Exploring the Ising–Heisenberg Crossover in Two-Dimensional Magnetism through Monte Carlo Simulations

Onsager’s solution of the 2D Ising model shows long-range order at finite Tc​, while the Mermin–Wagner theorem forbids spontaneous symmetry breaking at T>0 in systems with continuous symmetries and short-range interactions. This implies that ideal 2D Heisenberg systems cannot sustain long-range orders. 2D materials stress that magnetic order could appear in real systems that are neither pure Ising nor Heisenberg systems where factors such as finite flake size and substrate interactions can stabilize order. Finite anisotropy, e.g. through substrate interaction, breaks strict Heisenberg symmetry and lifts the Mermin–Wagner constraint. We use Monte Carlo simulations with a Metropolis algorithm focusing on heat capacity simulation and finite size scaling analysis to study how the critical temperature of a classical anisotropic nearest-neighbor Heisenberg system evolves from the Ising limit at large anisotropy to Tc​=0 in the isotropic limit. The determination of the functional form of the approach to Tc​=0 remained elusive yet would provide key new insights. Our results aim at understanding and designing 2D magnetic materials. They predict the sample size needed before long-wavelength spin-wave excitations dominate and quantify how substrates can create effective anisotropy.