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arXiv:cond-mat/0604583 · Physical Review B

Mapping of strongly correlated steady-state nonequilibrium system to an effective equilibrium

Jong E. Han

Published Mar 30, 2007• 24 citations
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Abstract

By mapping steady-state nonequilibrium to an effective equilibrium, we formulate nonequilibrium problems within an equilibrium picture. The Hamiltonian in the open system is rewritten in terms of scattering states with appropriate boundary condition. We first study the analytic properties of many-body scattering states, impose the boundary-condition operator in a statistical operator and prove that this mapping is equivalent to the linear-response theory in the low-bias limit. In an example of infinite-$U$ Anderson impurity model, we approximately solve the scattering state creation operators, based on which we derive the bias operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ to construct the nonequilibrium ensemble in the form of the Boltzmann factor ${e}^{\ensuremath{-}\ensuremath{\beta}(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{-}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y})}$. The resulting effective Hamiltonian is solved by noncrossing approximation. We obtain the $I\text{\ensuremath{-}}V$ features of Kondo anomaly conductance at zero bias, inelastic transport via the charge excitation on the quantum dot, and significant inelastic current background over a wide range of bias.

DOI
10.1103/physrevb.75.125122

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By mapping steady-state nonequilibrium to an effective equilibrium, we formulate nonequilibrium problems within an equilibrium picture. The Hamiltonian in the open system is rewritten in terms of scattering states with appropriate boundary condition. We first study the analytic properties of many-body scattering states, impose the boundary-condition operator in a statistical operator and prove that this mapping is equivalent to the linear-response theory in the low-bias limit. In an example of infinite-UUU Anderson impurity model, we approximately solve the scattering state creation operators, based on which we derive the bias operator Y\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}Y\ifmmode^\elseˆ\fi​ to construct the nonequilibrium ensemble in the form of the Boltzmann factor e\ensuremath−\ensuremathβ(H\ifmmode^\elseˆ\fi\ensuremath−Y\ifmmode^\elseˆ\fi){e}^{\ensuremath{-}\ensuremath{\beta}(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{-}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y})}e\ensuremath−\ensuremathβ(H\ifmmode^\elseˆ\fi​\ensuremath−Y\ifmmode^\elseˆ\fi​). The resulting effective Hamiltonian is solved by noncrossing approximation. We obtain the I\ensuremath-VI\text{\ensuremath{-}}VI\ensuremath-V features of Kondo anomaly conductance at zero bias, inelastic transport via the charge excitation on the quantum dot, and significant inelastic current background over a wide range of bias.
We develope a method of mapping quantum nonequilibrium steady-state to an effective equilibrium system and present an algorithm to calculate electron-transport using an equilibrium technique. A systematic implementation of boundary conditions in steady-state nonequilibrium is made in the statistical operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ constructed from scattering state operators. We explicitly demonstrate the equivalence of this method to nonequilibrium Green function techniques for a noninteracting quantum dot model. In electron-phonon coupled quantum dot systems, we formulate an algorithm to construct the statistical bias operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ and perform a full many-body calculation with the quantum Monte Carlo technique. The results coherently demonstrate various transport behaviors such as phonon dephasing, $I\ensuremath{-}V$ staircase, and phonon-assisted tunneling phenomena. This formulation makes the existing computational quantum many-body techniques applicable to quantum steady-state nonequilibrium problems, which will complement the theories based on the diagrammatic approach.
The theoretical description of strongly correlated quantum systems out of equilibrium presents several challenges and a number of open questions persist. In this paper we focus on nonlinear electronic transport through an interacting quantum dot maintained at finite bias using a concept introduced by Hershfield [Phys. Rev. Lett. 70, 2134 (1993)] whereby one can express such nonequilibrium quantum impurity models in terms of the system's Lippmann-Schwinger operators. These scattering operators allow one to reformulate the nonequilibrium problem as an effective equilibrium problem associated with a modified Hamiltonian. In this paper we provide a pedagogical analysis of the core concepts of the effective equilibrium theory. First, we demonstrate the equivalence between observables computed using the Schwinger-Keldysh framework and the effective equilibrium approach, and relate the Green's functions in the two theoretical frameworks. Second, we expound some applications of this method in the context of interacting quantum impurity models. We introduce a novel framework to treat effects of interactions perturbatively while capturing the entire dependence on the bias voltage. For the sake of concreteness, we employ the Anderson model as a prototype for this scheme. Working at the particle-hole symmetric point, we investigate the fate of the Abrikosov-Suhl resonance as a function of bias voltage and magnetic field.
Numerical renormalization group (NRG) is formulated for nonequilibrium steady-state by converting finite-lattice many-body eigenstates into scattering states. Extension of the full-density-matrix NRG for a biased Anderson impurity model, simplified by formulating with the original orbital basis as the Hamiltonian, enables detailed studies of the sub-Kondo spectral evolution in the zero-temperature limit, confirming the double-resonance structure at bias of the Kondo energy scale TKT_KTK​. The distribution shows distinct multi-scale spectral features at energy ωωω below the Kondo scale (ω≲TKω\lesssim T_Kω≲TK​) and near the bias (ω≳Vω\gtrsim Vω≳V), leading to the nonequilibrium temperature TlocT_{\rm loc}Tloc​ local to the Kondo dot scaling as kBTloc≈Vk_BT_{\rm loc}\approx VkB​Tloc​≈V for V≫TKV\gg T_KV≫TK​. The current-voltage relation in the low-temperature limit (T≪TKT\ll T_KT≪TK​) deviates from the unitary limit as the bias exceeds the Kondo scale (V/2≳TKV/2\gtrsim T_KV/2≳TK​) and reaches the current saturation regime.
We develop an exact nonperturbative framework to compute steady-state properties of quantum impurities subject to a finite bias. We show that the steady-state physics of these systems is captured by nonequilibrium scattering eigenstates which satisfy an appropriate Lippman-Schwinger equation. Introducing a generalization of the equilibrium Bethe ansatz--the nonequilibrium Bethe ansatz--we explicitly construct the scattering eigenstates for the interacting resonance level model and derive exact, nonperturbative results for the steady-state properties of the system.
We present improvements of a recently introduced numerical method [E. Arrigoni et al., Phys. Rev. Lett. 110, 086403 (2013)] to compute steady-state properties of strongly correlated electronic systems out of equilibrium. The method can be considered as a nonequilibrium generalization of exact diagonalization based dynamical mean-field theory (DMFT). The key modification for the nonequilibrium situation consists in addressing the DMFT impurity problem within an auxiliary system consisting of the correlated impurity, Nb{N}_{b}Nb​ uncorrelated bath sites, and two Markovian environments (sink and reservoir). Algorithmic improvements in the impurity solver allow to treat efficiently larger values of Nb{N}_{b}Nb​ than previously in DMFT. This increases the accuracy of the results and is crucial for a correct description of the physical behavior of the system in the relevant parameter range including a semiquantitative description of the Kondo regime. To illustrate the approach, we consider a monoatomic layer of correlated orbitals, described by the single-band Hubbard model, attached to two metallic leads. The nonequilibrium situation is driven by a bias voltage applied to the leads. For this system, we investigate the spectral function and the steady-state current-voltage characteristics in the weakly as well as in the strongly interacting limit. In particular, we investigate the nonequilibrium behavior of quasiparticle excitations within the Mott gap of the correlated layer. We find for low-bias voltage Kondo-type behavior in the vicinity of the insulating phase. In particular, we observe a splitting of the Kondo resonance as a function of the bias voltage.
We present a numerical method for the study of correlated quantum impurity problems out of equilibrium, which is particularly suited to address steady-state properties within dynamical mean field theory. The approach, recently introduced by Arrigoni et al. [Phys. Rev. Lett. 110, 086403 (2013)], is based upon a mapping of the original impurity problem onto an auxiliary open quantum system, consisting of the interacting impurity coupled to bath sites as well as to a Markovian environment. The dynamics of the auxiliary system is governed by a Lindblad master equation whose parameters are used to optimize the mapping. The accuracy of the results can be readily estimated and systematically improved by increasing the number of auxiliary bath sites, or by introducing a linear correction. Here, we focus on a detailed discussion of the proposed approach including technical remarks. To solve for the Green's functions of the auxiliary impurity problem, a non-Hermitian Lanczos diagonalization is applied. As a benchmark, results for the steady-state current-voltage characteristics of the single-impurity Anderson model are presented. Furthermore, the bias dependence of the single-particle spectral function and the splitting of the Kondo resonance are discussed. In its present form, the method is fast, efficient, and features a controlled accuracy.