  Exact analytic calculations in spin-1/2 XY chains, show the presence of
long-time tails in the asymptotic dynamics of spatially inhomogeneous
excitations. The decay of inhomogeneities, for $t\to \infty $, is given in the
form of a power law \[ (t/\tau_{Q}) ^{-\nu_{Q}} \] where the relaxation time
$\tau_{Q}$ and the exponent $\nu_{Q}$ depend on the wave vector $Q$,
characterizing the spatial modulation of the initial excitation. We consider
several variants of the XY model (dimerized, with staggered magnetic field,
with bond alternation, and with isotropic and uniform interactions), that are
grouped into two families, whether the energy spectrum has a gap or not. Once
the initial condition is given, the non-equilibrium problem for the
magnetization is solved in closed form, without any other assumption. The
long-time behavior for $t\to \infty $ can be obtained systematically in a form
of an asymptotic series through the stationary phase method. We found that
gapped models show critical behavior with respect to $Q$, in the sense that
there exist critical values $Q_{c}$, where the relaxation time $\tau_{Q}$
diverges and the exponent $\nu_{Q}$ changes discontinuously. At those points, a
slowing down of the relaxation process is induced, similarly to phenomena
occurring near phase transitions. Long-lived excitations are identified as
incommensurate spin density waves that emerge in systems undergoing the Peierls
transition. In contrast, gapless models do not present the above anomalies as a
function of the wave vector $Q$.
