Impurity spin relaxation inS=12XXchains

Abstract

Dynamic autocorrelations $〈{S}_{i}^{\ensuremath{\alpha}}{(t)S}_{i}^{\ensuremath{\alpha}}〉(\ensuremath{\alpha}=x,z)$ of an isolated impurity spin in a $S=\frac{1}{2}\mathrm{XX}$ chain are calculated. The impurity spin, defined by a local change in the nearest-neighbor coupling, is either in the bulk or at the boundary of the open-ended chain. The exact numerical calculation of the correlations employs the Jordan-Wigner mapping from spin operators to Fermi operators; the effects of finite system size can be eliminated. Two distinct temperature regimes are observed in the long-time asymptotic behavior of the bulk correlations. At $T=0$ only power laws are present. At high T the x correlation decays exponentially (except at short times), while the z correlation still shows an asymptotic power law (different from the one at $T=0)$ after an intermediate exponential phase. The boundary impurity correlations ultimately follow power laws at all T, with intermediate exponential phases at high T. The power laws for the z correlation and the boundary correlations can be derived from the impurity-induced changes in the properties of the Jordan-Wigner fermion states.