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 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.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 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.

arXiv:2505.06159Physical Review ResearchJan 2, 20261 citation

Algebraic properties of the 3-state Potts model

A. K. Kwaśniewski

Reports on Mathematical PhysicsOct 1, 19882 citations

Theory of Generalized 3-State Potts Model and Antiferromagnetic Potts Model

Takehiko Oguchi, Takuma Ishikawa, Haruo Yoshida

The spin Hamiltonian with spin-1 system including a single parameter is proposed which expresses the 3-state Potts model (PM) or antiferromagnetic Potts model (AFPM) according as a value of the parameter. They are investigated by a new method which is the extension of the effective Hamiltonian method in a unified way. The first-order transition or the second-order transition are obtained for the PM or the AFPM, respectively. In the AFPM, the ordered phase has a residual entropy, nevertheless the ordered phase can exist stably.The spin Hamiltonian with spin-1 system including a single parameter is proposed which expresses the 3-state Potts model (PM) or antiferromagnetic Potts model (AFPM) according as a value of the parameter. They are investigated by a new method which is the extension of the effective Hamiltonian method in a unified way. The first-order transition or the second-order transition are obtained for the PM or the AFPM, respectively. In the AFPM, the ordered phase has a residual entropy, nevertheless the ordered phase can exist stably.

Journal of the Physical Society of JapanMar 1, 19827 citations

Critical exponents of the three-state potts model

Bambi Hu

Physics Letters ASep 1, 198010 citations

On the Equivalent Theory of the Generalized τ^{(2)}-model and the Chiral Potts Model with two Alternating Vertical Rapidities

Shi-shyr Roan

By the Baxter's $Q_{72}$ -operator method, we demonstrate the equivalent theory between the generalized $τ^{(2)}$ -model (other than two special cases with a pseudovacuum state) and the $N$ -state chiral Potts model with two alternating vertical rapidities, where the degenerate models are included. As a consequence, the theory of the XXZ chain model associated to cyclic representations (with the parameter $ς$ ) of $U_{\sf q}(sl_2)$ with ${\sf q}^N=1$ for odd $N$ is identified with either (for $ς^N=1$ ) the chiral Potts model with two superintegrable vertical rapidities, or (for $ς^N \neq 1$ ) the degenerate model for the selfdual solution of the star-triangle relation. In all these identifications, the transfer matrices $T, \hat{T}$ of the chiral Potts model (including the degenerate ones) serve as the $Q_R, Q_L$ -operators of the corresponding $τ^{(2)}$ -model, so that the functional relations hold as in the solvable $N$ -state chiral Potts model.By the Baxter's $Q_{72}$-operator method, we demonstrate the equivalent theory between the generalized $τ^{(2)}$-model (other than two special cases with a pseudovacuum state) and the $N$-state chiral Potts model with two alternating vertical rapidities, where the degenerate models are included. As a consequence, the theory of the XXZ chain model associated to cyclic representations (with the parameter $ς$) of $U_{\sf q}(sl_2)$ with ${\sf q}^N=1$ for odd $N$ is identified with either (for $ς^N=1$) the chiral Potts model with two superintegrable vertical rapidities, or (for $ς^N \neq 1$) the degenerate model for the selfdual solution of the star-triangle relation. In all these identifications, the transfer matrices $T, \hat{T}$ of the chiral Potts model (including the degenerate ones) serve as the $Q_R, Q_L$-operators of the corresponding $τ^{(2)}$-model, so that the functional relations hold as in the solvable $N$-state chiral Potts model.

arXiv:0710.2764Oct 15, 20070 citations

Microcanonical simulation for the three-state Potts model

Jean-Michel Drouffe, Kevin J. M. Moriarty

Computer Physics CommunicationsMay 1, 19910 citations

By the Baxter's

Q_{72}

-operator method, we demonstrate the equivalent theory between the generalized

τ^{(2)}

-model (other than two special cases with a pseudovacuum state) and the

N

-state chiral Potts model with two alternating vertical rapidities, where the degenerate models are included. As a consequence, the theory of the XXZ chain model associated to cyclic representations (with the parameter

ς

) of

U_{\sf q}(sl_2)

with

{\sf q}^N=1

for odd

N

is identified with either (for

ς^N=1

) the chiral Potts model with two superintegrable vertical rapidities, or (for

ς^N \neq 1

) the degenerate model for the selfdual solution of the star-triangle relation. In all these identifications, the transfer matrices

T, \hat{T}

of the chiral Potts model (including the degenerate ones) serve as the

Q_R, Q_L

-operators of the corresponding

τ^{(2)}

-model, so that the functional relations hold as in the solvable

N

-state chiral Potts model.