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arXiv:0911.0587 · Physical Review B

Weak-coupling quantum Monte Carlo calculations on the Keldysh contour: Theory and application to the current-voltage characteristics of the Anderson model

Philipp Werner, Takashi Oka, Martin Eckstein, Andrew J. Millis

Published Jan 14, 2010• 136 citations

Abstract

We present optimized implementations of the weak-coupling continuous-time Monte Carlo method defined for nonequilibrium problems on the Keldysh contour. We describe and compare two methods of preparing the system before beginning the real-time calculation: the ``interaction quench'' and the ``voltage quench,'' which are found to be suitable for large and small voltage biases, respectively. We also discuss technical optimizations which increase the efficiency of the real-time measurements. The methods allow the accurate simulation of transport through quantum dots over wider interaction ranges and longer times than have heretofore been possible. The current-voltage characteristics of the particle-hole symmetric Anderson-impurity model is presented for interactions $U$ up to ten times the intrinsic level width $\ensuremath{\Gamma}$. We compare the Monte Carlo results to fourth-order perturbation theory, finding that perturbation theory is accurate up to $U\ensuremath{\approx}4\ensuremath{\Gamma}$ or for a voltage bias $V\ensuremath{\gtrsim}4\ensuremath{\Gamma}$. The interplay of voltage and temperature and the Coulomb blockade conductance regime are studied.

DOI: 10.1103/physrevb.81.035108

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We present optimized implementations of the weak-coupling continuous-time Monte Carlo method defined for nonequilibrium problems on the Keldysh contour. We describe and compare two methods of preparing the system before beginning the real-time calculation: the interaction quench'' and the voltage quench,'' which are found to be suitable for large and small voltage biases, respectively. We also discuss technical optimizations which increase the efficiency of the real-time measurements. The methods allow the accurate simulation of transport through quantum dots over wider interaction ranges and longer times than have heretofore been possible. The current-voltage characteristics of the particle-hole symmetric Anderson-impurity model is presented for interactions

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U\ensuremath{\approx}4\ensuremath{\Gamma}

or for a voltage bias

V\ensuremath{\gtrsim}4\ensuremath{\Gamma}

. The interplay of voltage and temperature and the Coulomb blockade conductance regime are studied.

The nonequilibrium spectral properties of the Anderson impurity model with a chemical potential bias are investigated within a numerically exact real-time quantum Monte Carlo formalism. The two-time correlation function is computed in a form suitable for nonequilibrium dynamical mean field calculations. Additionally, the evolution of the model's spectral properties are simulated in an alternative representation, defined by a hypothetical but experimentally realizable weakly coupled auxiliary lead. The voltage splitting of the Kondo peak is confirmed and the dynamics of its formation after a coupling or gate quench are studied. This representation is shown to contain additional information about the dot's population dynamics. Further, we show that the voltage-dependent differential conductance gives a reasonable qualitative estimate of the equilibrium spectral function, but significant qualitative differences are found including incorrect trends and spurious temperature dependent effects.

We propose a numerically exact approach to nonequilibrium real-time dynamics that is applicable to quantum impurity models coupled to biased noninteracting leads, such as those relevant to quantum transport in nanoscale devices. The method is based on a diagrammatic Monte Carlo sampling of the real-time perturbation theory along the Keldysh contour. We benchmark the method on a noninteracting resonant-level model and, as a first nontrivial application, we study zero-temperature nonequilibrium transport through a vibrating molecule.

We present a simple, general purpose, quantum Monte Carlo algorithm for out-of-equilibrium interacting nanoelectronic systems. It allows one to systematically compute the expansion of any physical observable (such as current or density) in powers of the electron-electron interaction coupling constant

U

. It is based on the out-of-equilibrium Keldysh Green's function formalism in real-time and corresponds to evaluating all the Feynman diagrams to a given order

{U}^{n}

(up to

n=15

in the present work). A key idea is to explicitly sum over the Keldysh indices in order to enforce the unitarity of the time evolution. The coefficients of the expansion can easily be obtained for long-time, stationary regimes, even at zero temperature. We then illustrate our approach with an application to the Anderson model, an archetype interacting mesoscopic system. We recover various results of the literature such as the spin susceptibility or the ``Kondo ridge'' in the current-voltage characteristics. In this case, we found the Monte Carlo free of the sign problem even at zero temperature, in the stationary regime and in absence of a particle-hole symmetry. The main limitation of the method is the lack of convergence of the expansion in

U

for large

U

, i.e., a mathematical property of the model rather than a limitation of the Monte Carlo algorithm. Standard extrapolation methods of divergent series can be used to evaluate the series in the strong correlation regime.

One of the main open problems in the field of transport in strongly interacting nanostructures is the understanding of currents beyond the linear response regime. In this work, we consider the single-impurity Anderson model and use the adaptive time-dependent density matrix renormalization group method to compute real-time currents out of equilibrium. We first focus on the particle-hole symmetric point where Kondo correlations are the strongest and then extend the study of the nonequilibrium transport to the mixed-valence regime. As a main result, we present accurate data for the current-voltage characteristics of this model.

We extend the recently developed real-time diagrammatic Monte Carlo method, in its hybridization expansion formulation, to the full Kadanoff-Baym-Keldysh contour. This allows us to study real-time dynamics in correlated impurity models starting from an arbitrary, even interacting, initial density matrix. As a proof of concept, we apply the algorithm to study the nonequilibrium dynamics after a local quantum quench in the Anderson impurity model. Being a completely general approach to real-time dynamics in quantum impurity models, it can be used as a solver for nonequilibrium dynamical mean field theory.