Complex Langevin: Boundary terms at poles of the drift

TL;DR

This paper investigates boundary terms at poles of the drift in the complex Langevin method, showing they appear at finite Langevin time but vanish at long times, impacting the method's formal justification.

Contribution

It provides a detailed analysis of boundary terms at drift poles in simple models, clarifying their transient nature and implications for complex Langevin simulations.

Findings

Boundary terms at poles arise after finite Langevin time.

These boundary terms vanish as time approaches infinity.

Contrast with boundary terms at infinity, which persist in the long run.

Abstract

The complex Langevin method is a general method to treat systems with complex action, such as QCD at nonzero density. The formal justification relies on the absence of certain boundary terms, both at infinity and at the unavoidable poles of the drift force. Here I focus on the boundary terms at these poles for simple models, which so far have not been discussed in detail. The main result is that those boundary terms (for the "un-evolved" observables) arise after running the Langevin process for a finite time and vanish again as the Langevin time goes to infinity. This is in contrast to the boundary terms at infinity, which can be found to occur in the long time limit (cf. the contribution by D\'enes Sexty).