Does the complex Langevin method give unbiased results?

We investigate whether the stationary solution of the Fokker-Planck equation of the complex Langevin algorithm reproduces the correct expectation values. When the complex Langevin algorithm for an action $S(x)$ is convergent, it produces an equivalent complex probability distribution $P(x)$ which ideally would coincide with $e^{-S(x)}$. We show that the projected Fokker-Planck equation fulfilled by $P(x)$ may contain an anomalous term whose form is made explicit. Such a term spoils the relation $P(x)=e^{-S(x)}$, introducing a bias in the expectation values. Through the analysis of several periodic and nonperiodic one-dimensional problems, using either exact or numerical solutions of the Fokker-Planck equation on the complex plane, it is shown that the anomaly is present quite generally. In fact, an anomaly is expected whenever the Langevin walker needs only a finite time to go to infinity and come back, and this is the case for typical actions. We conjecture that the anomaly is the rule rather than the exception in the one-dimensional case; however, this could change as the number of variables involved increases.

Figure 1

Velocity fields. Left: $S=i\cos (z)$ with $z\in [-\pi ,\pi ]\times [-\pi ,\pi ]$. Right: $S=\cos (z)+iz$ with $z\in [0,2\pi ]\times [-\pi ,\pi ]$. The two horizontal lines at $y=\pm \mathrm{arcsinh}(1)$ indicate the strip where $v_y(z)\le 0$ for all $x$.

Figure 2

Velocity field of $S=\cos (z)+iz$ in variables ($X$, $Y$) in the range $[-2,2]\times [-2,2]$. The fixed point is at $(-1,0)$. The circle indicates the boundary of the support of the equilibrium solution, i.e., the locus of $y=-y_0$. The diffusion moves the walker along circles centered at the origin (not displayed in the figure).

Figure 3

Function $\sigma (X,Y)$ for $\beta =0.5$ and $m=1$. Left: $\sigma$ on the ($X$, $Y$) plane. The support is located on $R\le R_m=\beta /m$. The numerical calculation uses 64 Fourier modes in $\phi$ and 64 points in the mesh of the variable $R$. Numerical derivatives in $R$ were obtained using 5 points. Right: Sections of $\sigma (X,Y)$ along the $X$ (solid blue line) and $Y$ (dashed black line) axes. $\sigma (0)=0.310$.

Figure 4

Stationary solution of the Fokker-Planck equation of $S=\beta \cos (x)$ for $\beta =0.25+i0.5$. Left: $\rho (x,y)$ for $0\le x\le 2\pi$. This function is periodic in $x$ and $\rho (x,y)=\rho (-x,-y)$. Right: $\sigma (X,Y)$. In the numerical solution the matching was set at $R_c=2$. For $x\equiv \phi$, 64 Fourier modes were used and 64 points in the $R$ and $y$ grids. Numerical derivatives in $R$ and $y$ were obtained using 5 points.

Figure 5

The functions $\rho (x,y)$ (left) and $\sigma (X,Y)$ (right) for $S=\beta x^4/4$ with $\beta =0.1+i0.25$ and $\lambda =0$. The matching is taken at $r_c=1.414$. A 64 point mesh is used for the interval $0<r<r_c$ and 128 for $0<R<R_c$. Thirty-two Fourier modes are used for $\arg Z$ (corresponding to 64 modes for $\arg z$). Numerically we find $\sigma (0)=0.011$ for this action.

Figure 6

Same as Fig. 5 for $\lambda =0.5$. Numerically $\sigma (0)=0.292$.

Figure 7

For $\beta =0.1+i0.25$ and $\lambda =0.5$, values of the density $\sigma (Z)$ plotted against the scaling variable $u=X/{\beta }_R-Y/{\beta }_I$. The 444 values shown correspond to a subset of the points used in the numerical solution of the differential equation (except that the 32 Fourier modes have been traded for 32 angular directions) defined by the condition $|y|\ge 3$. For these parameters $\sigma (0)=0.292$.

Figure 8

For the solution of Fig. 6: (a) Ratio $\rho /{\rho }_s=\sigma /\sigma (0)$ on the $z$ plane. $\rho$ is close to ${\rho }_s$ in the asymptotic region. (b) Marginal probabilities $N(y)$ of $\rho$ (solid line, blue) and ${\rho }_s$ (dashed line, black). They only differ in the region $|y|<2$.

Figure 9

Velocity field of $S=\beta x^4/4$ in the variable $Z=1/(2\beta z^2)$, for $\beta =0.1+i0.25$. The solid curves correspond to lines of $x=\text{constant}$, for $x=2$, 3, and 5. The diffusion randomly moves the Langevin walkers along these lines allowing them to visit a neighborhood of $Z=0$. This suggests that $\sigma (0)$, or at least its limit from negative $X$ if this function is not continuous at $Z=0$, will not be exactly zero. [The peak of $\sigma (X,Y)$ displayed in Fig. 5 falls at $Z=0.32-i0.18$ here.]