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

TL;DR

This paper investigates the effectiveness of the multi-grid Monte Carlo algorithm in reducing critical slowing-down in a one-dimensional $O(4)$-symmetric nonlinear sigma model, finding logarithmic growth in autocorrelation times.

Contribution

It provides empirical analysis of MGMC's dynamic behavior on large lattices for the $O(4)$ model, highlighting its partial success in mitigating critical slowing-down.

Findings

Autocorrelation time grows logarithmically with lattice size.

MGMC partially reduces critical slowing-down.

The reason for incomplete elimination of slowing-down remains unclear.

Abstract

We study the dynamic critical behavior of the multi-grid Monte Carlo (MGMC) algorithm with piecewise-constant interpolation and a W-cycle, applied to the one-dimensional $O(4)$-symmetric nonlinear $\sigma$-model [= $SU(2)$ principal chiral model], on lattices from $L=128$ to $L=16384$. Our data for the integrated autocorrelation time $\tau_{int,{\cal M}^2}$ are well fit by a logarithmic growth. We have no idea why the critical slowing-down is not completely eliminated.