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

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 ${\tau }_{int,{\mathcal{M}}^2}$ are well fit by a logarithmic growth. We have no idea why the critical slowing-down is not completely eliminated.