Necessary and sufficient conditions for correctness of complex Langevin

TL;DR

This work derives a two-part correctness criterion for complex Langevin simulations: (i) the Schwinger–Dyson equations must hold for a designated observable space, and (ii) a uniform bound relating the expectation of observables to a weighted norm must be satisfied. When combined with a nonvanishing holomorphic density and polynomial-growth assumptions, these conditions are shown to be not only necessary but also sufficient for reproducing the target density on the real manifold via a probability measure on its complexification. The authors validate the criterion on 1D and 2D toy models, including cases with zeros in the density, and demonstrate that it can detect incorrect convergence even when boundary-term analyses fail. While not a practical proof for real-world models (requiring verification over infinitely many bounds), the framework provides a powerful diagnostic to rule out incorrect convergence and offers a path toward more realistic applications of complex Langevin methods.

Abstract

We derive a family of correctness conditions for complex Langevin simulations. In particular, we show that if in a given theory the expectation values of all observables within a particular space satisfy the theory's Schwinger-Dyson equations as well as certain bounds, then these expectation values are necessarily correct. In fact, these findings are not only valid in the context of complex Langevin simulations, but they also hold for general probability densities on complex manifolds, given an initial complex density on a real manifold. We stress that, while the proposed conditions are necessary and sufficient in a mathematical sense, their practical use is not to prove the correctness of obtained simulation results. Rather, they are mainly useful for detecting incorrect convergence. In particular, we test these criteria in a few simple one- and two-dimensional toy models and find that they are indeed capable of ruling out incorrect results without the need of exact solutions.

Figures (3)

• Figure 1: Complex Langevin results for the model \ref{['eq:quartic_1d']}, with $\lambda=e^{5\mathrm{i}\pi/6}$ and a kernel of the form \ref{['eq:kernel_quartic_1d']}, as a function of the parameter $m_H$. We only show results for values of $m_H$ for which boundary terms are consistent with zero. (left): Validity of the SDEs \ref{['eq:dse_quartic_1d']} with $k=1,\dots,5$. Note that we only plot the real parts of $\left\langle Az^k\right\rangle$ here; the imaginary parts show similar behavior. The dashed horizontal line marks zero. (right): Absolute value of the expectation value of the control observable $f(z)$ in \ref{['eq:control_variable_quartic_1d']}. The solid horizontal line shows the bound $C\Vert f\Vert_w$ from \ref{['bounds']}.
• Figure 2: Similar to \ref{['fig:quartic_1d']} but for the model \ref{['eq:quartic_2d']}. We again use $\lambda=e^{5\mathrm{i}\pi/6}$ and a kernel proportional to the identity matrix, its entries given by \ref{['eq:kernel_quartic_1d']}. (left): Validity of \ref{['eq:dse_quartic_2d']} with $k_1=0$, $k_2=1$ and vice versa. (right): Expectation value of $f(z)$ in \ref{['eq:control_variable_quartic_2d']} and comparison with $C\Vert f\Vert_w$ from \ref{['bounds']}.
• Figure 3: Validity of the SDEs \ref{['eq:dse_one_pole']} in the model \ref{['eq:one_pole']} with $\beta=1.6$, $z_p=\mathrm{i}$, $n_p=2$, $k=1,\dots,8$ and without a kernel. The inset shows a close-up view of the smallest few $k$. The data points have been offset vertically for visual clarity.

Theorems & Definitions (12)

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• Lemma 1
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