Thermodynamic Consistency as a Reliability Test for Complex Langevin Simulations

TL;DR

This work addresses the reliability of the complex Langevin method in systems with complex actions by introducing a configurational-temperature estimator built from the gradient and Hessian of the action. The estimator yields a per-configuration value $\hat{\beta}$ and a mean $\beta_{\mathrm{M}}$, which should coincide with the input inverse temperature $\beta$ for correctly sampled configurations, and it scales predictably if the sampling weight differs by a factor $e^{-\alpha\mathcal{S}}$. Applied to one-dimensional PT-symmetric models, the method reproduces $\beta$ to within a few percent, detects controlled algorithmic errors, and tracks thermalization and discretization effects with high sensitivity. When compared with existing diagnostics based on the Langevin operator and drift distributions, the configurational-temperature criterion proves more sensitive to subtle deviations, offering a robust cross-check relevant to higher-dimensional scalar and gauge theories, including lattice QCD at finite density.

Abstract

The complex Langevin method (CLM) is a promising tool to address the sign problem in quantum field theories with complex actions. However, it can converge to incorrect results even when simulations appear stable, highlighting the need for robust diagnostics. Existing checks, such as monitoring drift distributions, are useful but indirect. We propose a complementary test based on the configurational temperature, constructed from the gradient and Hessian of the complex action. Unlike drift-based criteria, this estimator directly probes thermodynamic consistency and provides a physically interpretable cross-check of CLM dynamics. Using one-dimensional PT-symmetric models, we show that it reproduces the input temperature with high precision and sensitively detects algorithmic errors, step-size artifacts, and incomplete thermalization. While demonstrated in simple systems, the method extends naturally to higher-dimensional scalar and gauge theories. Since temperature is tied to the bare coupling in many lattice theories, configurational monitoring can also provide an independent check on coupling-dependent observables. Our results indicate that configurational temperature can enhance CLM reliability across a broad range of applications, including lattice QCD at finite density.

Figures (4)

• Figure 1: The estimated configurational temperature $\beta_M$ and the error in estimated temperature, $(\beta - \beta_M)/\beta$ as functions of input $\beta$.
• Figure 2: Thermalization behavior of the one-point function $G_1 \equiv \langle \phi \rangle$ (imaginary part) and the configurational temperature estimator $\beta_M$ during the initial Langevin evolution. The data are for the PT-symmetric model with $\delta = 1$ (left) and $\delta = 2$ (right) at coupling $g = 1.0$.
• Figure 3: The Langevin time history of $L G_1$. The data are for the PT-symmetric model with $\delta = 1$ and coupling $g = 1.0$.
• Figure 4: Decay of the drift terms. The simulations are for the PT-symmetric model with $\delta = 1$ (left) and $\delta = 2$ (right) and coupling $g = 1.0$.