Abstract
One reason for the well known fact that the Complex Langevin (CL) method sometimes fails to converge or converges to the wrong limit has been identified long ago: it is insufficient decay of the probability density either near infinity or near poles of the drift, leading to boundary terms that spoil the formal argument for correctness. To gain a deeper understanding of this phenomenon, in a previous paper we have studied the emergence of such boundary terms thoroughly in a simple model, where analytic results can be compared with numerics. Here we continue this type of analysis for more physically interesting models, focusing on the boundaries at infinity. We start with abelian and non-abelian one-plaquette models, then we proceed to a Polyakov chain model and finally to high density QCD (HDQCD) and the 3D XY model. We show that the direct estimation of the systematic error of the CL method using boundary terms is in principle possible.
Highlights
Complex Langevin simulations are a very general method which can in principle be applied to any model with complex action, allowing an analytic continuation into the complexification of the original configuration space
We show that the direct estimation of the systematic error of the complex Langevin (CL) method using boundary terms is in principle possible
This paper extends to realistic lattice models the study of boundary terms [1] which occur in some complex Langevin (CL) simulations and have the undesired effect of spoiling correctness
Summary
Complex Langevin simulations are a very general method which can in principle be applied to any model with complex action, allowing an analytic continuation into the complexification of the original configuration space. The setup is straightforward and needs no preliminary steps, such as model-dependent design or approximations These features motivate the work to ensure the reliability of complex Langevin simulations, since the resulting stochastic processes in the complexified configuration space require care due to their mathematical subtleties. This paper extends to realistic lattice models the study of boundary terms [1] which occur in some complex Langevin (CL) simulations and have the undesired effect of spoiling correctness. Z hOiρðtÞ 1⁄4 ρðx; tÞOðxÞdNx ð4Þ computed using an evolution of the complex density ρðtÞ on the original real configuration space M determined by the partial differential equation (“complex FPE”). FOðt; 0Þ 1⁄4 hOiPðtÞ; FOðt; tÞ 1⁄4 hOiρðtÞ; ð8Þ such that correctness of the evolution is guaranteed if
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