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

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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 σ-model [= SU(2) principal chiral model], on lattices up to 256 × 256. We find a dynamic critical exponent zint, M² = 0.60 ± 0.07 for the W-cycle and zint, M² = 1.13 ± 0.11 for the V-cycle, compared to zint, M² = 2.0 ± 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 × 256 lattice, W-cycle MGMC is about 35 times as efficient as a single-site heat-bath algorithm.