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Alain Billoire

Publications and source records attributed to Alain Billoire.

At least 19 recordsLinked to original sources
arXiv2019

Strong ergodicity breaking in aging of mean field spin glasses

Out of equilibrium relaxation processes show aging if they become slower as time passes. Aging processes are ubiquitous and play a fundamental role in the physics of glasses and spin glasses and in other applications (e.g. in algorithms minimizing complex cost/loss functions). The theory of aging in the out of equilibrium dynamics of mean-field spin glass models has achieved a fundamental role, thanks to the asymptotic analytic solution found by Cugliandolo and Kurchan. However this solution is based on assumptions (e.g. the weak ergodicity breaking hypothesis) which have never been put under a strong test until now. In the present work we present the results of an extraordinary large set of numerical simulations of the prototypical mean-field spin glass models, namely the Sherrington-Kirkpatrick and the Viana-Bray models. Thanks to a very intensive use of GPUs, we have been able to run the latter model for more than $2^{64}$ spin updates and thus safely extrapolate the numerical data both in the thermodynamical limit and in the large times limit. The measurements of the two-times correlation functions in isothermal aging after a quench from a random initial configuration to a temperature $T<T_c$ provides clear evidence that, at large times, such correlations do not decay to zero as expected by assuming weak ergodicity breaking. We conclude that strong ergodicity breaking takes place in mean-field spin glasses aging dynamics which, asymptotically, takes place in a confined configurational space. Theoretical models for the aging dynamics need to be revised accordingly.

cond-mat.dis-nn↗
arXiv2019

Maximum-energy records in glassy energy landscapes

We study the evolution of the maximum energy $E_\max(t)$ reached between time $0$ and time $t$ in the dynamics of simple models with glassy energy landscapes, in instant quenches from infinite temperature to a target temperature $T$. Through a detailed description of the activated dynamics, we are able to describe the evolution of $E_\max(t)$ from short times, through the aging regime, until after equilibrium is reached, thus providing a detailed description of the long-time dynamics. Finally, we compare our findings with numerical simulations of the $p$-spin glass and show how the maximum energy record can be used to identify the threshold energy in this model.

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arXiv2015

Numerical study of the dynamics of some long range spin glass models

We present results of a Monte Carlo study of the equilibrium dynamics of the one dimensional long-range Ising spin glass model. By tuning a parameter $σ$, this model interpolates between the mean field Sherrington-Kirkpatrick model and a proxy of the finite dimensional Edward-Anderson model. Activated scaling fits for the behavior of the relaxation time $τ$ as a function of the number of spins $N$ (Namely $\ln(τ)\propto N^ψ$) give values of $ψ$ that are not stable against inclusion of subleading corrections. Critical scaling ($τ\propto N^ρ$) gives more stable fits, at least in the non mean field region. We also present results on the scaling of the time decay of the critical remanent magnetization of the Sherrington-Kirkpatrick model, a case where the simulation can be done with quite large systems and that shows the difficulties in obtaining precise values for dynamical exponents in spin glass models.

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arXiv2014

Rare events analysis of temperature chaos in the Sherrington-Kirkpatrick model

We investigate the question of temperature chaos in the Sherrington-Kirkpatrick spin glass model, applying to existing Monte Carlo data a recently proposed rare events based data analysis method. Thanks to this new method, temperature chaos is now observable for this model, even with the limited size systems that can be currently simulated.

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arXiv2011

Dynamics in the Sherrington-Kirkpatrick Ising spin glass at and above Tg

A detailed numerical study is made of relaxation at equilibrium in the Sherrington-Kirkpatrick Ising spin glass model, at and above the critical temperature Tg. The data show a long time stretched exponential relaxation q(t) ~ exp[-(t/tau(T))^beta(T)] with an exponent beta(T) tending to ~ 1/3 at Tg. The results are compared to those which were observed by Ogielski in the 3d ISG model, and are discussed in terms of a phase space percolation transition scenario.

cond-mat.dis-nn↗
arXiv2011

What makes slow samples slow in the Sherrington-Kirkpatrick model

Using results of a Monte Carlo simulation of the Sherrington-Kirkpatrick model, we try to characterize the slow disorder samples, namely we analyze visually the correlation between the relaxation time for a given disorder sample $J$ with several observables of the system for the same disorder sample. For temperatures below $T_c$ but not too low, fast samples (small relaxation times) are clearly correlated with a small value of the largest eigenvalue of the coupling matrix, a large value of the site averaged local field probability distribution at the origin, or a small value of the squared overlap $ $. Within our limited data, the correlation remains as the system size increases but becomes less clear as the temperature is decreased (the correlation with $ $ is more robust) . There is a strong correlation between the values of the relaxation time for two distinct values of the temperature, but this correlation decreases as the system size is increased. This may indicate the onset of temperature chaos.

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arXiv2010

Distribution of time scales in the Sherrington-Kirkpatrick model

Numerical data on the probability distribution of the equilibrium relaxation time of the Sherrington-Kirkpatrick model are obtained by means of dynamical Monte Carlo simulation, for several values of the system size $N$ and temperature $T$. Proper care is taken that the thermal fluctuations on the relaxation time estimates are totally negligible compared to the disorder induced fluctuations. The probability distribution of $\lnτ-\overline{\lnτ}$ scales with the scaling variable $N^{1/3} (T_c-T)$ strengthening the belief that $\overline{\lnτ}\propto N^{1/3}$ in the whole spin glass phase.

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arXiv2010

Large random correlations in individual mean field spin glass samples

We argue that complex systems must possess long range correlations and illustrate this idea on the example of the mean field spin glass model. Defined on the complete graph, this model has no genuine concept of distance, but the long range character of correlations is translated into a broad distribution of the spin-spin correlation coefficients for almost all realizations of the random couplings. When we sample the whole phase space we find that this distribution is so broad indeed that at low temperatures it essentially becomes uniform, with all possible correlation values appearing with the same probability. The distribution of correlations inside a single phase space valley is also studied and found to be much narrower.

cond-mat.dis-nn↗
arXiv2009

Predictive power of MCT: Numerics and Finite size scaling for a mean field spin glass

The aim of this paper is to test numerically the predictions of the Mode Coupling Theory (MCT) of the glass transition and study its finite size scaling properties in a model with an exact MCT transition, which we choose to be the fully connected Random Orthogonal Model. Surprisingly, some predictions are verified while others seem clearly violated, with inconsistent values of some MCT exponents. We show that this is due to strong pre-asymptotic effects that disappear only in a surprisingly narrow region around the critical point. Our study of Finite Size Scaling (FSS) show that standard theory valid for pure systems fails because of strong sample to sample fluctuations. We propose a modified form of FSS that accounts well for our results. {\it En passant,} we also give new theoretical insights about FSS in disordered systems above their upper critical dimension. Our conclusion is that the quantitative predictions of MCT are exceedingly difficult to test even for models for which MCT is exact. Our results highlight that some predictions are more robust than others. This could provide useful guidance when dealing with experimental data.

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arXiv2007

Some aspects of infinite range models of spin glasses: theory and numerical simulations

These notes give an introduction to the physics of the infinite range version of the Edwards--Anderson model, the so-called Sherrington--Kirkpatrick model. In a first part, I motivate and introduce the Edwards--Anderson and Sherrington--Kirkpatrick models. In the second part, I explain the analytical solution of the Sherrington--Kirkpatrick model, following Giorgio Parisi. I next give the physical interpretation of this solution. This is a vast subject, and I concentrate on the major points and give references for more details. The third part presents the simulation approach and compare its results to theoretical expectations: thermodynamics, finite size scaling, determination of the critical temperature using "parameters" like the "Binder parameter", and fluctuations. The last part gives a summary of our current understanding of finite size effects for the free energy and internal energy of the Sherrington--Kirkpatrick model

cond-mat.dis-nn↗
arXiv2005

The mean field infinite range p=3 spin glass: equilibrium landscape and correlation time scales

We investigate numerically the dynamical behavior of the mean field 3-spin spin glass model: we study equilibrium dynamics, and compute equilibrium time scales as a function of the system size V. We find that for increasing volumes the time scales $τ$ increase like $\ln τ\propto V$. We also present an accurate study of the equilibrium static properties of the system.

cond-mat.dis-nn↗
arXiv2003

On the Tail of the Overlap Probability Distribution in the Sherrington--Kirkpatrick Model

We investigate the large deviation behavior of the overlap probability density in the Sherrington--Kirkpatrick model from several analytical perspectives. First we analyze the spin glass phase using the coupled replica scheme. Here generically $\frac1N \log P_N(q)$ $\approx$ $- {\cal A}$ $((|q|-q_{EA})^3$, and we compute the first correction to the expansion of $\A$ in powers of $T_c-T$. We study also the $q=1$ case, where $P(q)$ is know exactly. Finally we study the paramagnetic phase, where exact results valid for all $q$'s are obtained. The overall agreement between the various points of view is very satisfactory. Data from large scale numerical simulations show that the predicted behavior can be detected already on moderate lattice sizes.

cond-mat.stat-mech↗
arXiv2003

Numerical study of the SK Model in magnetic field

We study numerically the Sherrington--Kirkpatrick model as function of the magnetic field h, with fixed temperature T=0.6 Tc. We investigate the finite size scaling behavior of several quantities, such as the spin glass susceptibility, looking for numerical evidences of the transition on the De Almeida Thouless line. We find strong corrections to scaling which make difficult to locate the transition point. This shows, in a simple case, the extreme difficulties of spin glass simulations in non-zero magnetic field. Next, we study various sum rules (consequences of stochastic stability) involving overlaps between three and four replicas, which appear to be numerically well satisfied, and in a non-trivial way. Finally, we present data on P(q) for a large lattice size (N=3200) at low temperature T=0.4 Tc, where, for the first time, the shape predicted by the RSB solution of the model for non-zero magnetic field is visible.

cond-mat.stat-mech↗
arXiv2002

Magnetic field chaos in the SK Model

We study the Sherrington--Kirkpatrick model, both above and below the De Almeida Thouless line, by using a modified version of the Parallel Tempering algorithm in which the system is allowed to move between different values of the magnetic field h. The behavior of the probability distribution of the overlap between two replicas at different values of the magnetic field h_0 and h_1 gives clear evidence for the presence of magnetic field chaos already for moderate system sizes, in contrast to the case of temperature chaos, which is not visible on system sizes that can currently be thermalized.

cond-mat.stat-mech↗
arXiv2002

Overlap Among States at Different Temperatures in the SK Model

We discuss the issue of temperature chaos in the Sherrington--Kirkpatrick spin glass mean field model. We numerically compute probability distributions of the overlap among (equilibrium) configurations at two different values of the temperature, both in the spin glass phase. The situation on our medium size systems is clearly non-chaotic, but a weak form of chaos could be emerging on very large lattices.

cond-mat.dis-nn↗
arXiv2002

Overlap Distribution of the Three-Dimensional Ising Model

We study the Parisi overlap probability density P_L(q) for the three-dimensional Ising ferromagnet by means of Monte Carlo (MC) simulations. At the critical point P_L(q) is peaked around q=0 in contrast with the double peaked magnetic probability density. We give particular attention to the tails of the overlap distribution at the critical point, which we control over up to 500 orders of magnitude by using the multi-overlap MC algorithm. Below the critical temperature interface tension estimates from the overlap probability density are given and their approach to the infinite volume limit appears to be smoother than for estimates from the magnetization.

cond-mat.stat-mech↗
arXiv2001

Multi-overlap simulations of spin glasses

We present results of recent high-statistics Monte Carlo simulations of the Edwards-Anderson Ising spin-glass model in three and four dimensions. The study is based on a non-Boltzmann sampling technique, the multi-overlap algorithm which is specifically tailored for sampling rare-event states. We thus concentrate on those properties which are difficult to obtain with standard canonical Boltzmann sampling such as the free-energy barriers F^q_B in the probability density P_J(q) of the Parisi overlap parameter q and the behaviour of the tails of the disorder averaged density P(q) = [P_J(q)]_av.

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