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

TL;DR

The paper provides robust numerical evidence that the two‑dimensional classical Heisenberg (O(3)) model exhibits asymptotic freedom, demonstrated by an exponential divergence of the correlation length $\xi$ with inverse temperature $\beta$ and convergence of thermodynamic extrapolations to three‑loop perturbative predictions. A finite‑size scaling ansatz $\xi(\beta,L)=L\tilde{\xi}(\beta-\frac{1}{2\pi}\ln L)$ yields a data collapse across system sizes and a pseudo‑critical point $\beta_L$ that diverges logarithmically with $L$, arguing against a finite‑temperature phase transition or a BKT scenario. Analysis of magnetic susceptibility, $\chi$, and specific heat further supports perturbative scaling and rules out BKT scaling, with $\chi$ obeying $\chi\sim \xi^{2}[\ln(\xi/c_{ξ})]^{-2}$ in the thermodynamic limit and $\chi_L\sim aL^{2}[\ln(L/L_0)]^{-2}$ in the FSS regime. Overall, the work resolves long‑standing debates about the model by combining large‑scale simulations, extrapolation methods, and finite‑size scaling to align lattice results with continuum asymptotic‑freedom predictions, with implications for understanding nonperturbative phenomena in analogous theories.

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.

Figures (8)

• Figure 1: Exponential growth of correlation length at low temperatures. The correlation length $\xi$ is plotted as a function of $\beta$ in a semi-log coordinate for various system sizes. The growth of $\xi$ can be well described by the blue line representing $A e^{2\pi\beta}$ with $A \approx 0.0008$. The black line, representing $e^{-1.7}\cdot\exp(1.628/\sqrt{T-0.509})$, fitted by Ref. 10.21468/SciPostPhys.11.5.098, refers to the diverging curve of $\xi$ near the possible BKT phase transition point $T_c=0.509$. This line aligns well with our data points at higher temperatures but deviates significantly as the temperature decreases.
• Figure 2: The ratio between the numerical value $\xi$ and the three-loop theoretical result in Eq. \ref{['3loop']}. The ratio shows a strong deviation from $1$ initially. However, as $\beta$ increases, it gradually approaches $1$, suggesting that the two may align with each other at lower temperatures.
• Figure 3: Verification of the FSS ansatz \ref{['extrapolation_eq']}. The vertical axis is the correlation-length ratio, $\xi(\beta,2L)/\xi(\beta, L)$, of two systems with size $2L$ and $L$, and the horizontal axis is the dimensionless ratio $\xi(\beta, L)/L$. Data points of various system sizes collapse well onto a single red curve $F_\xi(x)$, which is obtained through the fitting of Eq. \ref{['extra_fit']}. The blue line refers to the perturbation result in Eq. \ref{['perturbative result']} for large $x$.
• Figure 4: (a) The extrapolated thermodynamic value of the correlation length $\xi_\infty$ as a function of $\beta$. The blue squares represent the values obtained directly from the simulation at sufficiently large systems, which can reflect the thermodynamic limit. The green dots represent further extrapolations of $\xi$. The red line refers to the three-loop perturbative result in Eq. \ref{['3loop']}. (b) The ratio of data with the three-loop perturbative result versus $\beta$, demonstrating an excellent agreement for large $\beta$.
• Figure 5: (a) Finite-size scaling of the ratio $\xi/L$ as a function of $x=\beta-\ln(L)/2\pi$ with no free parameters involved. The main plot shows that the data points collapse on a single curve $\Tilde{\xi}(x)$ for various system sizes ranging from $L=64$ to $16384$. The black and red lines refer to the asymptotic behaviors in Eq. \ref{['xi_asym']} and \ref{['xi_asym2']} respectively, with $A=0.0008$, $c=-0.21$, and they both align well with the data points. (b) The pseudo-critical point $\beta_L$ as a function of $aL$, with $\xi/L$ fixed at different constants. The ratio $\xi/L$ is fixed at $1,0.75,0.5,0.25$ with $a=1,0.0074,0.00037,0.0001$ respectively. The data points for various system sizes ranging from $L=16$ to $8192$ are included. The blue and black lines refer to the zero- and first-order results in Eq. \ref{['xiL_correction']} respectively, with $c=2.8\times10^5$. It can be seen that the $L$-dependent behavior of $\beta_L$ has approximately three ranges: at high temperature with $\beta_L \lesssim 1.5$ it significantly deviates from the field theoretical prediction, in intermediate temperature in range $(1.5,2.4)$ it approximately follows a logarithmic growth, and at low temperature with $\beta_L >2.4$ the subleading log-log correction appears. This interesting feature is similar to that in Fig. \ref{['fig1']}.