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arXiv:2610.41500

Characterization of Phase Transitions with the Annealed Importance Sampling Algorithm

A. Prat Pou, E. Romero, J. Marti, F. Mazzanti

Published Oct 30, 2026• 0 citations
arXivHTMLPDF

Abstract

We We explore the use of the Annealed Importance Sampling (AIS) algorithm as a numerical tool to study phase transitions and critical phenomena. By mapping physical systems such as Ising and spin-glass models directly onto Restricted Boltzmann Machines (RBMs), and extending the analysis to both synthetic and trained architectures, we use a unified RBM framework to compare the performance of AIS against standard Monte Carlo methods in regimes characterized by strong fluctuations and complex energy landscapes. The unified RBM formulation allows for massively parallel sampling, offering a scalable and precise alternative to traditional Monte Carlo methods. Building on a simple, heuristic mean-field AIS initialization derived from the weights of the network, the proposed scheme enhances sampling efficiency and accuracy across all systems considered, proving its effectiveness as a tool to detect and characterize phase transitions in discrete spin and spin-like systems. Our results show that a properly initialized AIS reliably estimates thermodynamic observables, including the partition function and magnetization, while accurately detecting both paramagnetic-ferromagnetic and paramagnetic-spin-glass phase transitions across various system sizes.

DOI
10.48550/arxiv.2610.41500

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We We explore the use of the Annealed Importance Sampling (AIS) algorithm as a numerical tool to study phase transitions and critical phenomena. By mapping physical systems such as Ising and spin-glass models directly onto Restricted Boltzmann Machines (RBMs), and extending the analysis to both synthetic and trained architectures, we use a unified RBM framework to compare the performance of AIS against standard Monte Carlo methods in regimes characterized by strong fluctuations and complex energy landscapes. The unified RBM formulation allows for massively parallel sampling, offering a scalable and precise alternative to traditional Monte Carlo methods. Building on a simple, heuristic mean-field AIS initialization derived from the weights of the network, the proposed scheme enhances sampling efficiency and accuracy across all systems considered, proving its effectiveness as a tool to detect and characterize phase transitions in discrete spin and spin-like systems. Our results show that a properly initialized AIS reliably estimates thermodynamic observables, including the partition function and magnetization, while accurately detecting both paramagnetic-ferromagnetic and paramagnetic-spin-glass phase transitions across various system sizes.
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We train a set of restricted Boltzmann machines (RBMs) on one- and two-dimensional Ising spin configurations at various values of temperature, generated using Monte Carlo simulations. We validate the training procedure by monitoring several estimators, including measurements of the log likelihood, with the corresponding partition functions estimated using annealed importance sampling. The effects of various choices of hyperparameters on training the RBM are discussed in detail, with a generic prescription provided. Finally, we present a closed-form expression for extracting the values of couplings, for every nnn-point interaction between the visible nodes of an RBM, in a binary system such as the Ising model. We aim at using this study as the foundation for further investigations of less well-known systems.
Machine learning methods are powerful in distinguishing different phases of matter in an automated way and provide a new perspective on the study of physical phenomena.We train a restricted Boltzmann machine (RBM) on data constructed with spin configurations sampled from the Ising Hamiltonian at different values of temperature and external magnetic field using Monte Carlo methods.From the trained machine we obtain the flow of iterative reconstruction of spin state configurations to faithfully reproduce the observables of the physical system.We find that the flow of the trained RBM approaches the spin configurations of the maximal possible specific heat which resemble the near-criticality region of the Ising model.In the special case of the vanishing magnetic field the trained RBM converges to the critical point of the renormalization group (RG) flow of the lattice model.Our results suggest an explanation of how the machine identifies the physical phase transitions, by recognizing certain properties of the configuration like the maximization of the specific heat, instead of associating directly the recognition procedure with the RG flow and its fixed points.Then from the reconstructed data we deduce the critical exponent associated with the magnetization to find satisfactory agreement with the actual physical value.We assume no prior knowledge about the criticality of the system and its Hamiltonian.
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This study presents a comprehensive Monte Carlo simulation of the two-dimensional (2-D) Ising model using the Metropolis algorithm to investigate critical phenomena and thermodynamic behavior in spin-lattice systems. The model, implemented in MATLAB with periodic boundary conditions, explores equilibrium properties such as magnetization, internal energy, specific heat, and susceptibility across a range of temperatures. By employing various initial spin configurations—ordered and random—the simulations demonstrate the system's ergodicity and convergence to thermal equilibrium. Key results include a sharp decline in magnetization and a pronounced peak in specific heat near the critical temperature, consistent with second-order phase transition behavior. The simulation captures microscopic domain evolution, highlighting the transition from ferromagnetic to paramagnetic phases as thermal fluctuations increase. The study further evaluates algorithmic efficiency, discusses the impact of lattice size on statistical accuracy, and proposes improvements using advanced cluster algorithms and parallel computing frameworks. The findings validate theoretical predictions from Onsager's solution and underscore the versatility of Monte Carlo techniques in modeling collective behavior in magnetic systems. The simulation framework offers a robust foundation for analyzing critical dynamics and extends its relevance to broader applications in material science, computational physics, and complex systems modeling.
A Boltzmann machine is a stochastic neural network that has been extensively used in the layers of deep architectures for modern machine learning applications. In this paper, we develop a Boltzmann machine that is capable of modeling thermodynamic observables for physical systems in thermal equilibrium. Through unsupervised learning, we train the Boltzmann machine on data sets constructed with spin configurations importance sampled from the partition function of an Ising Hamiltonian at different temperatures using Monte Carlo (MC) methods. The trained Boltzmann machine is then used to generate spin states, for which we compare thermodynamic observables to those computed by direct MC sampling. We demonstrate that the Boltzmann machine can faithfully reproduce the observables of the physical system. Further, we observe that the number of neurons required to obtain accurate results increases as the system is brought close to criticality.