Multigrid Monte Carlo method. Conceptual foundations

We present details of a stochastic generalization of the multigrid method, called multigrid Monte Carlo (MGMC), that reduces critical slowing down in Monte Carlo computations of lattice field theories. For Gaussian (free) fields, critical slowing down is completely eliminated. For a ${\phi }^4$ model, numerical experiments show a factor of ≈ 10 reduction, over a standard heat-bath algorithm, in the CPU time needed to achieve a given accuracy. For the two-dimensional $\mathrm{XY}$ model, experiments show a factor of ≈ 10 reduction on the high-temperature side of criticality, growing to an unbounded reduction in the low-temperature regime. The algorithm is also applicable to nonlinear $\sigma$ models, and to lattice gauge theories with or without bosonic matter fields.