TY - JOUR

T1 - Frustration-induced complexity in order-disorder transitions of the J1−J2−J3 Ising model on the square lattice

AU - Subert, Rodolfo

AU - Mulder, Bela

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

U2 - 10.1103/PhysRevE.106.014105

DO - 10.1103/PhysRevE.106.014105

M3 - Article

SN - 1539-3755

VL - 106

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

M1 - 014105

ER -