Long-time tails and anomalous slowing down in the relaxation of spatially inhomogeneous excitations in quantum spin chains

Abstract

Exact analytic calculations in spin-1/2 $\mathrm{XY}$ chains show the presence of long-time tails in the asymptotic dynamics of spatially inhomogeneous excitations. The decay of inhomogeneities for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty},$ is given in the form of a power law $(t/{\ensuremath{\tau}}_{Q}{)}^{\ensuremath{-}{\ensuremath{\nu}}_{Q}},$ where the relaxation time ${\ensuremath{\tau}}_{Q}$ and the exponent ${\ensuremath{\nu}}_{Q}$ depend on the wave vector Q, characterizing the spatial modulation of the initial excitation. We consider several variants of the $\mathrm{XY}$ 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 nonequilibrium problem for the magnetization is solved in closed form, without any other assumption. The long-time behavior for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ 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 ${\ensuremath{\tau}}_{Q}$ diverges and the exponent ${\ensuremath{\nu}}_{Q}$ 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 gapped systems, as a consequence of both approximate nesting of the spectrum and the degeneracy of some stationary points. In contrast, gapless models do not present the above anomalies as a function of the wave vector Q.