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arXiv:cond-mat/9712019 · Physical Review Letters

Boundary Interaction Changing Operators and Dynamical Correlations in Quantum Impurity Problems

Frédéric Lesage, Hubert Saleur

Published May 18, 1998• 89 citations
arXivHTMLPDF

Abstract

Recent developments have made possible the computation of equilibrium dynamical correlators in quantum impurity problems. In many situations, however, one is interested in correlators subject to a nonequilibrium initial preparation. This is the case, for instance, for the occupation probability $P(t)$ in the double well problem of dissipative quantum mechanics. We show here how to handle this situation in the framework of integrable quantum field theories. This allows us to obtain new exact results for $P(t)$: for instance, we find that at large times (or small $g$), the leading behavior for $g<\frac{1}{2}$ is $P(t)\ensuremath{\propto}{e}^{\ensuremath{-}\ensuremath{\Gamma}t}\mathrm{cos}\ensuremath{\Omega}t$, with the universal ratio $\frac{\ensuremath{\Omega}}{\ensuremath{\Gamma}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{cot}\frac{\ensuremath{\pi}g}{2(1\ensuremath{-}g)}$.

DOI
10.1103/physrevlett.80.4370

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Recent developments have made possible the computation of equilibrium dynamical correlators in quantum impurity problems. In many situations, however, one is interested in correlators subject to a nonequilibrium initial preparation. This is the case, for instance, for the occupation probability P(t)P(t)P(t) in the double well problem of dissipative quantum mechanics. We show here how to handle this situation in the framework of integrable quantum field theories. This allows us to obtain new exact results for P(t)P(t)P(t): for instance, we find that at large times (or small ggg), the leading behavior for g<12g<\frac{1}{2}g<21​ is P(t)\ensuremath∝e\ensuremath−\ensuremathΓtcos\ensuremathΩtP(t)\ensuremath{\propto}{e}^{\ensuremath{-}\ensuremath{\Gamma}t}\mathrm{cos}\ensuremath{\Omega}tP(t)\ensuremath∝e\ensuremath−\ensuremathΓtcos\ensuremathΩt, with the universal ratio \ensuremathΩ\ensuremathΓ=cot\ensuremathπg2(1\ensuremath−g)\frac{\ensuremath{\Omega}}{\ensuremath{\Gamma}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{cot}\frac{\ensuremath{\pi}g}{2(1\ensuremath{-}g)}\ensuremathΓ\ensuremathΩ​=cot2(1\ensuremath−g)\ensuremathπg​.
We discuss in more details the theory of low energy excitations in quantum impurity problems with an external field at vanishing temperature, giving further support to results of the previous paper. We then extend these results to the next order at low frequency, obtaining in particular the exact expression, as a function of the bias, of the first two derivatives of the response function χ ′ ′ (ω) at ω = 0 in the double well problem of dissipative quantum mechanics. We also extend our approach to the case of non vanishing temperature and no external field. Fendley et al. had obtained in that case an expression for the Hall conductance with a single impurity using a Landauer-Büttiker type approach. We recover their result in the framework of linear response theory using renormalized form-factors. We also obtain, as a function of the temperature, the first derivative of the response function χ ′ ′ (ω) at ω = 0 in the double well problem of dissipative quantum mechanics. 12/96
The equilibrium physics of quantum impurities frequently involves a universal crossover from weak to strong reservoir-impurity coupling, characterized by single-parameter scaling and an energy scale TK{T}_{K}TK​ (Kondo temperature) that breaks scale invariance. For the noninteracting resonant level model, the nonequilibrium time evolution of the Loschmidt echo after a local quantum quench was recently computed explicitly [R. Vasseur, K. Trinh, S. Haas, and H. Saleur, Phys. Rev. Lett. 110, 240601 (2013)]. It shows single-parameter scaling with variable TKt{T}_{K}tTK​t. Here, we scrutinize whether similar universal dynamics can be observed in various interacting quantum impurity systems. Using density matrix and functional renormalization group approaches, we analyze the time evolution resulting from abruptly coupling two noninteracting Fermi or interacting Luttinger liquid leads via a quantum dot or a direct link. We also consider the case of a single Luttinger liquid lead suddenly coupled to a quantum dot. We investigate whether the field-theory predictions for the universal scaling as well as for the large-time behavior successfully describe the time evolution of the Loschmidt echo and the entanglement entropy of microscopic models. Our study shows that for the considered local quench protocols the above quantum impurity models fall into a class of problems for which the nonequilibrium dynamics can largely be understood based on the knowledge of the corresponding equilibrium physics.
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 Keldysh boundary problem in a nonequilibrium Falicov-Kimball model in infinite dimensions is studied within the truncated and self-consistent perturbation theories and the dynamical mean-field theory. Within the model the system is started in equilibrium, and later a uniform electric field is turned on. The Kadanoff-Baym-Wagner equations for the nonequilibrium Green's functions are derived and numerically solved. The contributions of initial correlations are studied by monitoring the system evolution. It is found that the initial correlations are essential for establishing the full electron correlations of the system and are independent of the starting time of preparing the system in equilibrium. By examining the contributions of the initial correlations to the electric current and the double occupation, we find that the contributions are small in relation to the total value of those physical quantities when the interaction is weak, and significantly increase when the interaction is strong. The neglect of initial correlations may cause artifacts in the nonequilibrium properties of the system, especially in the strong-interaction case.
In the past few years there has been a resurgence of interest in dynamical impurity problems, as a result of developments in the theory of correlated electron systems. The general dynamical impurity problem is a set of conduction electrons interacting with an impurity which has internal degrees of freedom. The simplest and earliest example, the Kondo problem, has attracted interest since the mid-sixties not only because of its physical importance but also as an example of a model displaying logarithmic divergences order by order in perturbation theory. It provided one of the earliest applications of the renormalization group method, which is designed to deal with just such a situation. As we shall see, the antiferromagnetic Kondo model is controlled by a strong-coupling fixed point, and the essence of the renormalization group solution is to carry out the global renormalization numerically starting from the original (weak-coupling) Hamiltonian. In these lectures, we shall describe an alternative route in which we identify an exactly solvable model which renormalizes to the same fixed point as the original dynamical impurity problem. This approach is akin to determining the critical behavior at a second order phase transition point by solving any model in a given universality class.
We study different aspects of integrable boundary quantum field theories, focusing mostly on the ``boundary sine-Gordon model'' and its applications to condensed matter physics. The first part of the review deals with formal problems. We analyze the classical limit and perform semi-classical quantization. We show that the non-relativistic limit corresponds to the Calogero-Moser model with a boundary potential. We construct a lattice regularization of the problem via the XXZ chain. We classify boundary bound states. We generalize the Destri de Vega method to compute the ground state energy of the theory on a finite interval. The second part deals with some applications to condensed matter physics. We show how to compute analytically time and space dependent correlations in one-dimensional quantum integrable systems with an impurity. Our approach is based on a description of these systems in terms of massless scattering of quasiparticles. Correlators follow then from matrix elements of local operators between multiparticle states -- the massless form-factors. Although, in general an infinite sum of these form-factors has to be considered, we find that for the current, spin and energy operators only a few (two or three) are necessary to obtain an accuracy of more than 1\%. Our results hold for arbitrary impurity strength, in contrary to the perturbative expansions in the coupling constants. As an example, we compute the frequency dependent condunctance, at zero temperature, in a Luttinger liquid with an impurity, and also discuss the susceptibility in the Kondo model and the time-dependent properties of the two-state problem with dissipation.