Complex Langevin dynamics as a cure for the sign problem. - Frank James (2012)
Theories with a chemical potential are difficult to treat numerically because the action is complex and therefore methods based on a probability interpretation of the weight break down. This is an issue known as the sign problem. Complex Langevin dynamics was first proposed in the early 1980s and does not rely in a probability interpretation of the weight, so it can in principle l)e applied even where there is a severe sign problem. However, the combined problems of numerical instabilities and incorrect convergence impeded such early studies. In this work, the problem of runaway trajectories is cured l)y the use of a general adaptive stepsize procedure, which can be applied to both abelian and non-abelian theories. A study of the three-dimensional XY model at non-zero chemical potential follows, in which the problem of incorrect convergence is encountered. A formal justification of complex Langevin dynamics is given, from which a set of criteria are derived which can be used to test the validity of results. These ideas are applied to the SU(3) spin model, which is found to pass them all and therefore give correct results. An improved integration algorithm, which eliminates leading order step size corrections, is outlined and shown to give improved results.