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

Abstract

Machine learning 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 robust 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 machine learning-assisted framework for the 2D XY model that uses standard network outputs to extract the size-dependent sequence of 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 resulting temperature sequence shows a systematic finitesize drift consistent with BKT-type behavior and remains in the same fluctuation window as the susceptibility peak, supporting its interpretation as a finite-size pseudo-critical temperature. More broadly, this framework provides a practical route for converting standard neural-network outputs into physically interpretable finite-size observables in systems with strong crossover or topological transition signatures

Figures (4)

• Figure 1: ResNet18-based framework for extracting size-dependent temperature scales in the 2D XY model. Monte Carlo configurations are generated from the Quasi-ordered Phase, the Disordered Phase, and the intervening Critical Region. The network is trained using only samples from the Quasi-ordered Phase and the Disordered Phase, and the pseudo-critical temperature $T^*(L)$ is determined from the 50% crossing of the corresponding probability curves in the Critical Region.
• Figure 2: Thermodynamic quantities for the $32\times32$ system as functions of temperature. (a) Average energy per site. (b) Finite-size magnetization. (c) Specific heat. (d) Susceptibility.
• Figure 3: Neural-network probability curves and BKT finite-size scaling analysis for the 2D XY model across the studied sizes. (a) Probabilities assigned to the Quasi-ordered Phase and the Disordered Phase as functions of temperature. (b) Probability-curve collapse based on the BKT rescaled variable, with the inset showing the finite-size fit of the 50% probability crossing temperature $T^*(L)$.
• Figure 4: Comparison of size-dependent pseudo-critical temperatures $T^*(L)$ extracted from the classifier response with susceptibility-peak temperatures $T_{\chi}^{\mathrm{peak}}(L)$. (a) $T^*(L)$ and $T_{\chi}^{\mathrm{peak}}(L)$ as functions of system size. (b) Susceptibility as a function of temperature for the studied system sizes, with peak positions and 50% probability crossing temperatures marked.