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arXiv:1903.11646 · Physical Review X

Reconstructing Nonequilibrium Regimes of Quantum Many-Body Systems from the Analytical Structure of Perturbative Expansions

Corentin Bertrand, Serge Florens, Olivier Parcollet, Xavier Waintal

Published Oct 8, 2019• 62 citations
arXivHTMLPDF

Abstract

We propose a systematic approach to the nonequilibrium dynamics of strongly interacting many-body quantum systems, building upon the standard perturbative expansion in the Coulomb interaction.Highorder series are derived from the Keldysh version of the determinantal diagrammatic quantum Monte Carlo algorithm, and the reconstruction beyond the weak-coupling regime of physical quantities is obtained by considering them as analytic functions of a complex-valued interaction U. Our advances rely on two crucial ingredients: (i) a conformal change of variable, based on the approximate location of the singularities of these functions in the complex U plane, and (ii) a Bayesian inference technique, that takes into account additional known nonperturbative relations, in order to control the amplification of noise occurring at large U.This general methodology is applied to the strongly correlated Anderson quantum impurity model and is thoroughly tested both in and out of equilibrium.In the situation of a finite voltage bias, our method is able to extend previous studies, by bridging with the regime of unitary conductance and by dealing with energy offsets from particle-hole symmetry.We also confirm the existence of a voltage splitting of the impurity density of states and find that it is tied to a nontrivial behavior of the nonequilibrium distribution function.Beyond impurity problems, our approach could be directly applied to Hubbard-like models, as well as other types of expansions.

DOI
10.1103/physrevx.9.041008

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We propose a systematic approach to the nonequilibrium dynamics of strongly interacting many-body quantum systems, building upon the standard perturbative expansion in the Coulomb interaction.Highorder series are derived from the Keldysh version of the determinantal diagrammatic quantum Monte Carlo algorithm, and the reconstruction beyond the weak-coupling regime of physical quantities is obtained by considering them as analytic functions of a complex-valued interaction U. Our advances rely on two crucial ingredients: (i) a conformal change of variable, based on the approximate location of the singularities of these functions in the complex U plane, and (ii) a Bayesian inference technique, that takes into account additional known nonperturbative relations, in order to control the amplification of noise occurring at large U.This general methodology is applied to the strongly correlated Anderson quantum impurity model and is thoroughly tested both in and out of equilibrium.In the situation of a finite voltage bias, our method is able to extend previous studies, by bridging with the regime of unitary conductance and by dealing with energy offsets from particle-hole symmetry.We also confirm the existence of a voltage splitting of the impurity density of states and find that it is tied to a nontrivial behavior of the nonequilibrium distribution function.Beyond impurity problems, our approach could be directly applied to Hubbard-like models, as well as other types of expansions.
We present a general-purpose algorithm to extrapolate a low-rank function of two variables from a small domain to a larger one. It is based on the cross-interpolation formula. We apply it to reconstruct physical quantities in some quantum many-body perturbative expansions in the real-time Keldysh formalism, considered as a function of time ttt and interaction UUU. These functions are of remarkably low rank. This property, combined with the convergence of the perturbative expansion in UUU both at finite ttt (for any UUU), and small UUU (for any ttt), is sufficient for our algorithm to reconstruct the physical quantity at long time, strong coupling regime. Our method constitutes an alternative to standard resummation techniques in perturbative methods, such as diagrammatic quantum Monte Carlo. We benchmark it on the single impurity Anderson model and show that it is successful even in some regime where standard conformal mapping resummation techniques fail.
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In this thesis, we describe methods for solving quantum impurity models out of equilibrium, and consider several applications of these methods. Quantum impurity models describe systems in which a small subsystem (the impurity) is coupled to a large external environment (the bath). These models are useful in describing a wide variety of physical systems, including quantum dots, absorbed adatoms, and molecular junctions. They are also important for the study of lattice models, such as the Hubbard model, where they appear as auxiliary problems within the framework of dynamical mean field theory (DMFT). Many interesting phenomena in these systems, such as transport through a quantum dot, or optical pumping of a Mott insulator, occur away from equilibrium. Theoretical description of these phenomena requires methods that are capable of handling both real time evolution and strong correlation effects presenting a formidable challenge. In this thesis, we describe several methods for the solution of non-equilibrium quantum impurity models which meet this challenge. These methods are based on an expansion in the coupling between the impurity and the bath known as the hybridization expansion. Chapter 1 gives an introduction to the problems addressed in this thesis. Chapter 2 provides a brief overview of Keldysh formalism for nonequilibrium problems. Chapter 3 describes the bare hybridization expansion for the Anderson impurity model as well as its bold extensions. In particular we derive the non-crossing approximation (NCA), the one-crossing approximation (OCA), the inchworm expansion, and a bold expansion for the impurity Green's function. Chapter 4 describes the implementation of a quantum Monte Carlo algorithm based on the inchworm expansion. The rest of the thesis describes applications of these algorithms to problems of physical interest. Chapters 5 and 6 apply the methods previously described to understanding the Kondo effect in quantum dot systems. The first chapter examines the behavior of the Kondo effect under an applied voltage bias, a classic non-equilibrium strongly-correlated problem. The second chapter examines different ways of measuring the cloud of electrons which screen the impurity spin in the Kondo regime. Chapters 7 and 8 apply the same methods to the Hubbard model using the Dynamical mean-field theory (DMFT) mapping. The first chapter studies the Hubbard model in equilibrium using the numerically exact inchworm quantum Monte Carlo method. The second chapter studies the possibility of dynamic switching between the metal and Mott-insulator states of the Hubbard model using the one-crossing approximation (OCA). Finally chapter 9 gives some overall conclusions and suggests future directions for the development of hybridization expansion methods.
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.
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We develop a method for studying the real-time dynamics of Heisenberg operators in strongly interacting nonequilibrium quantum impurity models. Our method is applicable to a wide range of interaction strengths and to bias voltages beyond the linear-response regime, works at zero temperature, and overcomes the finite-size limitations faced by other numerical methods. We compare our method with quantum Monte Carlo simulations at a strong interaction strength, at which no analytical method is applicable up to now. We find a very good coincidence of the results at high-bias voltage, and in the short-time period at low-bias voltage. We discuss the possible reason of the deviation in the long-time period at low-bias voltage. We also find a good coincidence of our results with the perturbation results at weak interactions.