Order-Disorder Transition of a Range-3 Ising Model on the Square Lattice

Abstract

The Ising model is arguably the most studied model in the history of physics, yet it is not fully understood in all its facets. In the present talk we will focus on the frustration effects induced by long range, competing anti-ferromagnetic interactions. To study this behaviour, we focus on a minimal model retaining the phenomenology: the range-3 Ising model on the square lattice. Previous investigation highlighted the role of the third range coupling in tuning the infinitely many modulated phases. Our approach relies on a fresh take on the standard Mean Field Theory framework: we introduce a systematic prescription to enumerate phases of increasing complexity as solutions of the mean field self-consistency equations. The essential idea is that constraining the lattice with Periodic Boundary Conditions enforces the emergence of those phases whose period is commensurate with the lattice size. We will show how these premises provide the region of stability of the paramagnetic phase as a convex polytope, where each face is the bifurcation surface to a unique modulated phase. Finally, we use Monte Carlo simulations to probe the theoretical predictions on the shape of the phase diagram.