Suwa Hidemaro
Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA and Department of Physics, University of Tokyo, Tokyo 113-0033, Japan.
Phys Rev E. 2021 Jan;103(1-1):013308. doi: 10.1103/PhysRevE.103.013308.
The worm algorithm is a versatile technique in the Markov chain Monte Carlo method for both classical and quantum systems. The algorithm substantially alleviates critical slowing down and reduces the dynamic critical exponents of various classical systems. It is crucial to improve the algorithm and push the boundary of the Monte Carlo method for physical systems. We here propose a directed worm algorithm that significantly improves computational efficiency. We use the geometric allocation approach to optimize the worm scattering process: worm backscattering is averted, and forward scattering is favored. Our approach successfully enhances the diffusivity of the worm head (kink), which is evident in the probability distribution of the relative position of the two kinks. Performance improvement is demonstrated for the Ising model at the critical temperature by measurement of exponential autocorrelation times and asymptotic variances. The present worm update is approximately 25 times as efficient as the conventional worm update for the simple cubic lattice model. Surprisingly, our algorithm is even more efficient than the Wolff cluster algorithm, which is one of the best update algorithms. We estimate the dynamic critical exponent of the simple cubic lattice Ising model to be z≈0.27 in the worm update. The worm and the Wolff algorithms produce different exponents of the integrated autocorrelation time of the magnetic susceptibility estimator but the same exponent of the asymptotic variance. We also discuss how to quantify the computational efficiency of the Markov chain Monte Carlo method. Our approach can be applied to a wide range of physical systems, such as the |ϕ|^{4} model, the Potts model, the O(n) loop model, and lattice QCD.
蠕虫算法是马尔可夫链蒙特卡罗方法中一种适用于经典和量子系统的通用技术。该算法极大地缓解了临界慢化现象，并降低了各种经典系统的动态临界指数。改进该算法并拓展蒙特卡罗方法在物理系统中的应用边界至关重要。我们在此提出一种定向蠕虫算法，它能显著提高计算效率。我们采用几何分配方法来优化蠕虫散射过程：避免蠕虫反向散射，促进正向散射。我们的方法成功提高了蠕虫头部（扭结）的扩散率，这在两个扭结相对位置的概率分布中很明显。通过测量指数自相关时间和渐近方差，证明了在临界温度下伊辛模型的性能提升。对于简单立方晶格模型，当前的蠕虫更新效率约为传统蠕虫更新的25倍。令人惊讶的是，我们的算法甚至比沃尔夫团簇算法更高效，而沃尔夫团簇算法是最佳更新算法之一。我们估计在蠕虫更新中简单立方晶格伊辛模型的动态临界指数为z≈0.27。蠕虫算法和沃尔夫算法产生的磁化率估计器积分自相关时间的指数不同，但渐近方差的指数相同。我们还讨论了如何量化马尔可夫链蒙特卡罗方法的计算效率。我们的方法可应用于广泛的物理系统，如|ϕ|⁴模型、Potts模型、O(n)环模型和晶格量子色动力学。
