Relevant--marginal field mixing resolves the four-state Potts endpoint in the square-lattice $J_1$--$J_2$ Ising model

Abstract

Identifying endpoint criticality is challenging because pseudo-first-order signatures can persist over accessible system sizes and obscure the underlying scaling behavior. This difficulty is especially acute in the frustrated square-lattice \(J_1\)--\(J_2\) Ising model, where the location and even the nature of the stripe-ordering endpoint have remained controversial for decades. Here we treat this endpoint as a relevant--marginal two-field scaling problem and develop a generalized field-mixing construction to resolve the endpoint ambiguity. Mixed-histogram symmetry defines a finite-size pseudotransition manifold, transforming pseudo-first-order finite-size structure into a scaling trajectory on which the normal relevant detuning is removed and the residual marginal drift is exposed. Along this manifold, Binder analysis locates the endpoint as a distinct singular point, while mixed-histogram scaling, comparison with the Ashkin--Teller Potts point, and complementary Binder scaling establish four-state Potts endpoint criticality with logarithmic corrections.

9 pages, 7 figures