Numerical simulation study of the dynamical behavior of the Niedermayer algorithm

TL;DR

This study analyzes the dynamic critical behavior of the Niedermayer algorithm for 2D Ising and XY models, comparing it to Metropolis and Wolff algorithms, and finds Wolff to be optimal.

Contribution

It provides a detailed calculation of the dynamic critical exponent for the Niedermayer algorithm across different parameters, clarifying its efficiency relative to established algorithms.

Findings

Niedermayer algorithm behaves like Metropolis for large lattices.

Wolff algorithm outperforms Niedermayer for cluster updates.

Autocorrelation time grows faster than power law for certain parameters.

Abstract

We calculate the dynamic critical exponent for the Niedermayer algorithm applied to the two-dimensional Ising and XY models, for various values of the free parameter $E_0$. For $E_0=-1$ we regain the Metropolis algorithm and for $E_0=1$ we regain the Wolff algorithm. For $-1<E_0<1$, we show that the mean size of the clusters of (possibly) turned spins initially grows with the linear size of the lattice, $L$, but eventually saturates at a given lattice size $\widetilde{L}$, which depends on $E_0$. For $L>\widetilde{L}$, the Niedermayer algorithm is equivalent to the Metropolis one, i.e, they have the same dynamic exponent. For $E_0>1$, the autocorrelation time is always greater than for $E_0=1$ (Wolff) and, more important, it also grows faster than a power of $L$. Therefore, we show that the best choice of cluster algorithm is the Wolff one, when compared to the Nierdermayer generalization. We also obtain the dynamic behavior of the Wolff algorithm: although not conclusive, we propose a scaling law for the dependence of the autocorrelation time on $L$.