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We revisit the field-free Ising model on a square lattice with up tothird-neighbour (nnnn) interactions, also known as the$J_{1}$--$J_{2}$--$J_{3}$ model, in the mean-field approximation. Using asystematic enumeration procedure, we show that the region of phase space inwhich the high-temperature disordered phase is stable against all modesrepresenting periodic magnetisation patterns up to a given size is a convexpolytope that can be obtained by solving a standard vertex enumeration problem.Each face of this polytope corresponds to a set of coupling constants for whicha single set of modes, equivalent up to a symmetry of the lattice, bifurcatesfrom the disordered solution. While the structure of this polytope is simple inthe halfspace $J_{3}>0$, where the nnnn-interaction is ferromagnetic, itbecomes increasingly complex in the halfspace $J_{3}<0$, where theantiferromagnetic nnnn-interaction induces strong frustration. We characterizea few salient properties of these `disorder polytopes' in terms of the geometryof the space of contributing modes. We then consider the limit$N\rightarrow\infty$ giving a closed form description of the order-disordersurface in the thermodynamic limit, which shows that for $J_3 <0$ the emergentordered phases will have a `devil's surface'-like mode structure. Finally,using Monte Carlo simulations, we show that for small periodic systems themean-field analysis correctly predicts the dominant modes of the ordered phasesthat develop for coupling constants associated with the centroid of the facesof the disorder polytope.

Geometry of the order-disorder surface of the mean-field square lattice Ising model with up to third-neighbor interactions