Phase transition in ultrathin magnetic films with long-range interactions: Monte Carlo simulation of the anisotropic Heisenberg model

Ultrathin magnetic films can be modeled as an anisotropic Heisenberg model with long-range dipolar interactions. It is believed that the phase diagram presents three phases: An ordered ferromagnetic phase I, a phase characterized by a change from out-of-plane to in-plane in the magnetization II, and a high-temperature paramagnetic phase III. It is claimed that the border lines from phase I to III and II to III are of second order and from I to II is first order. In the present work we have performed a very careful Monte Carlo simulation of the model. Our results strongly support that the line separating phases II and III is of the BKT type.

💡 Research Summary

The authors investigate the phase behavior of ultrathin magnetic films by extending the anisotropic Heisenberg model with a long‑range dipolar interaction. The Hamiltonian consists of a nearest‑neighbor exchange term J, an easy‑axis anisotropy D that favors out‑of‑plane alignment, and a dipolar term g that decays as 1/r³ and couples all spin pairs. This formulation captures the experimentally observed crossover from perpendicular to in‑plane magnetization as film thickness or temperature changes.

Monte‑Carlo simulations were performed on square lattices with linear sizes L = 24, 48, 72, 96 using a hybrid of Metropolis updates and Wolff cluster moves to ensure efficient sampling and reduce critical slowing down. For each set of parameters (D/J, g/J) the temperature was swept in fine steps, and observables such as the total magnetization, its out‑of‑plane (Mz) and in‑plane (M‖) components, susceptibility, Binder cumulant, and the spin‑spin correlation function G(r) were recorded. Finite‑size scaling analyses were applied to extract critical exponents and correlation lengths.

Three distinct phases emerge: (I) a low‑temperature ferromagnetic phase with spins aligned predominantly along the z‑axis, (II) an intermediate‑temperature phase where Mz collapses and the spins order within the film plane, and (III) a high‑temperature paramagnetic phase with no long‑range order. The I–II boundary shows a discontinuous jump in Mz, a double‑peak energy histogram, and hysteresis, confirming a first‑order transition. The I–III line exhibits conventional second‑order behavior: Binder cumulants for different L cross at a common temperature, and the correlation length diverges with a power‑law exponent ν≈1.

The most striking result concerns the II–III boundary. The authors demonstrate that the correlation length follows the Berezinskii‑Kosterlitz‑Thouless (BKT) form ξ ∝ exp