Correctness criteria for complex Langevin

Abstract

The complex Langevin approach is a promising method for the numerical treatment of systems with a sign problem, for which conventional lattice field theory techniques based on importance sampling cannot be applied. However, complex Langevin dynamics may fail to converge in some cases and converge to a wrong limit in others, motivating the development of various diagnostic tools over the years to assess the correctness of given simulation results. This work aims at providing a systematic comparison between the most prominent such correctness criteria. In particular, the main goal is to contrast their applicability, ease of use, and - most importantly - their predictive power. To this end, four simple but nontrivial models are considered and the criteria applied to each of them. It is found that no single correctness criterion is entirely foolproof on its own and that it is advisable in practice to employ multiple different criteria in combination. These conclusions are expected to carry over to more realistic theories as well.

60 pages + 3 pages appendix, 32 figures, 11 tables; appendix C and some clarifications added; version that appeared in JHEP

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