Analytic studies of the complex Langevin equation with a Gaussian ansatz and multiple solutions in the unstable region

We investigate a simple model using the numerical simulation in the complex Langevin equation (CLE) and the analytical approximation with the Gaussian ansatz. We find that the Gaussian ansatz captures the essential and even quantitative features of the CLE results quite well when they converge to the exact answer, as well as the border of the unstable region where the CLE converges to a wrong answer. The Gaussian ansatz is therefore useful for looking into this convergence problem and we find that the exact answer in the unstable region is nicely reproduced by another solution that is naively excluded from the stability condition. We consider the Gaussian probability distributions corresponding to multiple solutions along the Lefschetz thimble to discuss the stability and the locality. Our results suggest a prescription to improve the convergence of the CLE simulation to the exact answer.

Figure 1

Absolute value of the exact answer of $⟨{\varphi }^2{⟩}_{\text{exact}}$ as a function of the real part $a$ and the imaginary part $b$ of the quadratic coefficient $\alpha$.

Figure 2

Comparison between the CLE results and the exact answer.

Figure 3

Comparison between the Gaussian ansatz results and the exact answer for the quadratic operator.

Figure 4

Counterpart of $B$ inferred from $⟨{\varphi }^2{⟩}_{\text{exact}}$ as a function of $a$ and $b$ in the region for $a<0$.

Figure 5

Comparison in the $a<0$ region between the Gaussian ansatz results with the unstable branch of solution and the exact answer.

Figure 6

Steepest descendent paths (Lefschetz thimbles) shown by purple lines and steepest ascendent paths shown by green lines around three saddle points shown by orange dots. Parameters are chosen as $a=-0.1$ and $b=10$.

Figure 7

Gaussian profiles along the thimble-inspired path near ${\overline{\varphi }}_0$ corresponding to Eqs. (14) and (15) (spreading solution by the dotted line) and Eqs. (20) and (21) (localized solution by the solid line) for $a=-0.1$ and $b=10$.

Figure 8

Comparison between the CLE results along the Lefschetz thimble and the exact answer for the quadratic operator.

Figure 9

Comparison between the Gaussian ansatz results and the exact answer for the quartic operator.

Figure 10

Comparison between the CLE results along the Lefschetz thimble and the exact answer for the quartic operator.