Multi-Grid Monte Carlo III. Two-Dimensional O(4)-Symmetric Nonlinear $\sigma$-Model

TL;DR

This study evaluates the efficiency of multi-grid Monte Carlo algorithms in simulating the two-dimensional O(4) nonlinear sigma model, showing significant reduction in critical slowing-down compared to local algorithms.

Contribution

It provides the first detailed analysis of the dynamic critical behavior of MGMC algorithms applied to the 2D O(4) sigma model, demonstrating their effectiveness in reducing critical slowing-down.

Findings

W-cycle MGMC has a dynamic critical exponent of 0.60.

V-cycle MGMC has a dynamic critical exponent of 1.13.

W-cycle MGMC is about 35 times more efficient than single-site heat-bath algorithm.

Abstract

We study the dynamic critical behavior of the multi-grid Monte Carlo (MGMC) algorithm with piecewise-constant interpolation applied to the two-dimensional O(4)-symmetric nonlinear $\sigma$-model [= SU(2) principal chiral model], on lattices up to $256 \times 256$. We find a dynamic critical exponent $z_{int,{\cal M}^2} = 0.60 \pm 0.07$ for the W-cycle and $z_{int,{\cal M}^2} = 1.13 \pm 0.11$ for the V-cycle, compared to $z_{int,{\cal M}^2} = 2.0 \pm 0.15$ for the single-site heat-bath algorithm (subjective 68% confidence intervals). Thus, for this asymptotically free model, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated. For a $256 \times 256$ lattice, W-cycle MGMC is about 35 times as efficient as a single-site heat-bath algorithm.