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Berichte der Bunsengesellschaft für physikalische Chemie

Dynamics of an Impurity Spin Coupled to a Spin‐Boson Dissipative System

Sushanta Dattagupta, Tabish Qureshi

Published Mar 1, 1991• 2 citations
Publisher

Abstract

Abstract The dynamics of an impurity spin coupled to a spin‐boson dissipative system is studied using the resolvent expansion technique. In one particular physical realization the model simulates a highly anisotropic Kondo system. The transverse correlation function of the impurity spin is calculated and the result obtained is analyzed in various limiting cases. NMR lineshapes are also calculated for various temperatures which are relevant in another physical realization of the model. Motional narrowing is observed at high temperature.

DOI
10.1002/bbpc.19910950336

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Dynamics of an impurity spin coupled to a spin-boson dissipative system

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Wilson's momentum shell renormalization group method is used to solve for the dynamics of the dissipative two-state system. We utilize the mapping of the spin-boson model onto the anisotropic Kondo model (AKM) and solve for the dynamics of the latter. We find that the AKM captures the physics of the dissipative two-state system for dissipation strength $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}4$ corresponding to $\ensuremath{\infty}\ensuremath{\ge}{J}_{\ensuremath{\Vert}}\ensuremath{\ge}\ensuremath{-}\ensuremath{\infty}$ in the AKM. The dynamics of the AKM shows a smooth crossover between two strong coupling regimes corresponding to weak and strong dissipation in the spin-boson model.

arXiv:cond-mat/9601107•Physical Review Letters•Mar 4, 1996•109 citations
Abstract The dynamics of an impurity spin coupled to a spin‐boson dissipative system is studied using the resolvent expansion technique. In one particular physical realization the model simulates a highly anisotropic Kondo system. The transverse correlation function of the impurity spin is calculated and the result obtained is analyzed in various limiting cases. NMR lineshapes are also calculated for various temperatures which are relevant in another physical realization of the model. Motional narrowing is observed at high temperature.
The authors study the dynamics of an impurity spin coupled to a spin-boson dissipative system. They calculate the transverse correlation function of the impurity spin and compare the result with that obtained from an equivalent stochastic model. A phenomenological rate parameter in the stochastic model is represented in terms of microscopically obtained quantities. The authors also calculate the NMR lineshapes for various temperatures which are relevant in the context of a particular physical realization of the model. Motional narrowing of lines is observed at high temperatures.
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.
We consider a prototypical system of an infinite range transverse field Ising model coupled to a bosonic bath. By integrating out the bosonic degrees, an effective anisotropic Heisenberg model is obtained for the spin system. The phase diagram of the latter is calculated as a function of coupling to the heat bath and the transverse magnetic field. Collective excitations at low temeratures are assessed within a spin-wave like analysis that exhibits a vanishing energy gap at the quantum critical point. We also consider another limit where the system reduces to a generalized spin-boson model of two interacting spins. By increasing the coupling strength with the heat bath, the two-spin wavefunction changes from an entangled state to a factorized state of two spins which are aligned along the transverse field. We also discuss the possible realization and application of the model to different physical systems.
It is well known that the free electron Kondo problem can be described by a one-dimensional (1D) model because only the s-wave part of the electronic wavefunction is affected by the Kondo coupling. Moreover, since only the spin degrees of freedom are involved in the Kondo interaction, and due to spin–charge separation in 1D, the universal low energy long distance physics of the Kondo model also arises when a magnetic impurity is coupled to the end of a gapless antiferromagnetic J 1 –J 2 spin- chain, where J 1 ( J 2 ) is the (next) nearest neighbor coupling. Experimental realizations of such spin chain models are possible and, using various analytical and numerical techniques, we present a detailed and quantitative comparison between the usual free electron Kondo model and such spin chain versions of the Kondo problem. For the gapless J 1 –J 2 spin chain two cases are studied, with zero next nearest neighbor coupling, J 2 = 0, and with a critical second-neighbor coupling, J 2 = J 2 c . We first focus on the spin chain impurity model with a critical second-neighbor antiferromagnetic exchange where a bulk marginal coupling present in the spin chain model for J 2 < J 2 c vanishes . There, the usual Kondo physics is recovered in the spin chain model in the low energy regime (up to negligible corrections, dropping as powers of inverse length or energy). At J 2 c the spin chain model is not exactly solvable and we demonstrate the equivalence to the Kondo problem by comparing density matrix renormalization group calculations on the frustrated spin chain model with exact Bethe ansatz calculations of the electronic Kondo problem. We then analyze the nearest neighbor model ( J 2 = 0) where a new kind of Kondo effect occurs due to the presence of the bulk marginal coupling. This marginal coupling slightly alters the β function for the Kondo coupling, leading to a slower variation of the Kondo temperature, T K , with the bare Kondo coupling. In their exact Bethe ansatz solution of this spin chain impurity model ( J 2 = 0), Frahm and Zvyagin noted this relation as well as the connection to the Kondo problem. Here, by numerically solving the Bethe ansatz equations we provide further evidence for the connection to Kondo physics and in addition we present low temperature quantum Monte Carlo results for the impurity susceptibility that further support this connection.
Recent work is reviewed on continuous quantum phase transitions in impurity models, both with fermionic and bosonic baths – these transitions are interesting realizations of boundary critical phenomena at zero temperature. The models with fermion bath are generalizations of the standard Kondo model, with the common feature that Kondo screening of the localized spin can be suppressed due to competing processes. The models with boson bath are related to the spin–boson model of dissipative two-level systems, where the interplay between tunneling and friction results in multiple phases. The competition inherent to all models can generate unstable fixed points associated with quantum phase transitions, where the impurity properties undergo qualitative changes. Interestingly, certain impurity transitions feature both lower-critical and upper-critical “dimensions” and allow for epsilon-type expansions. Results are presented for a number of observables, obtained by both analytical and numerical renormalization group techniques, and connections made to experiments.
Wilson's momentum shell renormalization group method is used to solve for the dynamics of the dissipative two-state system. We utilize the mapping of the spin-boson model onto the anisotropic Kondo model (AKM) and solve for the dynamics of the latter. We find that the AKM captures the physics of the dissipative two-state system for dissipation strength $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}4$ corresponding to $\ensuremath{\infty}\ensuremath{\ge}{J}_{\ensuremath{\Vert}}\ensuremath{\ge}\ensuremath{-}\ensuremath{\infty}$ in the AKM. The dynamics of the AKM shows a smooth crossover between two strong coupling regimes corresponding to weak and strong dissipation in the spin-boson model.