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arXiv:2607.20243

Dynamical Criticality of a Machine-learning-assisted Monte Carlo algorithm for a Mean-Field Spin Glass model

Seiya Miyamoto, Masayuki Ohzeki, Yoshihiko Nishikawa

Published Jul 22, 2026• 0 citations
arXivHTMLPDF

Abstract

We critically assess the performance of an autoregressive generative neural network model by applying it to an antiferromagnetic Ising model on a random regular graph. We train the network on equilibrium configurations in the low-temperature spin-glass phase of the model and perform Monte Carlo simulations using spin configurations generated by the network. The relaxation time of the Monte Carlo simulations drastically decreases with increasing the size of the training dataset and converges to an optimal value. The dynamical exponent characterizing the growth of the optimal relaxation time as a function of the system size is slightly reduced compared to the local Monte Carlo dynamics. However, we find that the size of the training dataset to achieve the optimal performance grows much faster with the system size than the relaxation time, implying that the total training cost eventually hinders a practical use of the method at large system sizes.

DOI
10.48550/arxiv.2607.20243

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We critically assess the performance of an autoregressive generative neural network model by applying it to an antiferromagnetic Ising model on a random regular graph. We train the network on equilibrium configurations in the low-temperature spin-glass phase of the model and perform Monte Carlo simulations using spin configurations generated by the network. The relaxation time of the Monte Carlo simulations drastically decreases with increasing the size of the training dataset and converges to an optimal value. The dynamical exponent characterizing the growth of the optimal relaxation time as a function of the system size is slightly reduced compared to the local Monte Carlo dynamics. However, we find that the size of the training dataset to achieve the optimal performance grows much faster with the system size than the relaxation time, implying that the total training cost eventually hinders a practical use of the method at large system sizes.
Monte Carlo methods are essential across diverse scientific fields, yet their efficiency is frequently hampered by critical slowing down-a sharp increase in autocorrelation times near phase transitions. Although deep learning approaches, such as neural-network-based samplers, have been proposed to alleviate this issue, they face another serious problem: the difficulty of training the models. This difficulty partially stems from the overly general nature of original machine-learning architectures, which often ignore underlying physical symmetries and force networks to relearn them from scratch. In this paper, we demonstrate that incorporating physical priors into the model significantly enhances performance. Building upon existing strategies that integrate spin-spin interactions, we propose a framework that utilizes a prior probability distribution as a starting point for training. Our results for the Ising model, as well as for the Edwards-Anderson spin glass model, suggest that moving away from `blank slate' models in favor of physics-informed priors reduces the training burden and facilitates the simulation of larger system sizes in discrete spin 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 , 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 as a function of the number of spins N (Namely ) give values of that are not stable against inclusion of subleading corrections. Critical scaling ( ) 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.
We investigate the performance of machine learning algorithms trained exclusively with configurations obtained from importance sampling Monte Carlo simulations of the two-dimensional Ising model with conserved magnetization. For supervised machine learning, we use convolutional neural networks and find that the corresponding output not only allows to locate the phase transition point with high precision, it also displays a finite-size scaling characterized by an Ising critical exponent. For unsupervised learning, restricted Boltzmann machines (RBM) are trained to generate new configurations that are then used to compute various quantities. We find that RBM generates configurations with magnetizations and energies forbidden in the original physical system. The RBM generated configurations result in energy density probability distributions with incorrect weights as well as in wrong spatial correlations. We show that shortcomings are also encountered when training RBM with configurations obtained from the non-conserved Ising model.
We investigate by means of Monte Carlo simulations the fully connected p -state Potts model for different system sizes in order to see how the static and dynamic properties of a finite model compare with the, exactly known, behaviour of the system in the thermodynamic limit. Using p = 10 we are able to study the equilibrium dynamics for system sizes as large as N = 2560. We find that the static quantities, such as the energy, the entropy, the spin glass susceptibility as well as the distribution of the order parameter P ( q ) show very strong finite-size effects. From P ( q ) we calculate the fourth-order cumulant g 4 ( N , T ) and the Guerra parameter G ( N , T ) and show that these quantities cannot be used to locate the static transition temperature for the system sizes investigated. Also the spin-autocorrelation function C ( t ) shows strong finite-size effects in that it does not show a plateau even for temperatures around the dynamical critical temperature T D . We show that the dependence on N and T of the α-relaxation time can be understood by means of a dynamical finite-size scaling ansatz. C ( t ) does not obey the time-temperature superposition principle for temperatures around T D , but does so for significantly lower T . Finally we study the relaxation dynamics of the individual spins and show that their dependence on time depends strongly on the chosen spin, i.e. that the system is dynamically very heterogeneous, which explains the non-exponentiality of C ( t ).
We describe results of Monte Carlo simulation studies on the Ising spin glass in four dimensions on a hypercubic lattice with nearest neighbor bonds. Studies of the equilibrium static properties show that the system undergoes a genuine phase transition to an ordered spin glass phase. Critical dynamical behavior is analyzed to obtain the dynamic exponent. Finally, we describe results on the spin glass phase, in particular the finite size scaling of the order parameter distribution function, and compare it with existing models of the spin glass phase, namely the droplet model and the Parisi solution for the low temperature phase of the infinite range spin glass.
We study the critical properties of an Ising model with nearest neighbor antiferromagnetic interactions, placed on a triangular lattice, where each site is occupied by a spin, only with x probability. Randomness and frustration, two essential ingredients of spin-glasses, are present in this model. The entropy is obtained, by a transfer matrix calculation, as a function of x at low temperature (T=0.3 J/kB), for systems on a lattice of 10×19 sites. A fairly shallow minimum in the entropy appears near x=0.9, which suggests that this case is the most likely to show a transition into an ordered state at low temperature. We therefore simulate systems on lattices of 50×50 sites for x=0.9 and x=1 by the Monte Carlo method. The specific heat has a broad maximum at T=0.9 J/kB (1.2) for x=0.9 (1). The relaxation time (τ) obtained for T≥0.6 J/kB, turns out to be proportional to exp[A/(T−T0)c], and 0<T0≲0.4 J/kB; a fit with T0=0 yields c≂1 for x=1 but c≂2 for x=0.9.