Long time tails in the dynamics of the spatially inhomogeneous magnetization of dimerized isotropic XY chains for spin s=1/2

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 --> infinity, 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/tau)(-v) power law. It is shown that both, the inverse of the time scale tau(-1) and the exponent v, have critical-like behavior with respect to the wave-vector Q characterizing the spatial inhomogeneity of the initial state. The value of tau(-1) goes to zero at \Q\ --> Q(ci), where Q(ci) (i = 1, 2, 3) are critical values of Q determined by parameters of the main Hamiltonian only. Just at points Q(c1), Q(c2), the exponent v changes its value discontinuously from v = 1/2 to v = 1/4. 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.