems. This approach is capable of describing quantitatively Kondo resonance and Fermi-liquid characteristics, achieving the accuracy of the latest high-level numerical renormalization group approach, as demonstrated on single-impurity Anderson model systems. Its application to a two-impurity Anderson model results in differential conductance versus external bias, which correctly reproduces the continuous transition from Kondo states of individual impurity to singlet spin states formed between two impurities. The outstanding performance on characterizing both equilibrium and nonequilibrium properties of quantum impurity systems makes the hierarchical equations of motion approach potentially useful for addressing strongly correlated lattice systems in the framework of dynamical mean-field theory.
In this paper, we present an extended dissipaton equation of motion for studying the dynamics of electronic impurity systems. Compared with the original theoretical formalism, the quadratic couplings are introduced into the Hamiltonian accounting for the interaction between the impurity and its surrounding environment. By exploiting the quadratic fermionic dissipaton algebra, the proposed extended dissipaton equation of motion offers a powerful tool for studying the dynamical behaviors of electronic impurity systems, particularly in situations where nonequilibrium and strongly correlated effects play significant roles. Numerical demonstrations are carried out to investigate the temperature dependence of the Kondo resonance in the Kondo impurity model.
We propose a hierarchical dynamics approach for evaluation of nonequilibrium dynamic response properties of quantum impurity systems. It is based on a hierarchical equations of motion formalism, in conjunction with a linear response theory established in the hierarchical Liouville space. This provides an accurate and universal tool for characterization of a variety of response and correlation functions of local impurities, as well as transport related response properties. The practicality of our proposed approach is demonstrated via the evaluation of various dynamical properties of a single-impurity Anderson model. These include the impurity spectral density, local charge fluctuation, local magnetic susceptibility, and current-voltage admittance, in both equilibrium and nonequilibrium situations. The numerical results are considered to be quantitatively accurate, as long as they converge quantitatively with respect to the truncation of the hierarchy.
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Memory effects play a key role in the dynamics of strongly correlated systems driven out of equilibrium. In this paper, we explore the nature of memory in the nonequilibrium Anderson impurity model. The Nakajima-Zwanzig-Mori formalism is used to derive an exact generalized quantum master equation for the reduced density matrix of the interacting quantum dot, which includes a non-Markovian memory kernel. A real-time path integral formulation is developed in which all diagrams are stochastically sampled in order to numerically evaluate the memory kernel. We explore the effects of temperature down to the Kondo regime, as well as the role of source-drain-bias voltage and bandwidth on the memory. Typically, the memory decays on time scales significantly shorter than the dynamics of the reduced density matrix itself, yet under certain conditions, it develops a low magnitude but long-ranged tail. In addition, we address the conditions required for the existence, uniqueness, and stability of a steady state.
Quantum impurity models with strong electron correlations, such as the paradigmatic Anderson Impurity Model (AIM), are central to our understanding of a range of physical phenomena including local moment formation, Coulomb blockade and Kondo screening. They describe magnetic atoms and molecules on surfaces, quantum dot circuits, and correlated materials through dynamical mean field theory. The physics of such systems in strongly non-equilibrium conditions is particularly complex and challenging to capture, whereas Gaussian models of free fermions can be easily solved. Here we show that the time-evolving dynamics of the AIM after a quench can be described by a completely non-interacting version of the model, at the expense of coupling to additional static auxiliary degrees of freedom. Starting from the full solution of the quenched AIM using ED and DMRG, we study the properties of this mapping using numerical optimization, and uncover intriguing structure in the auxiliary system. The method allows us to understand interacting non-equilibrium dynamics through the simpler lens of an effective non-interacting system of larger dimension.