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

TL;DR

The paper addresses why Ising-trained models can identify phase transitions in Potts systems using reduced spin representations and introduces a family of alternative (quasi)order parameters based on occupancies of subdominant spin states and a magnetization-like measure from two-component reductions. By performing finite-size scaling on $q=3$ and $q=4$ Potts models, it derives exact relationships among order-parameter-like quantities and shows that these forms reproduce known $T_c$, $ u$, and $ abla_ ext{σ}$ values, with $M_{ ext{map}}$ serving as a magnetization analogue in reduced representations. The results demonstrate that critical information persists in non-traditional descriptors and provide a principled explanation for the generalization of Ising-trained ML to Potts systems, connecting to $Z_q$ universality and the $Z_5$ clock model in the $q=3$ case. Overall, the work offers a concrete route to discover quasiorder parameters in broader spin systems and highlights the interpretability of data-driven approaches in capturing fundamental thermodynamic transitions.

Abstract

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 $q=3$ and $q=4$ 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 magnetization-like 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, non-traditional forms, thereby opening avenues for discovering analogous order parameters in broader spin systems.

Figures (11)

• Figure 1: (Color online) The mean (upper), standard error of mean (SEM) (middle), and relative standard error (RSE) (lower) of alternative order parameters $M_{\mathrm{max}}$, $M_{\mathrm{med}}$, and $M_{\mathrm{min}}$ and quasiorder parameter $M_{\mathrm{map}}$ as functions of temperature $T$ for the three-state Potts model at $d=2$, simulated on a square lattice of size $L=96$. The relation $M_{\mathrm{map}}=\frac{4}{9}(M_{\mathrm{max}}-M_{\mathrm{min}})$ is confirmed by the overlap of the curves in the upper panel. The blue horizontal dashed lines mark zero to indicate the sign, while the red vertical dashed lines denote the theoretical critical temperature $T_c$.
• Figure 2: (Color online) The mean (upper), standard error of mean (SEM) (middle), and relative standard error (RSE) (lower) of the alternative order parameters $M_{\mathrm{max}}^{}$, $M_{\mathrm{max}}^{\mathrm{sec}}$, $M_{\mathrm{min}}^{\mathrm{sec}}$, $M_{\mathrm{min}}^{}$, and quasiorder parameter $M_{\mathrm{map}}^{}$ as functions of temperature $T$ for the four-state Potts model at $d=2$, simulated on a square lattice of size $L=96$. The relation $M_{\mathrm{map}}=\frac{1}{8}(3(M_{\mathrm{max}}-M_{\mathrm{min}})+M_{\mathrm{max}}^{\mathrm{sec}}-M_{\mathrm{min}}^{\mathrm{sec}})$ is confirmed by the overlap of the curves in the upper panel. The blue horizontal dashed lines mark zero to indicate the sign, while the red vertical dashed lines denote the theoretical critical temperature $T_c$.
• Figure A1: (Color online) (Left) Order parameter $M_{\mathrm{max}}$ as a function of temperature $T$ for the $q=3$ Potts model on a square lattice with six different system sizes $L$. (Middle) Scaled order parameter $M_{\mathrm{max}}L^{\Delta_\sigma}$ versus $T$, with curves intersecting near the theoretical critical temperature $T_c$, marked by the red vertical dashed line. (Right) Scaled order parameter $M_{\mathrm{max}}L^{\Delta_\sigma}$ against the rescaled temperature $tL^{1/\nu}$, demonstrating data collapse consistent with finite-size scaling.
• Figure A2: (Color online) (Left) Quasiorder parameter $M_{\mathrm{map}}$ as a function of temperature $T$ for the $q=3$ Potts model on a square lattice with six different system sizes $L$. (Middle) Scaled quasiorder parameter $M_{\mathrm{map}}L^{\Delta_\sigma}$ versus $T$, with curves intersecting near the theoretical critical temperature $T_c$, marked by the red vertical dashed line. (Right) Scaled quasiorder parameter $M_{\mathrm{map}}L^{\Delta_\sigma}$ against the rescaled temperature $tL^{1/\nu}$, demonstrating data collapse consistent with finite-size scaling.
• Figure A3: (Color online) (Left) Order parameter $M_{\mathrm{min}}$ as a function of temperature $T$ for the $q=3$ Potts model on a square lattice with six different system sizes $L$. (Middle) Scaled order parameter $M_{\mathrm{min}}L^{\Delta_\sigma}$ versus $T$, with curves intersecting near the theoretical critical temperature $T_c$, marked by the red vertical dashed line. (Right) Scaled order parameter $M_{\mathrm{min}}L^{\Delta_\sigma}$ against the rescaled temperature $tL^{1/\nu}$, demonstrating data collapse consistent with finite-size scaling.