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arXiv:2409.20377

Learning thermodynamics and topological order of the 2D XY model with generative real-valued restricted Boltzmann machines

Kai Zhang

Published Sep 30, 2024• 0 citations
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Abstract

Detecting the topological Kosterlitz-Thouless (KT) transition in the prototypical 2D XY model using unsupervised machine learning methods has long been a challenging problem due to the lack of suitable order parameters. To address this issue, we begin with a conventional real-valued RBM (RBM-xy), which uses exponential conditional probabilities to generate visible units. We then develop a novel real-valued RBM (RBM-CosSin) featuring nonlinear cos/sin activation, whose visible units follow the von Mises distribution. Our findings reveal that RBM-CosSin effectively learns the underlying Boltzmann distribution of 2D XY systems and generate authentic XY configurations that accurately capture both thermodynamics and topological order (vortex). Furthermore, we demonstrate that it is possible to extract phase transition information, including the KT transition, from the weight matrices without relying on prior physics knowledge.

DOI
10.48550/arxiv.2409.20377

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Detecting the topological Kosterlitz-Thouless (KT) transition in the prototypical 2D XY model using unsupervised machine learning methods has long been a challenging problem due to the lack of suitable order parameters. To address this issue, we begin with a conventional real-valued RBM (RBM-xy), which uses exponential conditional probabilities to generate visible units. We then develop a novel real-valued RBM (RBM-CosSin) featuring nonlinear cos/sin activation, whose visible units follow the von Mises distribution. Our findings reveal that RBM-CosSin effectively learns the underlying Boltzmann distribution of 2D XY systems and generate authentic XY configurations that accurately capture both thermodynamics and topological order (vortex). Furthermore, we demonstrate that it is possible to extract phase transition information, including the KT transition, from the weight matrices without relying on prior physics knowledge.
Detecting the topological Kosterlitz-Thouless (KT) transition in the prototypical two-dimensional (2D) XY model using unsupervised machine learning methods has long been a challenging problem due to the lack of suitable order parameters. To address this issue, we begin with a conventional real-valued restricted Boltzmann machine (RBM) (RBM-xy), which uses exponential conditional probabilities to generate visible units. We then develop a novel real-valued RBM (RBM-CosSin) featuring nonlinear cos/sincos/sincos/sin activation, whose visible units follow the von Mises distribution. Our findings reveal that RBM-CosSin effectively learns the underlying Boltzmann distribution of 2D XY systems and generates authentic XY configurations that accurately capture both thermodynamics and topological order (vortex). Furthermore, we demonstrate that it is possible to extract phase transition information, including the KT transition, from the weight matrices without relying on prior physics knowledge.
The restricted Boltzmann machine (RBM) is a two-layer energy-based model that uses its hidden-visible connections to learn the underlying distribution of visible units, whose interactions are often complicated by high-order correlations. Previous studies on the Ising model of small system sizes have shown that RBMs are able to accurately learn the Boltzmann distribution and reconstruct thermal quantities at temperatures away from the critical point Tc. How the RBM encodes the Boltzmann distribution and captures the phase transition are, however, not well explained. In this work, we perform RBM learning of the 2d and 3d Ising model and carefully examine how the RBM extracts useful probabilistic and physical information from Ising configurations. We find several indicators derived from the weight matrix that could characterize the Ising phase transition. We verify that the hidden encoding of a visible state tends to have an equal number of positive and negative units, whose sequence is randomly assigned during training and can be inferred by analyzing the weight matrix. We also explore the physical meaning of the visible energy and loss function (pseudo-likelihood) of the RBM and show that they could be harnessed to predict the critical point or estimate physical quantities such as entropy.
The application of generative modeling to many-body physics offers a promising pathway for analyzing high-dimensional state spaces of spin systems. However, unlike computer vision tasks where visual fidelity suffices, physical systems require the rigorous reproduction of higher-order statistical moments and thermodynamic quantities. While Score-Based Generative Models (SGMs) have emerged as a powerful tool, their standard formulation on Euclidean embedding space is ill-suited for continuous spin systems, where variables inherently reside on a manifold. In this work, we demonstrate that training on the Euclidean space compromises the model's ability to learn the target distribution as it prioritizes to learn the manifold constraints. We address this limitation by proposing the use of Manifold-Aware Score-Based Generative Modeling framework applied to the 64x64 2D XY model (a 4096-dimensional torus). We show that our method estimates the theoretical Boltzmann score with superior precision compared to standard diffusion models. Consequently, we successfully capture the Berezinskii-Kosterlitz Thouless (BKT) phase transition and accurately reproduce second-moment quantities, such as heat capacity without explicit feature engineering. Furthermore, we demonstrate zero-shot generalization to unseen lattice sizes, accurately recovering the physics of variable system scales without retraining. Since this approach bypasses domain-specific feature engineering, it remains intrinsically generalizable to other continuous spin systems.
Abstract Machine learning (ML) has become a useful tool for studying phase transitions in statistical systems. For the two-dimensional classical XY model, however, the topological character of the Berezinskii–Kosterlitz–Thouless (BKT) transition and pronounced finite-size effects make it nontrivial to extract reproducible size-dependent pseudo-critical temperatures from configuration data. Existing studies often stop at phase classification, leaving open how standard neural-network outputs can be turned into quantitatively testable observables. Here we develop a ML-assisted framework for the 2D XY model that uses standard network outputs to extract a size-dependent sequence of operational pseudo-critical temperatures T ∗ ( L ) . Specifically, we generate Monte Carlo configurations using embedded cluster updates, train a standard ResNet18 only on samples from the Quasi-ordered Phase and the Disordered Phase, and determine T ∗ ( L ) from bootstrap-averaged probability curves using the 50% crossing criterion. We then analyze the finite-size drift of this temperature sequence using BKT-motivated scaling and compare it with susceptibility-peak temperatures. The selected T ∗ ( L ) values display an overall downward finite-size drift compatible with BKT-motivated behavior and lie within the conventional susceptibility fluctuation regime. We further characterize and assess the extracted scale through threshold-sensitivity tests, window-variation analysis, probability calibration, comparison with a simple-physics baseline, vortex-density measurement, input-encoding comparison, and saliency mapping. These diagnostics reveal that while the network response is dominated by macroscopic physical observables, the extracted T ∗ ( L ) is conditional on representation, augmentation, and model-selection protocols. These results demonstrate that standard classifier outputs can be converted into an operational finite-size temperature scale on a well-characterized benchmark model, while also revealing the dependence of that scale on the adopted inference protocol.
In several physical systems, important properties characterizing the system itself are theoretically related with specific degrees of freedom. Although standard Monte Carlo simulations provide an effective tool to accurately reconstruct the physical configurations of the system, they are unable to isolate the different contributions corresponding to different degrees of freedom. Here we show that unsupervised deep learning can become a valid support to MC simulation, coupling useful insights in the phases detection task with good reconstruction performance. As a testbed we consider the 2D XY model, showing that a deep neural network based on variational autoencoders can detect the continuous Kosterlitz-Thouless (KT) transitions, and that, if endowed with the appropriate constrains, they generate configurations with meaningful physical content.
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.