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 XY chains show the presence of long-time tails in the asymptotic dynamics of spatially inhomogeneous excitations. The decay of inhomogeneities for t→ϱ, is given in the form of a power law (t/ Q ) Ϫ Q , where the relaxation time Q and the exponent 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 interactions͒, 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 t→ϱ 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 Q diverges and the exponent 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.

Logarithmic current fluctuations in nonequilibrium quantum spin chains

“…The relaxation from a large class of initial conditions was considered in numerous studies starting from late sixties; see [32,33,34,35] and [36] for a review of more recent work. Our focus, however, is on non-equilibrium states with a non-vanishing current.…”

Relaxation in theXXquantum chain

“…Consequently, in such systems there is no finite relaxation time. Moreover, it was shown that for certain inhomogeneously magnetised initial states, at special wave vectors, a slowing down of the magnetisation relaxation can occur for gapped free fermionic models ‡ Corresponding author: karevski@lpm.u-nancy.fr [9]. The effect of quenched disorder was also considered in the Ising quantum chain [27].…”

Non-ergodic states induced by impurity levels in quantum spin chains

“…as a linear combination of independent particular solutions r n and r −n . In (12) and subsequent relations, the index λ is omitted. Substitution in relation (11) leads to the spectrum:…”