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

TL;DR

This paper introduces a machine learning framework to accurately extract size-dependent pseudo-critical temperatures in the 2D XY model, addressing challenges posed by topological transitions and finite-size effects.

Contribution

It develops a method using neural networks trained on phase data to determine pseudo-critical temperatures and analyze their finite-size scaling behavior.

Findings

The temperature sequence exhibits finite-size drift consistent with BKT behavior.

The extracted pseudo-critical temperatures align with susceptibility peak temperatures.

The framework enables conversion of neural network outputs into physically meaningful observables.

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