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arXiv:2604.00386 · Physica Scripta

Machine-learning extraction of size-dependent temperature scales in the 2D XY model

Qingao Fan, Xu Li, Tingting Xue

Published Sep 30, 2026• 0 citations
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Abstract

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.

DOI
10.1088/1402-4896/aeaeb7

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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.
We propose two machine-learning methods based on neural networks, which we respectively call the phase-classification method and the temperature-identification method, for detecting different types of phase transitions in the XXZ models without prior knowledge of their critical temperatures. The XXZ models have exchange couplings which are anisotropic in the spin space where the strength is represented by a parameter $\mathrm{\ensuremath{\Delta}}(>0)$. The models exhibit second-order phase transition when $\mathrm{\ensuremath{\Delta}}>1$, but the Berezinskii-Kosterlitz-Thouless (BKT) phase transition when $\mathrm{\ensuremath{\Delta}}<1$. In the phase-classification method, the neural network is trained using spin or vortex configurations of well-known classical spin models other than the XXZ models, e.g., the Ising models and the XY models, to classify those of the XXZ models to corresponding phases. We demonstrate that the trained neural network successfully detects the phase transitions for both $\mathrm{\ensuremath{\Delta}}>1$ and $\mathrm{\ensuremath{\Delta}}<1$, and the evaluated critical temperatures coincide well with those evaluated by conventional numerical calculations. In the temperature-identification method, on the other hand, the neural network is trained so as to identify temperatures at which the input spin or vortex configurations are generated by the Monte Carlo thermalization. The critical temperatures are evaluated by analyzing the optimized weight matrix, which coincides with the result of numerical calculation for the second-order phase transition in the Ising-like XXZ model with $\mathrm{\ensuremath{\Delta}}=1.05$ but cannot be determined uniquely for the BKT transition in the XY-like XXZ model with $\mathrm{\ensuremath{\Delta}}=0.95$.
We test an improved finite-size scaling method for reliably extracting the critical temperature T BKT of a Berezinskii–Kosterlitz–Thouless (BKT) transition. Using known single-parameter logarithmic corrections to the spin stiffness ρ s at T BKT in combination with the Kosterlitz–Nelson relation between the transition temperature and the stiffness, ρ s ( T BKT ) = 2 T BKT /π, we define a size-dependent transition temperature T BKT ( L 1 , L 2 ) based on a pair of system sizes L 1 , L 2 , e.g., L 2 = 2 L 1 . We use Monte Carlo data for the standard two-dimensional classical XY model to demonstrate that this quantity is well behaved and can be reliably extrapolated to the thermodynamic limit using the next expected logarithmic correction beyond the ones included in defining T BKT ( L 1 , L 2 ). For the Monte Carlo calculations we use GPU (graphical processing unit) computing to obtain high-precision data for L up to 512. We find that the sub-leading logarithmic corrections have significant effects on the extrapolation. Our result T BKT = 0.8935(1) is several error bars above the previously best estimates of the transition temperature, T BKT ≈ 0.8929. If only the leading log-correction is used, the result is, however, consistent with the lower value, suggesting that previous works have underestimated T BKT because of the neglect of sub-leading logarithms. Our method is easy to implement in practice and should be applicable to generic BKT transitions.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is a typical topological phase transition defined between binding and unbinding states of vortices and antivortices, which is not accompanied by spontaneous symmetry breaking. It is known that the BKT transition is difficult to detect from thermodynamic quantities such as specific heat and magnetic susceptibility because of the absence of anomaly in free energy and significant finite-size effects. Therefore, methods based on statistical mechanics which are commonly used to discuss phase transitions cannot be easily applied to the BKT transition. In recent years, several attempts to detect the BKT transition using machine-learning methods based on image recognition techniques have been reported. However, it has turned out that the detection is difficult even for machine learning methods because of the absence of trivial order parameters and symmetry breaking. Most of the methods proposed so far require prior knowledge about the models and/or preprocessing of input data for feature engineering, which is problematic in terms of the general applicability. In this article, we introduce recent development of the machine-learning methods to detect the BKT transitions in several spin models. Specifically, we demonstrate the success of two new methods named temperature-identification method and phase-classification method for detecting the BKT transitions in the q-state clock model and the XXZ model. This progress is expected to sublimate the machine-learning-based study of spin models for exploring new physics beyond simple benchmark test.
In this paper, we apply machine learning methods to study phase transitions in certain statistical mechanical models on the two-dimensional lattices, whose transitions involve nonlocal or topological properties, including site and bond percolations, the XY model, and the generalized XY model. We find that using just one hidden layer in a fully connected neural network, the percolation transition can be learned and the data collapse by using the average output layer gives correct estimate of the critical exponent ν. We also study the Berezinskii-Kosterlitz-Thouless transition, which involves binding and unbinding of topological defects, vortices and antivortices, in the classical XY model. The generalized XY model contains richer phases, such as the nematic phase, the paramagnetic and the quasi-long-range ferromagnetic phases, and we also apply machine learning method to it. We obtain a consistent phase diagram from the network trained with only data along the temperature axis at two particular parameter Δ values, where Δ is the relative weight of pure XY coupling. Aside from using the spin configurations (either angles or spin components) as the input information in a convolutional neural network, we devise a feature engineering approach using the histograms of the spin orientations in order to train the network to learn the three phases in the generalized XY model and demonstrate that it indeed works. The trained network by using system size L×L can be used to the phase diagram for other sizes (L^{'}×L^{'}, where L^{'}≠L) without any further training.
Using a supervised neural network (NN) trained once on a one-dimensional lattice of 200 sites, we calculate the Berezinskii–Kosterlitz–Thouless phase transitions of the two-dimensional (2D) classical XY and the 2D generalized classical XY models. In particular, both the bulk quantities Binder ratios and the spin states of the studied systems are employed to construct the needed configurations for the NN prediction. By applying semiempirical finite-size scaling to the relevant data, the critical points obtained by the NN approach agree well with the known results established in the literature. This implies that for each of the considered models, the determination of its various phases requires only a little information. The outcomes presented here demonstrate convincingly that the employed universal NN is not only valid for the symmetry breaking related phase transitions, but also works for calculating the critical points of the phase transitions associated with topology. The efficiency of the used NN in the computation is examined by carrying out several detailed benchmark calculations.
The classical XY model has been consistently studied since it was introduced more than six decades ago. Of particular interest has been the two-dimensional spin model's exhibition of the Berezinskii-Kosterlitz-Thouless (BKT) transition. This topological phenomenon describes the transition from bound vortex-antivortex pairs at low temperatures to unpaired or isolated vortices and antivortices above some critical temperature. In this work we propose a machine learning based method to determine the emergence of this phase transition. Generating unique states can be difficult due to the U(1) symmetry present. We introduce an auxiliary field (analogous to a vortex density field) corresponding to a given state in order to eliminate the unwanted symmetry. An autoencoder was used to map these auxiliary fields into a lower-dimensional latent space. Samples were taken from this latent space to determine the thermal average of the vortex density, which was then used to determine the critical temperature of the phase transition.