512 Nuclear Physics B (Proc. Suppl.) 9 (1989) 512-515 North-Holland, Amsterdam MULTIGRID MONTE CARLO IN AN EXTERNAL U(1) GAUGE FIELD* Arian Hulsebos, Jan Smit and Jeroen C. Vink t Institute of Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands We test a Multigrid Monte Carlo algorithm on a two dimensional scalar field in an external U(1) gauge field. Multigrid methods may be essential for enabling us to perform Monte Carlo simulations in quantum field theory on really large lattices. An important question is how to deal with gauge invariance. One approach to Multigrid Monte Carlo is by us- ing effective actions, obtained by approximate renor- malization group transformations. However, it may be rather difficult to control the approximations in- volved. We wish to consider algorithms free of system- atic errors, in which the original action of the system determines the acceptance probability of a newly pro- posed field configuration. In such an approach, there is the danger that update proposals for the long distance properties of a configuration are rejected by e.g. the Metropolis algorithm, as changes in the action tend to be large for global updates. Furthermore, the algo- rithm should be efficient for the computer with respect to vectorization or parallellism. We studied a very simple system: a scalar field in an external U(1) gauge field in two dimensions. The action is of the form *Presented by A. Hulsebos. S = ~zVz~v + h*~z + *h * ~ ~ , Ci) where summation over repeated indices is understood and where ~ is a complex and unconstrained scalar field. The matrix V is hermitian and given by tt U~ is the U(1) lattice gauge field , ~-i is an inverse hopping parameter and h~ -- 0. The reason for adding the h-terms in (1) will become clear shortly. Our Multigrid method is essentially a variant of that of refi 1. We divide the lattice into 2 x 2 blocks B~-, labelled by ~ (see fig. 1) and make a transforma- tion of variables Here f~,z is such that ~f~,~ represents the long wave- lengths at the scale of the blocks B~. The f~,~ are determined only by V~ v and they are orthogonal to r~ f;,~ ~rz = 0. (4) iPresent address: NIKHEF-H, P.O. Box 41882, 1009 DB Amsterdam, The Netherlands. 0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)