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Physica A Statistical Mechanics and its Applications

Stretched exponential-like impurity relaxation in the chain: a numerical study

Surajit Sen, Thomas D. Blersch

Published May 1, 1998• 13 citations
Publisher
DOI
10.1016/s0378-4371(98)00051-x

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The relaxation of observables to their nonequilibrium steady states in a disordered XX chain subjected to dephasing at every site has been intensely studied in recent years. We comprehensively analyze the relaxation of staggered magnetization, i.e., imbalance, in such a system, starting from the N\'eel initial state. We analytically predict emergence of several timescales in the system and extract results which match with large-system numerics without any extra fitting parameter until a universal timescale. An often reported stretched exponential decay is just one of the regimes which holds in a finite window of time and is therefore in fact not a true stretched exponential decay. Subsequently, the asymptotic decay of imbalance is governed by a power law irrespective of the disorder. We show that this emerges from the continuum limit of the low magnitude eigenspectrum of the Liouvillian. However, for finite systems, due to discreteness of the spectrum, the final phase of relaxation is governed by the relevant smallest Liouvillian gap.
Nonequilibrium relaxation behaviors in the Ising model on a square lattice based on the Wolff algorithm are totally different from those based on local-update algorithms. In particular, the critical relaxation is described by the stretched-exponential decay. We propose a novel scaling procedure to connect nonequilibrium and equilibrium behaviors continuously, and find that the stretched-exponential scaling region in the Wolff algorithm is as wide as the power-law scaling region in local-update algorithms. We also find that relaxation to the spontaneous magnetization in the ordered phase is characterized by the exponential decay, not the stretched-exponential decay based on local-update algorithms.
Abstract The autocorrelation function in many complex systems shows a crossover in the form of its decay: from a stretched exponential relaxation (SER) at short times to a power law at long times. Studies of the mechanisms leading to such multiple relaxation patterns are rare. Additionally, the inherent complexity of these systems makes it hard to understand the underlying mechanism leading to the crossover. Here we develop a simple one-dimensional spin model, which we call a domain wall (DW) to doublon model, that shows such a crossover as the nature of the excitations governing the relaxation dynamics changes with temperature and time. The relevant excitations are DWs and bound pairs of DWs, which we term ‘doublons’. The diffusive motion of the DWs governs the relaxation at short times, whereas the diffusive motion of the doublons yields the long-time decay. This change of excitations and their relaxation leads to a crossover from SER to a power law in the decay pattern of the autocorrelation function. We augment our numerical results with simple physical arguments and analytic derivations.