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arXiv:1703.09806 · Physical review. B./Physical review. B

Configuration memory in patchwork dynamics for low-dimensional spin glasses

Jie Yang, A. Alan Middleton

Published Dec 27, 2017• 2 citations
arXivHTMLPDF

Abstract

A patchwork method is used to study the dynamics of loss and recovery of an initial configuration in spin glass models in dimensions $d=1$ and $d=2$. The patchwork heuristic is used to accelerate the dynamics to investigate how models might reproduce the remarkable memory effects seen in experiment. Starting from a ground-state configuration computed for one choice of nearest-neighbor spin couplings, the sample is aged up to a given scale under new random couplings, leading to the partial erasure of the original ground state. The couplings are then restored to the original choice and patchwork coarsening is again applied, in order to assess the recovery of the original state. Eventual recovery of the original ground state upon coarsening is seen in two-dimensional Ising spin glasses and one-dimensional clock models, while one-dimensional Ising spin systems neither lose nor gain overlap with the ground state during the recovery stage. The recovery for the two-dimensional Ising spin glasses suggests scaling relations that lead to a recovery length scale that grows as a power of the aging length scale.

DOI
10.1103/physrevb.96.214208

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A patchwork method is used to study the dynamics of loss and recovery of an initial configuration in spin glass models in dimensions d=1d=1d=1 and d=2d=2d=2. The patchwork heuristic is used to accelerate the dynamics to investigate how models might reproduce the remarkable memory effects seen in experiment. Starting from a ground-state configuration computed for one choice of nearest-neighbor spin couplings, the sample is aged up to a given scale under new random couplings, leading to the partial erasure of the original ground state. The couplings are then restored to the original choice and patchwork coarsening is again applied, in order to assess the recovery of the original state. Eventual recovery of the original ground state upon coarsening is seen in two-dimensional Ising spin glasses and one-dimensional clock models, while one-dimensional Ising spin systems neither lose nor gain overlap with the ground state during the recovery stage. The recovery for the two-dimensional Ising spin glasses suggests scaling relations that lead to a recovery length scale that grows as a power of the aging length scale.
Spin glass behavior was first seen in metallic alloys with magnetic impurities dispersed randomly in the main non-magnetic component. Due to the simultaneous presence of quenched disorder and competing ferromagnetic and antiferromagnetic interactions, the nonequilibrium behaviors of spin glasses are intricate and difficult to understand. Experimental research on spin glass materials indicate extremely slow dynamics, suggesting that the direct numerical simulation using local spin flips will take impractical time scales to replicate the nonequilibrium behaviors. This thesis introduces efficient algorithms and heuristics for the numerical simulation of simple models for spin glasses and discusses significant simulation results. The results provide insights into understanding the nonequilibrium behaviors, especially aging and memory, through studying microscopic constituent configurations. The methodology and conclusion presented in this thesis can be potentially extended to address general questions about disordered materials at nonequilibrium. Chapter 1 provides a thorough overview of spin glasses. A detailed introduction on complex nonequilibrium behaviors, including slow dynamics and memory effects, seen in typical experiments of spin glass materials, is presented. A general discussion about the characterization and categorization of memory effects found in many out-of-equilibrium systems is highlighted. Two prevalent theoretical models on spin glasses, the droplet and replica symmetry breaking theory, are described. Lastly, algorithm development and simulation techniques used to study spin glasses are covered. In Chapter 2, a patchwork heuristic is applied to accelerate the relaxation of spin glass models in dimensions d=1d=1d=1 and d=2d=2d=2 with the goal of investigating the dynamics of loss and recovery of spin configurations. A well-designed simulation protocol, consists of a aging and recovery stage, is used to mimic the temperature cycling experiment, where the temperature variations are replaced by the variations of the spin coupling realization. Eventual recovery of memory upon coarsening is seen in two-dimensional Ising spin glasses and one-dimensional clock models, while one-dimensional Ising spin systems do not exhibit configuration memory. A central result is that the coarsening recovery length scale scs_{\mathrm c}sc​ grows faster than the aging length scale ℓ\ellℓ as a power law relation sc∼ℓ2.58s_{\mathrm c}\sim\ell^{2.58}sc​∼ℓ2.58. In Chapter 3, the microscopic dependence of spin glass equilibration on sample history, i.e., the initial configuration and the specific noise history of stochastic dynamics, is examined. The patchwork dynamics is used to simulate the glassy dynamics over a wide range of length scales. The dependence of the nearest-neighbor spin correlations at long time on initial spin configurations and on noise histories are compared. Most of the local spin correlations are independent of the details of evolution, due to the formation of rigid domains. The correlations on a fractal set of domain walls are found to have distinct dependence on the noise history and the initial state. The initial ``nature'' of the state is retained in the domain walls under coarsening more than the ``nurture'' of the noise, while the location of the domain walls at each scale is determined by the frozen disorder (``environment''). We also provide evidence that coarsening with local dynamics at finite temperature gives persistence of overlaps roughly consistent with coarsening at zero temperature, at least over short distances, supporting the use of zero-temperature heuristics to address finite-temperature dynamics and memory.
Slow dynamics in disordered materials prohibits direct simulation of their rich nonequilibrium behavior at large scales. ``Patchwork dynamics'' is introduced to mimic relaxation over a very broad range of time scales by directly equilibrating or optimizing on successive length scales. This dynamics is used to study coarsening and to replicate memory effects for spin glasses and random ferromagnets. It is also used to find, with high confidence, correct ground states in large toroidal samples.
Using Monte Carlo simulations, we have studied the relaxation of energy of the three-dimensional Ising spin-glass model in aging process. Our finite-size-scaling analysis on the isothermal energy decay after the quench suggests strongly that the energy relaxes by coarsening of domain walls in agreement with the droplet theory. We have also evaluated relaxation times required for spin configurations in small systems, which are regarded as isolated droplets, to flip globally. The results support the fundamental assumption of the droplet theory that coarsening of domain walls, in case of spin glasses, is driven by successive nucleation of thermally activated droplets.
In order to determine if the effects of an overlap length are visible in an Ising Edwards-Anderson spin-glass model, we have performed Monte Carlo simulations that mimic experimental procedures. The spin system was quenched from a random initial condition to a temperature T1{\mathit{T}}_{1}T1​, equilibrated in zero field for a time tw1{\mathit{t}}_{\mathit{w}1}tw1​, whereupon the system was subjected to a perturbation in field \ensuremath{\Delta}H, temperature \ensuremath{\Delta}T, or bond \ensuremath{\Delta}J with a duration tw2{\mathit{t}}_{\mathit{w}2}tw2​. After the perturbation of the original field, the temperature or bonds were restored. After waiting for a time tw3{\mathit{t}}_{\mathit{w}3}tw3​, a weak magnetic field H was applied. The susceptibility \ensuremath{\chi}(t)=[M+{\mathit{M}}_{+}M+​(t)-${\mathit{M}}_{\mathrm{\ensuremath{-}}}$(t)]/2H, where ${\mathit{M}}_{\ifmmode\pm\else\textpm\fi{}}$(t) is the magnetization due to positive or negative field, and the spin autocorrelation function q(t) were studied as functions of time. For small perturbations, \ensuremath{\Delta}HH0{\mathit{H}}_{0}H0​, \ensuremath{\Delta}TT0{\mathit{T}}_{0}T0​, or \ensuremath{\Delta}JJ0{\mathit{J}}_{0}J0​, respectively, the system is undisturbed. For larger perturbations, the correlations built up during tw1{\mathit{t}}_{\mathit{w}1}tw1​ are gradually destroyed. Similar results are found in all three types of simulations and the qualitative features are in agreement with experimental results. These features are found for both two-dimensional and three-dimensional spin-glass systems. The results can be interpreted in the domain models for spin glasses. \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ is found to be strongly dependent on the duration tw2{\mathit{t}}_{\mathit{w}2}tw2​ of the perturbation. If tw1{\mathit{t}}_{\mathit{w}1}tw1​ is constant and tw2{\mathit{t}}_{\mathit{w}2}tw2​ increases, \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ decreases. If tw1{\mathit{t}}_{\mathit{w}1}tw1​ is set equal to tw2{\mathit{t}}_{\mathit{w}2}tw2​ and tw1{\mathit{t}}_{\mathit{w}1}tw1​=tw2{\mathit{t}}_{\mathit{w}2}tw2​ is increased, \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ decreases, in agreement with the expected behavior. \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ was found to increase with increasing temperature T, which is in agreement with experimental results, but contrary to the theoretical expectations.
Memory and rejuvenation effects in the magnetic response of off-equilibrium spin glasses have been widely regarded as the doorway into the experimental exploration of ultrametricity and temperature chaos (maybe the most exotic features in glassy free-energy landscapes). Unfortunately, despite more than twenty years of theoretical efforts following the experimental discovery of memory and rejuvenation, these effects have thus far been impossible to simulate reliably. Yet, three recent developments convinced us to accept this challenge: first, the custom-built Janus II supercomputer makes it possible to carry out "numerical experiments" in which the very same quantities that can be measured in single crystals of CuMn are computed from the simulation, allowing for parallel analysis of the simulation/experiment data. Second, Janus II simulations have taught us how numerical and experimental length scales should be compared. Third, we have recently understood how temperature chaos materializes in aging dynamics. All three aspects have proved crucial for reliably reproducing rejuvenation and memory effects on the computer. Our analysis shows that (at least) three different length scales play a key role in aging dynamics, while essentially all theoretical analyses of the aging dynamics emphasize the presence and the crucial role of a single glassy correlation length.
By means of Monte Carlo simulations on the three-dimensional Ising spin-glass model, we have studied aging phenomena with various temperature(TTT)-change protocols. Particularly, a TTT-shift protocol, in which a system is first quenched to and aged for a period tw1t_{\rm w1}tw1​ at a temperature T1T_1T1​, and subsequently aged at a new temperature T2T_2T2​ is closely investigated. Most importantly, the mean size of domains, which is extracted from the replica-overlap function, was found to grow monotonically without any appreciable decrease by the TTT-change. We also found the relaxation of energy density and spin auto-correlation function during the TTT-shift process can be explained within the following picture: the dynamics finally crossovers to an isothermal aging of T2T_2T2​ at around t2≃tw1(eff)t_2 \simeq t_{\rm w1}^{\rm (eff)}t2​≃tw1(eff)​ after the TTT-change, where tw1(eff)t_{\rm w1}^{\rm (eff)}tw1(eff)​ is an effective waiting time related with tw1t_{\rm w1}tw1​, but in the transient regime t2<tw1(eff)t_2 \lt t_{\rm w1}^{\rm (eff)}t2​<tw1(eff)​, adjustment of the population of thermally active droplets from that at T1T_1T1​ to T2T_2T2​ takes place very slowly. Implications of the results on aging phenomena in other TTT-change protocols such as TTT-cycling and continuous TTT-change with an intermittent stop observed by simulations as well as experiments are discussed.