Complex Langevin method: When can it be trusted?

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.

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 $\mathrm{Re}⟨e^{iz}⟩$ (lower points) and $\mathrm{Re}⟨e^{-iz}⟩$ (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 (0.483 564 and 0.592 966, respectively).

Figure 4

Cutoff ($Y$) dependence of $\mathrm{Re}⟨e^{iz}⟩$ (lower data) and $\mathrm{Re}⟨e^{-iz}⟩$ (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 $\beta =1$, $N_I=1$.

Figure 6

$N_I$ dependence of $\mathrm{Im}⟨z⟩$ in the GP model at $\beta =1$.

Figure 7

Scatter plot in the GP model for $\beta =1$, $N_I=0$.

Figure 8

Distribution of $\mathrm{Re}z$ (top) and $\mathrm{Im}z$ (bottom) in the GP model with $\beta =1$, $N_I=0$.