The one-dimensional (1D) ${J}_{1}\text{\ensuremath{-}}{J}_{2}$
q-state Potts model is solved exactly for arbitrary
q by analytically block-diagonalizing the original ${q}^{2}\ifmmode\times\else\texttimes\fi{}{q}^{2}$ transfer matrix into a simple $2\ifmmode\times\else\texttimes\fi{}2$ maximally symmetric subspace, based on using OpenAI's reasoning model o3-mini-high to exactly solve the
q=3 case. Furthermore, by matching relevant subspaces, we map the Potts model onto a simpler effective 1D
q-state Potts model, where
J2 acts as the nearest-neighbor interaction and
J1 as an effective magnetic field, nontrivially generalizing a 56-year-old theorem previously limited to the simplest case (
q=2, the Ising model). Our exact results provide insights to phenomena such as atomic or electronic order stacking in layered materials and the emergence of dome-shaped phases in complex phase diagrams. This work is anticipated to fuel both research in 1D frustrated magnets for recently discovered finite-temperature application potentials and the fast moving topic area of AI in science.
The q-state Potts model with long-range interactions that decay as ${1/r}^{\ensuremath{\alpha}}$ subjected to a uniform magnetic field on d-dimensional lattices is analyzed for different values of q in the nonextensive regime $0<~\ensuremath{\alpha}<~d.$ We calculate the mean-field solution of the model for all q and performed, for some values of q, Monte Carlo simulations for the spontaneous magnetization in the one-dimensional case. We show that, using a derived scaling which properly describes the nonextensive thermodynamic behavior, both types of calculations present an excellent agreement for $0<~\ensuremath{\alpha}<d.$ We also consider the two-dimensional antiferromagnetic Ising model with competing antiferromagnetic long-range interactions and ferromagnetic first neighbor ones in the presence of a uniform magnetic field. We calculate the mean-field magnetization for this case and compare it with Monte Carlo numerical data from Sampaio et al. They also show a very good agreement for $\ensuremath{\alpha}<d.$ These results, together with some previous ones, led us to conjecture that the mean-field theory is exact for nonextensive classical spin models with $0<~\ensuremath{\alpha}<d.$