Asymptotic Freedom and Finite-size Scaling of Two-dimensional Classical Heisenberg Model

Abstract

The classical Heisenberg model is one of the most fundamental models in statistical and condensed matter physics. Extensive theoretical and numerical studies suggest that, in two dimensions, this model does not exhibit a finite-temperature phase transition but instead manifests asymptotic freedom. However, some research has also proposed the possibility of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition over the years. In this study, we revisit the classical two-dimensional (2D) Heisenberg model through large-scale simulations with linear system sizes up to $L=16384$. Our Monte-Carlo data, without any extrapolation, clearly reveal an exponential divergence of the correlation length $ξ$ as a function of inverse temperature $β$, a hallmark of asymptotic freedom. Moreover, extrapolating $ξ$ to the thermodynamic limit in the low-temperature regime achieves close agreement with the three-loop perturbative calculations. We further propose a finite-size scaling (FSS) ansatz for $ξ$, demonstrating that the pseudo-critical point $β_L$ diverges logarithmically with $L$. The thermodynamic and finite-size scaling behaviors of the magnetic susceptibility $χ$ are also investigated and corroborate the prediction of asymptotic freedom. Our work provides solid evidence for asymptotic freedom in the 2D Heisenberg model and advances understanding of finite-size scaling in such systems.