Thermodynamic Diagnostics for Complex Langevin Simulations: The Role of Configurational Temperature

Abstract: The complex Langevin method (CLM) is a promising approach to tackle the sign problem in quantum field theories with complex actions. However, it can converge to incorrect results even when simulations appear stable, thus underscoring the need for robust diagnostics. Existing criteria, such as monitoring the drift distribution or the Langevin-time operator, are valuable, but they remain indirect. In this work, we propose a complementary reliability test based on the configurational temperature. It is constructed from the gradient and Hessian of the complex action. Unlike drift-based checks, this estimator directly probes thermodynamic consistency and thus offers a physically interpretable cross-check of CLM dynamics. To demonstrate that the estimator reproduces the input temperature with high precision, we use one-dimensional PT-symmetric models as a controlled benchmark. We show that the estimator also sensitively detects algorithmic errors, step-size artifacts, and incomplete thermalization. While tested in low-dimensional systems, the method is readily extensible to higher-dimensional scalar and gauge theories, where scalable approximations make its implementation practical. Since temperature is tied to the bare coupling in many lattice theories, configurational monitoring can also serve as an independent check on coupling-dependent observables. Our results suggest that configurational temperature can be integrated into a hybrid diagnostic framework, thus enhancing the robustness of CLM across a wide range of applications, including future studies of lattice QCD at finite density.