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arXiv:1709.09829 · Journal of Statistical Mechanics Theory and Experiment

Dynamic variational study of chaos: spin glasses in three dimensions

Alain H. Billoire, L. A. Fernández, A. Maiorano, Enzo Marinari, Víctor Martín-Mayor, J. Moreno-Gordo, Giorgio Parisi A, Federico Ricci‐Tersenghi, Juan Jesus Ruiz-Lorenzo

Published Mar 1, 2018• 29 citations
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Abstract

Abstract We have introduced a variational method to improve the computation of integrated correlation times in parallel tempering dynamics, obtaining a better estimate (a lower bound, at least) of the exponential correlation time. Using this determination of the correlation times, we revisited the problem of the characterization of the chaos in temperature in finite dimensional spin glasses, by way of the study of correlations between various chaos indicators computed in the static and the correlation times of the parallel tempering dynamics. The sample-distribution of the characteristic time for the parallel tempering dynamics turns out to be fat-tailed, and to obey finite-size scaling.

DOI
10.1088/1742-5468/aaa387

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Abstract We have introduced a variational method to improve the computation of integrated correlation times in parallel tempering dynamics, obtaining a better estimate (a lower bound, at least) of the exponential correlation time. Using this determination of the correlation times, we revisited the problem of the characterization of the chaos in temperature in finite dimensional spin glasses, by way of the study of correlations between various chaos indicators computed in the static and the correlation times of the parallel tempering dynamics. The sample-distribution of the characteristic time for the parallel tempering dynamics turns out to be fat-tailed, and to obey finite-size scaling.
Some recent progress in Monte Carlo simulations of spin glasses will be presented. The problem of slow dynamics at low temperatures is partially alleviated by use of the parallel tempering (replica exchange) method. A useful technique to check for equilibration (applicable only for a Gaussian distribution) will be discussed. It will be argued that a finite size scaling analysis of the scaled correlation length of the system is a good approach with which to investigate phase transitions in spin glasses. This method will be used to study two questions: (i) whether there is a phase transition in zero field in the Heisenberg spin glass in three dimensions, and (ii) whether there is phase transition in a magnetic field in an Ising spin glass, also in three dimensions.
I present an analysis of the dynamic behavior of short-range Ising spin glasses observed in stochastic simulations. The time dependence of the order parameter q(t)=〈Sx{S}_{x}Sx​(0)Sx{S}_{x}Sx​(t)〉\ifmmode\bar\else\textasciimacron\fi{}---which is the same as that of the structure factor---and the time dependence of the related dynamic correlation functions have been recorded with good statistics and very long observation times. The spin-glass model with a symmetric distribution of discrete nearest-neighbor \ifmmode\pm\else\textpm\fi{}J interactions on a simple-cubic lattice was used. Simulations were performed with a special fast computer, allowing for the first-time investigation of the equilibrium dynamics for a wide range of temperatures (0.7\ensuremath{\le}kT/J\ensuremath{\le}5.0) and lattice sizes (83{8}^{3}83, 163{16}^{3}163, 323{32}^{3}323, and 643{64}^{3}643). I have found that the empirical formula q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$exp(-\ensuremath{\omega}${t}^{\ensuremath{\beta}}$) with temperature-dependent exponents x(T) and \ensuremath{\beta}(T) describes the decay very well at all temperatures above the spin-glass transition. In the spin-glass phase, only the algebraic decay q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$ could be observed, with different temperature dependences of the exponent x(T). The dynamic scaling hypothesis and finite-size scaling explain well the observed temperature and size dependence of the data, and the functional form of the correlation functions is com- patible with the scaling form if corrections to scaling are taken into account. The scaling behavior and the dynamic and static critical exponents found in my simulations are in reasonable agreement with recent experiments performed on insulating spin glasses, showing that despite its simplicity the discrete model of spin glasses analyzed in this work displays behavior similar to that seen in nature.
The statics-dynamics correspondence in spin glasses relate non-equilibrium results on large samples (the experimental realm) with equilibrium quantities computed on small systems (the typical arena for theoretical computations). Here we employ statics-dynamics equivalence to study the Ising spin-glass critical behavior in three dimensions. By means of Monte Carlo simulation, we follow the growth of the coherence length (the size of the glassy domains), on lattices too large to be thermalized. Thanks to the large coherence lengths we reach, we are able to obtain accurate results in excellent agreement with the best available equilibrium computations. To do so, we need to clarify the several physical meanings of the dynamic exponent close to the critical temperature.
Domain-wall free-energy δF\delta FδF, entropy δS\delta SδS, and the correlation function, CtempC_{\rm temp}Ctemp​, of δF\delta FδF are measured independently in the four-dimensional ±J\pm J±J Edwards-Anderson (EA) Ising spin glass. The stiffness exponent θ\thetaθ, the fractal dimension of domain walls dsd_{\rm s}ds​ and the chaos exponent ζ\zetaζ are extracted from the finite-size scaling analysis of δF\delta FδF, δS\delta SδS and CtempC_{\rm temp}Ctemp​ respectively well inside the spin-glass phase. The three exponents are confirmed to satisfy the scaling relation ζ=ds/2−θ\zeta=d_{\rm s}/2-\thetaζ=ds​/2−θ derived by the droplet theory within our numerical accuracy. We also study bond chaos induced by random variation of bonds, and find that the bond and temperature perturbations yield the universal chaos effects described by a common scaling function and the chaos exponent. These results strongly support the appropriateness of the droplet theory for the description of chaos effect in the EA Ising spin glasses.
The nature of static chaos in Ising spin glasses is studied. For the problem of chaos with magnetic field, scaling relations in the case of the SK model and short-range models are presented. Our results also suggest that if there is de Almeida-Thouless line then it is similar to that of mean-field theory for d=4 and close to the h=0 axis for d=3. We estimate the lower critical dimension to be in the range 2.7-2.9. Numerical studies at d=4 show that there is chaos against temperature changes dT and the correlation length diverges like 1/dT.
We discuss the problem of static chaos in spin glasses. In the case of magnetic-field perturbations, we propose a scaling theory for the spin-glass phase. Using the mean-field approach we argue that some pure states are suppressed by the magnetic field and their free-energy cost is determined by the finite-temperature fixed-point exponents. In this framework, numerical results suggest that mean-field chaos exponents are probably exact in finite dimensions. If we use the droplet approach, numerical results suggest that the zero-temperature fixed-point exponent \ensuremath{\theta} is very close to (d-3)/2. In both approaches d=3 is the lower critical dimension in agreement with recent numerical simulations.