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Stretched exponentials in the dynamics and at the ground state of the plaquette Ising model

Adam Lipowski

Published Aug 30, 2012• 0 citations
OpenAlex

Abstract

Simulations show that the autocorrelation function C(t) in the d=3 Ising model with a plaquette interaction has a stretched-exponential decay in a supercooled liquid phase. Such a decay characterizes also some ground-state probability distributions obtained from the numerically exact counting of up to 10^450 configurations. A related model with a strongly degenerate ground state but lacking glassy features does not exhibit such a decay. Tensionless modification of the droplet model might explain stretched-exponential decay of C(t) even in three-dimensional systems.

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Simulations show that the autocorrelation function C(t) in the d=3 Ising model with a plaquette interaction has a stretched-exponential decay in a supercooled liquid phase. Such a decay characterizes also some ground-state probability distributions obtained from the numerically exact counting of up to 10^450 configurations. A related model with a strongly degenerate ground state but lacking glassy features does not exhibit such a decay. Tensionless modification of the droplet model might explain stretched-exponential decay of C(t) even in three-dimensional systems.
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We study the out-of-equilibrium dynamics of the quantum Ising model with power-law interactions and positional disorder. For arbitrary dimension ddd and interaction range $\ensuremath{\alpha}\ensuremath{\ge}d$ we analytically find a stretched-exponential decay of the global magnetization and ensemble-averaged single-spin purity with a stretch power $\ensuremath{\beta}=d/\ensuremath{\alpha}$ in the thermodynamic limit. Numerically, we confirm that glassy behavior persists for finite system sizes and sufficiently strong disorder. We identify dephasing between disordered coherent pairs as the main mechanism leading to a relaxation of global magnetization, whereas genuine many-body interactions lead to a loss of single-spin purity which signifies the buildup of entanglement. The emergence of glassy dynamics in the quantum Ising model extends prior findings in classical and open quantum systems, where the stretched-exponential law is explained by a scale-invariant distribution of timescales, to both integrable and nonintegrable quantum systems.
The authors discuss the dynamics of an observable field related to a locally conserved field. They show that for a model of phase ordering dynamics and for a variation of model C critical dynamics, that the two-time momentum space correlation function decays asymptotically as a stretched exponential. They show that under some conditions the asymptotic behaviour is experimentally observable and argue that the non-exponential behaviour is not a feature specific to these models but should rather apply for any observable coupled to a locally conserved field.
We investigate nonequilibrium relaxations of Ising models at the critical point by using a cluster update. While preceding studies imply that nonequilibrium cluster-flip dynamics at the critical point are universally described by the stretched-exponential function, we find that the dynamics changes from the stretched exponential to the power function as the dimensionality is increased: The two-, three-, four-, and infinite-dimensional Ising models are numerically studied, and the four- and infinite-dimensional Ising models exhibit the power-law relaxation. We also show that the finite-size scaling analysis using the normalized correlation length is markedly effective for the analysis of relaxational processes rather than the direct use of the Monte Carlo step.
Using Monte Carlo techniques, we study the decay of magnetization in diluted two-dimensional Ising models at and below the critical temperature Tc of the undiluted Ising model, but above the critical temperature of the diluted system. Using damage spreading (or rather damage "healing"), we are able to measure down to much lower final magnetizations (10–9) and to much larger times than previous authors. Nevertheless, we do not yet find the predicted asymptotic behavior in the Griffiths phase T < Tc. But we can at least exclude a stretched exponential decay as found in previous papers, for T < Tc. Finally, we discuss the case T = Tc where a stretched exponential decay can be proven to hold, at least for p < pc. We indeed do see a stretched exponential for p = pc (and T = Tc), but we show that it cannot describe the asymptotic behavior either.
We study numerically the nonequilibrium dynamical behavior of an Ising model with mixed two-spin and four-spin interactions after a sudden quench from the high-temperature phase to the first-order phase transition point. The autocorrelation function is shown to approach its limiting value, given by the magnetization in the ordered phase at the transition point, m c , through a stretched exponential decay. On the other hand relaxation of the magnetization starting with an uncorrelated initial state with magnetization, m i , approaches either m c , for m i > 0.5, or zero, for m i < 0.5. For small m i and for m i slightly larger than 0.5 the relaxation of the magnetization shows an asymptotic power law time dependence, thus from a nonequilibrium point of view the transition is continuous.