Complex Langevin: Etiology and Diagnostics of its Main Problem

TL;DR

This work analyzes why complex Langevin simulations can converge to incorrect limits in sign-problem settings and develops a practical correctness diagnostic based on a set of identities tied to the Schwinger-Dyson structure. By reformulating the problem through the interpolated observable evolution $F(t,\tau)$ and its τ-independence, the authors derive a truncated criterion $\langle \tilde{L}\mathcal{O}\rangle=0$ for a chosen basis of holomorphic observables, testing it on two toy models: a U(1) one-link model and the Guralnik–Pehlevan polynomial model. They demonstrate that complex noise can violate the formal correctness due to super-exponential growth of observables in the imaginary direction and slow falloff of equilibrium measures, but that a carefully tuned finite set of identities can reliably flag correct results and even reproduce them when the cutoff is suitably chosen. The results suggest that, while not universally guaranteeing correctness, the truncated criterion provides a highly specific and sensitive diagnostic tool that can guide CLE applications, particularly with real noise or controlled cutoffs; further work aims to extend these insights to XY and nonabelian systems. $F(t,\tau)$ and related operators $L$, $L^T$, and $\tilde{L}$ play central roles in connecting real and complex measures and diagnosing convergence properties, with practical implications for diagnostics in complex-action simulations.

Abstract

The complex Langevin method is a leading candidate for solving the so-called 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.

Figures (15)

• Figure 1: $F(t,\tau)$ vs. $\tau$ for several values of $t$, with $0<\tau<t$, $N_I=0.1$ and the cutoff $Y=3.162$.
• Figure 2: As in Fig. 1, with $Y=1.582$.
• Figure 3: As in Fig. 1, with $Y=0.474$
• Figure 4: As in Fig. 1, with $Y=0.158$.
• Figure 5: $\tau$ derivative of $F(t,\tau)$ as a function of $\tau$ at $t=10$ for several values of the cutoff $Y$ and complex noise parameter $N_I$.