We revisit the field-free Ising model on a square lattice with up to third-neighbor (NNNN) interactions, also known as the J1-J2-J3 model, in the mean-field approximation. Using a systematic enumeration procedure, we show that the region of phase space in which the high-temperature disordered phase is stable against all modes representing periodic magnetization patterns up to a given size is a convex polytope that can be obtained by solving a standard vertex enumeration problem. Each face of this polytope corresponds to a set of coupling constants for which a single set of modes, equivalent up to a symmetry of the lattice, bifurcates from the disordered solution. While the structure of this polytope is simple in the half-space J3>0, where the NNNN interaction is ferromagnetic, it becomes increasingly complex in the half-space J3<0, where the antiferromagnetic NNNN interaction induces strong frustration. We then pass to the limit N→∞ giving a closed-form description of the order-disorder surface in the thermodynamic limit, which shows that for J3<0, the emergent ordered phases will have a "devil's surface"-like mode structure. Finally, using Monte Carlo simulations, we show that for small periodic systems, the mean-field analysis correctly predicts the dominant modes of the ordered phases that develop for coupling constants associated with the centroid of the faces of the disorder polytope.

The Netherlands Organisation for Scientific Research (NWO)
Phys. Rev. E
Theory of Biomolecular Matter

Subert, R., & Mulder, B. (2022). Frustration-induced complexity in order-disorder transitions of the J1-J2-J3 Ising model on the square lattice. Phys. Rev. E, 106(1), 014105: 1–17. doi:10.1103/PhysRevE.106.014105