The Role of Integration Cycles in Complex Langevin Simulations

Complex Langevin simulations are an attempt to solve the sign (or complex-action) problem encountered in various physical systems of interest. The method is based on a complexification of the underlying degrees of freedom and an evolution in an auxiliary time dimension. The complexification, however, does not come without drawbacks, the most severe of which is the infamous 'wrong convergence' problem, stating that complex Langevin simulations sometimes fail to produce correct answers despite their apparent convergence. It has long been realized that wrong convergence may - in principle - be fixed by the introduction of a suitable kernel into the complex Langevin equation, such that the conventional correctness criteria are met. However, as we discuss in this work, complex Langevin results may - especially in the presence of a kernel - still be affected by unwanted so-called integration cycles of the theory spoiling them. Indeed, we confirm numerically that in the absence of boundary terms the complex Langevin results are given by a linear combination of such integration cycles, as put forward by Salcedo & Seiler. In particular, we shed light on the way different choices of kernel affect which integration cycles are being sampled in a simulation and how this knowledge can be used to ensure correct convergence in simple toy models.