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arXiv:1406.5308 · Physical Review B

Universal quench dynamics of interacting quantum impurity systems

Dante Marvin Kennes, V. Meden, Romain Vasseur

Published Sep 2, 2014• 57 citations
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Abstract

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 ${T}_{K}$ (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 ${T}_{K}t$. 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.

DOI
10.1103/physrevb.90.115101

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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.
We study the real-time dynamics of the Kondo effect after a quantum quench in which a magnetic impurity is coupled to two metallic Hubbard chains. Using an effective field theory approach, we find that for noninteracting electrons the charge current across the impurity is given by a scaling function that involves the Kondo time. In the interacting case, we show that the Kondo time decreases with the strength of the repulsive interaction and the time dependence of the current reveals signatures of the Kondo effect in a Luttinger liquid. In addition, we verify that the relaxation of the impurity magnetization does not exhibit universal scaling behavior in the perturbative regime below the Kondo time. Our results highlight the role of nonequilibrium dynamics as a valuable tool in the study of quantum impurities in interacting systems.
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We present here the details of a method [A. B. Culver and N. Andrei, Many-body wavefunctions for quantum impurities out of equilibrium, Phys. Rev. B 103, L201103 (2021)] for calculating the time-dependent many-body wavefunction that follows a local quench. We apply the method to the voltage-driven nonequilibrium Kondo model to find the exact time-evolving wavefunction following a quench where the dot is suddenly attached to the leads at t=0t=0t=0. The method, which does not use Bethe ansatz, also works in other quantum impurity models and may be of wider applicability. We show that the long-time limit (with the system size taken to infinity first) of the time-evolving wavefunction of the Kondo model is a current-carrying nonequilibrium steady state that satisfies the Lippmann-Schwinger equation. We show that the electric current in the time-evolving wavefunction is given by a series expression that can be expanded either in weak coupling or in strong coupling, converging to all orders in the steady-state limit in either case. The series agrees to leading order with known results in the well-studied regime of weak antiferromagnetic coupling and also reveals a universal regime of strong ferromagnetic coupling with Kondo temperature ${T}_{K}^{(F)}=D{e}^{\ensuremath{-}\frac{3{\ensuremath{\pi}}^{2}}{8}\ensuremath{\rho}|J|}$ ($J<0, \ensuremath{\rho}|J|\ensuremath{\rightarrow}\ensuremath{\infty}$). In this regime, the differential conductance dI/dVdI/dVdI/dV reaches the unitarity limit 2e2/h2{e}^{2}/h2e2/h asymptotically at large voltage or temperature.
Whether a small quantum mechanical system is able to equilibrate with its environment once an external local perturbation drives it out of thermal equilibrium is a central question which cuts across many different fields of science. Here we consider such a problem for a correlated quantum impurity coupled to a fermionic reservoir and driven out of equilibrium by local quantum quenches such as those recently realized in optical absorption experiments on single quantum dots. We argue that equilibration in this problem is deeply connected to the occurrence of the Kondo effect at low energy and that a highly nontrivial dynamical behavior may emerge whenever a local quantum critical point intrudes between a conventional Kondo screened phase and a Kondo unscreened one. We discuss this issue in the context of the Anderson impurity model coupled to a pseudogap reservoir by using a correlated time-dependent variational wave function that is able to qualitatively describe this physics
We consider the entanglement between two one-dimensional quantum wires (Luttinger liquids) coupled by tunneling through a quantum impurity. The physics of the system involves a crossover between weak and strong coupling regimes characterized by an energy scale TB, and methods of conformal field theory therefore cannot be applied. The evolution of the entanglement in this crossover has led to many numerical studies, but has remained little understood, analytically or even qualitatively. We argue in this Letter that the correct universal scaling form of the entanglement entropy S (for an arbitrary interval of length L containing the impurity) is ∂S/∂ ln L=f(LTB). In the special case where the coupling to the impurity can be refermionized, we show how the universal function f(LTB) can be obtained analytically using recent results on form factors of twist fields and a defect massless-scattering formalism. Our results are carefully checked against numerical simulations.
We provide evidence that the relaxation dynamics of one-dimensional, metallic Fermi systems resulting out of an abrupt amplitude change of the two-particle interaction has aspects which are universal in the Luttinger liquid sense: the leading long-time behavior of certain observables is described by universal functions of the equilibrium Luttinger liquid parameter and the renormalized velocity. We analytically derive those functions for the Tomonaga-Luttinger model and verify our hypothesis of universality by considering spinless lattice fermions within the framework of the density-matrix renormalization group.