Machine-learning extraction of size-dependent temperature scales in the 2D XY model
Qingao Fan, Xu Li, Tingting Xue
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.
arXiv:2604.00386•Physica Scripta•Sep 30, 2026•0 citations