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arXiv:0911.2130 · Physical Review B

Nonequilibrium steady state in a periodically driven Kondo model

Markus Heyl, Stefan K. Kehrein

Published Apr 9, 2010• 23 citations
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Abstract

We investigate the Kondo model with time-dependent couplings that are periodically switched on and off. On the Toulouse line we derive exact analytical results for the spin dynamics in the steady state that builds up after an infinite number of switching periods. Remarkably, the universal long-time behavior of the spin-spin correlation function remains completely unaffected by the driving. In the limit of slow driving the dynamics becomes equivalent to that of a single interaction quench. In the limit of fast driving it is shown that the steady state cannot be described by some effective equilibrium Hamiltonian due to the observation that an incautious implementation of the Trotter formula is not correct. As a consequence, the steady state in the limit of fast switching serves as an example for the emergence of new quantum states not accessible in equilibrium.

DOI
10.1103/physrevb.81.144301

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We investigate the Kondo model with time-dependent couplings that are periodically switched on and off. On the Toulouse line we derive exact analytical results for the spin dynamics in the steady state that builds up after an infinite number of switching periods. Remarkably, the universal long-time behavior of the spin-spin correlation function remains completely unaffected by the driving. In the limit of slow driving the dynamics becomes equivalent to that of a single interaction quench. In the limit of fast driving it is shown that the steady state cannot be described by some effective equilibrium Hamiltonian due to the observation that an incautious implementation of the Trotter formula is not correct. As a consequence, the steady state in the limit of fast switching serves as an example for the emergence of new quantum states not accessible in equilibrium.
We study the nonequilibrium dynamics of a periodically driven anisotropic Kondo impurity model. The periodic time dependence is introduced for a local magnetic field which couples to the impurity spin and also for an in-plane exchange interaction. We obtain the exact results on the time evolution for arbitrary periodic time dependence at the special point in the parameter space known as the Toulouse limit. We first consider a specific case where the local magnetic field is periodically switched on and off. When the driving period is much shorter than the inverse of the Kondo temperature, an intriguing oscillating behavior (resonance phenomenon) emerges in the time average of the impurity spin polarization with increasing the local magnetic field intensity. By taking the high-frequency limit of the external driving, we elucidate that the system recovers the translational invariance in time and can be described by a mixture of the zero-temperature and infinite-temperature properties. In certain cases, the system is governed by either zero-temperature or infinite-temperature properties and, therefore, can be properly described by the corresponding equilibrium state.
We investigate the equilibration of a Kondo model that is initially prepared in a nonequilibrium state towards its equilibrium behavior. Such initial nonequilibrium states can, e.g., be realized in quantum-dot experiments with time-dependent gate voltages. We evaluate the nonequilibrium spin-spin correlation function at the Toulouse point of the Kondo model exactly and analyze the crossover between nonequilibrium and equilibrium behavior as the nonequilibrium initial state evolves as a function of the waiting time for the first spin measurement. Using the flow equation method, we extend these results to the experimentally relevant limit of small Kondo couplings.
The dynamics of the anisotropic Kondo model under the periodic driving is studied. The periodic driving we consider is the local magnetic field which couples with the impurity spin and the in-plane exchange interaction. At the special point in the parameter space (Toulouse limit), the anisotropic Kondo model can be transformed to the non-interacting resonant level model and thereby the exact results for arbitrary external fields can be obtained. It is shown that characteristic behavior of the dynamics changes with the field intensity, the field frequency, and the Kondo temperature. When the local magnetic field intensity is very large compared with the field frequency and the Kondo temperature, the time dependence of the impurity spin and the spin density of the fermionic bath shows an oscillation whose period is inversely proportional to the field intensity. We also find that another time scale, which is given by the inverse of the Kondo temperature, governs the response of the system to the external field.
Motivated by recent experiments on molecular quantum dots we investigate the relaxation of pure spin states when coupled to metallic leads. Under suitable conditions these systems are well described by a ferromagnetic Kondo model. Using two recently developed theoretical approaches, the time-dependent numerical renormalization group and an extended flow equation method, we calculate the real-time evolution of a Kondo spin into its partially screened steady state. We obtain exact analytical results which agree well with numerical implementations of both methods. Analytical expressions for the steady state magnetization and the dependence of the long-time relaxation on microscopic parameters are established. We find the long-time relaxation process to be much faster in the regime of anisotropic Kondo couplings. The steady state magnetization is found to deviate significantly from its thermal equilibrium value.
We present a method 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 (we include results for the interacting resonant level and the Anderson impurity model) and may be of wider applicability. In the particular case of the Kondo model, we show that the long-time limit (with the system size taken to infinity first) of the time-evolving wavefunction 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 another 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.
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.