Spin correlation functions and quasiparticle decay

We study one-dimensional anisotropic XXZ spin-$\frac{1}{2}$ model with ferromagnetic sign of the coupling and $z-z$ exchange constant $J_z = \Delta J$,where$\Delta < 1$,and$J$ is the coupling within XY spin plane. We calculate damping of low-energy excitations with $\omega \ll T$ due to their scattering from thermal excitation bath with temperature $T\ll J$, taking into account nonzero curvature of the excitation spectrum, $\epsilon(q) = u q + \delta\epsilon(q)$. We calculate also longitudinal spin-spin correlation function $⟨S^z(x,t)S^z(0,0)⟩$ at $x\approx ut$ and find the shape of the spreading “wave packet”.