Discovering quasiorder parameters in the Potts model: A bridge between machine learning and critical phenomena

Machine-learning (ML) models trained on Ising spin configurations have demonstrated surprising effectiveness in classifying phases of Potts models, even when processing severely reduced representations that retain only two spin states. To unravel this remarkable capability, we identify a family of alternative order parameters for the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>q</a:mi> <a:mo>=</a:mo> <a:mn>3</a:mn> </a:mrow> </a:math> and <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>q</b:mi> <b:mo>=</b:mo> <b:mn>4</b:mn> </b:mrow> </b:math> Potts models on a square lattice, constructed from the occupancies of secondary and minimal spin states rather than the conventional dominant-state order parameter. Through systematic finite-size scaling analyses, we demonstrate that these quantities, along with a magnetizationlike quantity derived from a reduced spin representation, accurately capture critical behavior, yielding critical temperatures and exponents consistent with established theoretical predictions and numerical benchmarks. Furthermore, we rigorously establish the fundamental relationships between these alternative (quasi)order parameters, demonstrating how they collectively encode criticality through different aspects of spin configurations. Our results clarify, within this specific setting, how reduced spin representations can retain the essential thermodynamic information needed for identifying critical behavior. Taken together, this work establishes a concrete bridge between Ising-trained ML models and critical phenomena in Potts systems by showing that Potts criticality can be encoded in more compact, nontraditional forms, thereby opening avenues for discovering analogous order parameters in broader spin systems.