Complex Langevin: etiology and diagnostics of its main problem

Abstract

The complex Langevin method is a leading candidate for solving the socalled sign problem occurring in various physical situations. Its most vexing problem is that in some cases it produces 'convergence to the wrong limit'. In the first part of the paper we go through the formal justification of the method, identify points at which it may fail and identify a necessary and sufficient criterion for correctness. This criterion would, however, require checking infinitely many identities, and therefore is somewhat academic. We propose instead a truncation to the check of a few identities; this still gives a necessary criterion, but a priori it is not clear whether it remains sufficient. In the second part we carry out a detailed study of two toy models: first we identify the reasons why in some cases the method fails, second we test the efficiency of the truncated criterion and find that it works perfectly at least in the toy models studied.

Stability of complex Langevin dynamics in effective models

“…This means that if initial conditions are chosen close to y = 0, the complex stochastic process will take place in a strip around y = 0. The probability distribution P (x, y) will be strictly zero outside this strip and the theoretical foundation of the complex Langevin method is justified [13,14]. For larger (complex) β the stable fixed points will move out further into the complex plane and the dynamics will eventually no longer be confined to a strip.…”

“…Such a modification, if chosen appropriately, can improve the falloff of the equilibrium distribution, which was identified in refs. [13,14] as the essential prerequisite for a correct CLE process. Some of those modifications, like kernels, were known already in the 1980s [25][26][27], others such as variable transformations, were encountered just recently.…”

“…Here we have discovered a generalization: 'configurationdependent complex diffusion'. But it is not clear how much this can help in solving the problems of the CLE, in view of the fact that it was found that introducing 'imaginary diffusion' (or complex noise) actually tends to create problems by leading to slow decay in the imaginary directions and convergence to the wrong limit [13,14]. So F j should at least be chosen in such a way that it vanishes sufficiently fast for |y j | → ∞, y = 1, .…”

“…[13,14] only uses the fact that the complex FP operator annihilates the complex measure exp(−S)dx. This is manifestly the case for the most general case of a holomorphic matrix kernel, see Eq.…”

“…Recently we have clarified question (1) in some detail by putting the mathematical foundation of complex Langevin dynamics on firmer footing and deriving a set of criteria for correctness that need to be satisfied [13,14]. These consistency conditions can be calculated during the complex Langevin simulation.…”