Long time tails in the dynamics of the spatially inhomogeneous magnetization of dimerized isotropic XY chains for spin

Exact results for the dynamics of the spatially inhomogeneous magnetization (SIM) in the one-dimensional dimerized isotropic XY model are obtained, and the long-time behavior of SIM is considered in detail. It is shown that in the asymptotic limit t → ∞, the time dependence of SIM can be represented as a sum of several components oscillating at different frequencies. Amplitudes of these components decrease according to the (t/τ−ν power law. It is shown that both, the inverse of the time scale τ−1 and the exponent ν, have critical-like behavior with respect to the wave-vector Q characterizing the spatial inhomogeneity of the initial state. The value of τ−1 goes to zero at Q → Qci, where Qci (i = 1,2,3) are critical values of Q determined by parameters of the main Hamiltonian only. Just at points Qci, Qc2, the exponent ν changes its value discontinuously from . This effect is very similar to the critical slowing down phenomena in phase transitions. Due to the long-time tails in the relaxation process, we critically discuss the validity of the spin temperature assumption in spin systems.