Complex Langevin: Correctness criteria, boundary terms, and spectrum

The Complex Langevin (CL) method to simulate "complex probabilities" ideally produces expectation values for the observables that converge to a limit equal to the expectation values obtained with the original complex "probability" measure. The situation may be spoiled in two ways: failure to converge and convergence to the wrong limit. It was found long ago that "wrong convergence" is caused by boundary terms; nonconvergence may arise from bad spectral properties of the various evolution operators related to the CL process. Here, we propose a class of criteria that allow one to rule out boundary terms and at the same time bad spectrum. Ruling out boundary terms in the equilibrium distribution arising from a CL simulation implies that the so-called convergence conditions are fulfilled. This in turn has been shown to guarantee that the expectation values of holomorphic observables are given by complex linear combinations of expo-S thorn over various integration cycles. If the spectrum is pathological, however, the CL simulation in general does not reproduce the integral over the desired real cycle.