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Physical Review Letters

Thermalization of a Magnetic Impurity in the Isotropic XY Model

Douglas B. Abraham, Eytan Barouch, Giovanni Gallavotti, Anders Martin‐Löf

Published Nov 16, 1970• 51 citations
Publisher

Abstract

We carry out an exact study of the relaxation of a single magnetic impurity embedded in a linear quantum chain. We obtain a rigorous result for the expectation value of any spin in the chain, and find that it relaxes to its equilibrium value as ${t}^{\ensuremath{-}1}$.

DOI
10.1103/physrevlett.25.1449

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We carry out an exact study of the relaxation of a single magnetic impurity embedded in a linear quantum chain. We obtain a rigorous result for the expectation value of any spin in the chain, and find that it relaxes to its equilibrium value as t\ensuremath−1{t}^{\ensuremath{-}1}t\ensuremath−1.
Some nonequilibrium properties of an impurity in a linear chain of spins with nearest-neighbor interaction of the isotropic $X\ensuremath{-}Y$ type are studied. It is shown that under certain conditions its behavior is in agreement with the general principles of statistical mechanics. The relaxation in the presence of a transverse field is also studied, and it is found that in the weak coupling region the approach to equilibrium is described by a sum of exponentials. However, the results do not agree with the solution of the Bloch equations.
Consider a closed, linear chain of NNN localized spins (each s=12s=\frac{1}{2}s=21​) with nearest-neighbor Ising interaction. Such a chain, exposed to a uniform external magnetic field, will be called the pure host. Replace a single spin in the pure host by an impurity spin which is also subjected to nearest-neighbor Ising interaction and to the external field. The magnitude of the impurity spin, the magnitude of its magnetic moment, and the magnitude and sign of its interaction with the host are allowed to differ from the corresponding values characterizing the host. For the bulk system ($N\ensuremath{\rightarrow}\ensuremath{\infty}$, for constant linear density of spins), the thermodynamic properties, such as the impurity magnetization, the (position-dependent) magnetization of the impurity-host system and the impurity-host spin correlation functions, are obtained exactly in terms of conventional, tabulated functions. Numerical results are presented for impurity spins of magnitude S=32,52S=\frac{3}{2},\frac{5}{2}S=23​,25​.
The characteristics of the nuclear magnetic resonance (NMR), the contributions to the spin-lattice relaxation rate, and Knight shift of the resonance position are calculated exactly for a model spin-1/21/21/2 XYXYXY chain with the magnetic impurity. It is shown that the deviations of the impurity's coupling to the chain and the effective impurity's ggg factor from the values in the host spin chain yield special features in the temperature and magnetic field behavior for these NMR characteristics. The role of impurity-induced local levels in the NMR low-energy spin dynamics of spin chains is determined.