Long time tails in the dynamics of the spatially inhomogeneous magnetization of dimerized isotropic XY chains for spin s = 12

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 ν = 12to ν = 14. 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.