The Complex Langevin method: When can it be trusted?

TL;DR

The paper analyzes when the complex Langevin method reliably solves sign problems by deriving formal connections between complex and real-measure expectations for holomorphic observables, while highlighting boundary-term loopholes that can invalidate those connections. Through detailed case studies of the U(1) one-link model and the Guralnik–Pehlevan model, it demonstrates that convergence to correct limits is delicate: with $N_I=0$ the method can reproduce correct results, but for $N_I>0$ boundary effects can drive convergence to wrong limits. The work emphasizes the crucial role of imaginary-direction decay in the equilibrium distribution and advocates practical safeguards—primarily using $N_I$ as small as feasible and validating against trusted methods—to ensure credible results in applications to more complex theories. Overall, it provides a pragmatic map of when CLE is trustworthy and how to diagnose and mitigate failures arising from boundary terms and poor decay in the imaginary directions.

Abstract

We analyze to what extent the complex Langevin method, which is in principle capable of solving the so-called sign problems, can be considered as reliable. We give a formal derivation of the correctness and then point out various mathematical loopholes. The detailed study of some simple examples leads to practical suggestions about the application of the method.

Figures (7)

• Figure 1: Scatter plot for the $U(1)$ one-link model at $\beta=1,\\\kappa=0,\ N_I=1$ with reduced noise (see text).
• Figure 2: Distributions $P(x,y;t\to\infty)$ in the U(1) one-link model, obtained from a numerical solution of the real FPE, for various values of $N_I$: $N_I=0.0001$ (top), $0.01$ (middle), $0.1$ (bottom). See the main text for further details.
• Figure 3: $N_I$ dependence of $\hbox{Re}\,\langle e^{iz}\rangle$ (lower points) and $\hbox{Re}\,\langle e^{-iz}\rangle$ (higher points) from FPE for various values of the cutoff $Y$. The bottom figure zooms in on smaller values of $N_I$. The lines are guides to the eye, the horizontal dotted lines indicate the correct results (0.483564 and 0.592966 respectively).
• Figure 4: Cutoff ($Y$) dependence of $\hbox{Re}\,\langle e^{iz}\rangle$ (lower data) and $\hbox{Re}\,\langle e^{-iz}\rangle$ (upper data) for various values of $N_I$, for FPE (open symbols) and CLE (full symbols, note that the errorbars are much smaller that the points). The lines are guides to the eye. The horizontal dotted lines indicate the correct results.
• Figure 5: Scatter plot in the GP model with $\beta=1$, $N_I=1$.