[Paper Review] Dynamic critical behavior of cluster algorithms for 2D Ashkin-Teller and Potts models

Jesús Salas|arXiv (Cornell University)|Apr 19, 2000

Theoretical and Computational Physics27 references3 citations

TL;DR

This paper investigates the dynamic critical behavior of cluster algorithms—specifically the Swendsen–Wang and a related algorithm for the 2D Ashkin–Teller model—finding that the Li–Sokal lower bound on autocorrelation times is not sharp, but nearly so, with the ratio of integrated autocorrelation time to specific heat growing logarithmically or as a small power (0.05–0.12) of system size. The exponential and integrated autocorrelation times for energy are shown to scale with the same dynamic critical exponent, indicating a near-optimal dynamic scaling behavior.

ABSTRACT

We study the dynamic critical behavior of two algorithms: the Swendsen-Wang algorithm for the two-dimensional Potts model with q=2,3,4 and a Swendsen-Wang-type algorithm for the two-dimensional symmetric Ashkin-Teller model on the self-dual curve. We find that the Li--Sokal bound on the autocorrelation time τ_{{ m int},{\cal E}} \geq const imes C_H is almost, but not quite sharp. The ratio τ_{{ m int},{\cal E}}/C_H appears to tend to infinity either as a logarithm or as a small power (0.05 \ltapprox p \ltapprox 0.12). We also show that the exponential autocorrelation time τ_{{ m exp},{\cal E}} is proportional to the integrated autocorrelation time τ_{{ m int},{\cal E}}.

Motivation & Objective

• To understand why the Swendsen–Wang algorithm performs well in some models but not others, particularly near critical points.
• To test the sharpness of the Li–Sokal lower bound on autocorrelation times, which relates dynamic critical exponents to static critical exponents.
• To determine whether the exponential and integrated autocorrelation times for energy scale with the same dynamic critical exponent in 2D cluster algorithms.
• To investigate the nature of the slow modes in the dynamics of non-local Monte Carlo algorithms.

Proposed method

• Numerical Monte Carlo simulations of the 2D q-state Potts model (q=2,3,4) and the symmetric Ashkin–Teller model on the self-dual curve using the Swendsen–Wang and a related cluster algorithm.
• Estimation of integrated and exponential autocorrelation times for the energy observable using high-precision data from long simulations.
• Application of finite-size scaling analysis to extract dynamic critical exponents from the scaling of autocorrelation times with system size L.
• Use of the scaling ansatz ρ_AA(t;L) ≈ |t|^{-p_A} h_A[t/τ_exp,A; ξ(L)/L] to test whether the autocorrelation function of the energy behaves as a pure exponential.
• Comparison of the ratio τ_int,ℰ / C_H with the Li–Sokal bound to assess its sharpness, using logarithmic and power-law fitting forms.
• Validation of the ansatz τ_int,ℰ ≈ τ_exp,ℰ by checking data collapse of ρ_ℰℰ(t) vs. t/τ_int,ℰ across different lattice sizes.

Experimental results

Research questions

• RQ1 Is the Li–Sokal lower bound on autocorrelation times sharp for cluster algorithms in two-dimensional systems?
• RQ2 Do the exponential and integrated autocorrelation times for the energy scale with the same dynamic critical exponent in the Swendsen–Wang algorithm?
• RQ3 How does the ratio τ_int,ℰ / C_H scale with system size L, and does it grow as a logarithm or a small power?
• RQ4 What is the nature of the slowest mode in the dynamics of the 2D Potts and Ashkin–Teller models?
• RQ5 Why is the Li–Sokal bound so close to being sharp in 2D but clearly not sharp in higher dimensions?

Key findings

• The Li–Sokal bound τ_int,ℰ ≥ const × C_H is not sharp, but the ratio τ_int,ℰ / C_H increases with system size L as either a logarithm or a small power (0.05 ≤ p ≤ 0.12).
• The dynamic critical exponent z_int,ℰ for the energy in the 2D Potts model (q=2,3,4) and the Ashkin–Teller model is consistent with z_int,ℰ ≈ z_exp,ℰ, indicating that the exponential and integrated autocorrelation times scale with the same exponent.
• For the 2D Ising model (q=2), the ratio τ_int,ℰ / C_H grows logarithmically, suggesting near-optimality of the algorithm’s dynamic scaling.
• The power p in the scaling τ_int,ℰ / C_H ∼ L^p increases from approximately 0.05 in the Ising model to 0.12 in the 4-state Potts model, indicating a continuous trend across models.
• The data collapse of ρ_ℰℰ(t) vs. t/τ_int,ℰ is excellent for L ≥ 64, supporting the ansatz that the energy autocorrelation function behaves as a pure exponential.
• The study suggests that in d=2, the energy and bond occupation are among the slowest modes, while in d>2, they are not, explaining the difference in the sharpness of the Li–Sokal bound.