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arXiv:1105.1175 · Physical Review B

Numerically exact long-time behavior of nonequilibrium quantum impurity models

Emanuel Gull, David R. Reichman, Andrew J. Millis

Published Aug 29, 2011• 75 citations
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Abstract

A Monte Carlo sampling of diagrammatic corrections to the noncrossing approximation is shown to provide numerically exact estimates of the long-time dynamics and steady-state properties of nonequilibrium quantum impurity models. This ``bold'' expansion converges uniformly in time and significantly ameliorates the sign problem that has heretofore limited the power of real-time Monte Carlo approaches to strongly interacting real-time quantum problems. The approach enables the study of previously intractable problems ranging from generic long-time nonequilibrium transport characteristics in systems with large on-site repulsion to the direct description of spectral functions on the real frequency axis in dynamical mean field theory.

DOI
10.1103/physrevb.84.085134

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A Monte Carlo sampling of diagrammatic corrections to the noncrossing approximation is shown to provide numerically exact estimates of the long-time dynamics and steady-state properties of nonequilibrium quantum impurity models. This ``bold'' expansion converges uniformly in time and significantly ameliorates the sign problem that has heretofore limited the power of real-time Monte Carlo approaches to strongly interacting real-time quantum problems. The approach enables the study of previously intractable problems ranging from generic long-time nonequilibrium transport characteristics in systems with large on-site repulsion to the direct description of spectral functions on the real frequency axis in dynamical mean field theory.
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.
This work is motivated by the fact that the investigation of non-equilibrium phenomena in strongly correlated electron systems has developed into one of the most active and exciting branches of condensed matter physics as it provides rich new insights that could not be obtained from the study of equilibrium situations. However, a theoretical description of those phenomena is missing. Therefore, in this thesis, we develop a numerical method that can be used to study two minimal models -- the Hubbard model and the Anderson impurity model with general parameter range and time dependence. We begin by introducing the theoretical framework and the general features of the Hubbard model. We then describe the dynamical mean field theory (DMFT), which was first invented by Georges in 1992. It provides a feasible way to approach strongly correlated electron systems and reduces the complexity of the calculations via a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. We employ the non-equilibrium extension of DMFT and map the Hubbard model to the single impurity Anderson model (SIAM). Since the fundamental component of the DMFT method is a solver of the single impurity Anderson model, we continue with a description of the formalism to study the real-time dynamics of the impurity model staring at its thermal equilibrium state. We utilize the non-equilibrium strong-coupling perturbation theory and derive semi-analytical approximation methods such as the non-crossing approximation (NCA) and the one-crossing approximation (OCA). We then use the Quantum Monte-Carlo method (QMC) as a numerically exact method and present proper measurements of local observables, current and Green's functions. We perform simulations of the current after a quantum quench from equilibrium by rapidly applying a bias voltage in a wide range of initial temperatures. The current exhibits short equilibrium times and saturates upon the decrease of temperature at all times, indicating Kondo behavior both in the transient regime and in the steady state. However, this bare QMC solver suffers from a dynamical sign problem for long time propagations. To overcome the limitations of this bare treatment, we introduce the ``Inchworm algorithm'', based on iteratively reusing the information obtained in previous steps to extend the propagation to longer times and stabilize the calculations. We show that this algorithm greatly reduces the required order for each simulation and re-scales the exponential challenge to quadratic in time. We introduce a method to compute Green's functions, spectral functions, and currents for inchworm Monte Carlo and show how systematic error assessments in real time can be obtained. We illustrate the capabilities of the algorithm with a study of the behavior of quantum impurities after an instantaneous voltage quench from a thermal equilibrium state. We conclude with the applications of the unbiased inchworm impurity solver to DMFT calculations. We employ the methods for a study of the one-band paramagnetic Hubbard model on the Bethe lattice in equilibrium, where the DMFT approximation becomes exact. We begin with a brief introduction of the Mott metal insulator phase diagram. We present the results of both real time Green's functions and spectral functions from our nonequilibrium calculations. We observe the metal-insulator crossover as the on-site interaction is increased and the formation of a quasi-particle peak as the temperature is lowered. We also illustrate the convergence of our algorithms in different aspects.
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.
We investigate the dynamical and steady-state spin response of the nonequilibrium Anderson model to magnetic fields, bias voltage, and temperature using a numerically exact method combining a bold-line quantum Monte Carlo technique with a memory function formalism. We obtain converged results in a range of previously inaccessible regimes, in particular calculating the spin dynamics for a range of temperatures down to the crossover to the Kondo domain. We provide predictions for nonequilibrium phenomena, including nonmonotonic temperature dependence of observables at high bias voltage and oscillatory quench dynamics at high magnetic fields.
We develop an iterative, numerically exact approach for the treatment of nonequilibrium quantum transport and dissipation problems that avoids the real-time sign problem associated with standard Monte Carlo techniques. The method requires a well-defined decorrelation time of the nonlocal influence functional for proper convergence to the exact limit. Since finite decorrelation times may arise either from temperature or from a voltage drop at zero temperature, the approach is well suited for the description of the real-time dynamics of single-molecule devices and quantum dots driven to a steady state via interaction with two or more electron leads. We numerically investigate two nontrivial models: the evolution of the nonequilibrium population of a two-level system coupled to two electronic reservoirs, and quantum transport in the nonequilibrium Anderson model. For the latter case, two distinct formulations are described. Results are compared to those obtained by other techniques.