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

TL;DR

The paper tackles the CLM sign problem by introducing a configurational-temperature diagnostic derived from the gradient and Hessian of the action, enabling a direct test of whether configurations are sampled with the correct weight $e^{-\mathcal{S}}$. It demonstrates the estimator on 1D PT-symmetric models, showing $\beta_{\mathrm{conf}}$ matches the input inverse temperature within $\sim 0.2\%-3\%$, and that the diagnostic sensitively detects algorithmic issues such as noise mis-scaling, step-size artifacts, and incomplete thermalization—often more effectively than Langevin-operator or drift-decay criteria. The work connects the estimator to Euclidean lattice field theory, discusses finite-size and discretization effects, and analyzes computational costs, offering practical strategies (trace-only, sparse, stochastic tracing) to scale to higher dimensions. It positions configurational temperature as a valuable supplement to existing CLM diagnostics, with potential impact on lattice QCD at finite density and other complex-action theories. The findings lay groundwork for robust, thermodynamically grounded monitoring in complex-action simulations and highlight directions for future high-dimensional benchmarking and algorithmic improvements.

Abstract

The complex Langevin method (CLM) offers a potential solution to the sign problem in quantum field theories with complex actions, but can converge to incorrect results even when simulations appear stable. Existing diagnostics monitor drift distributions or Langevin-time operators but do not explicitly test whether configurations are sampled with the correct Boltzmann statistical weight. We propose a complementary diagnostic based on configurational temperature, constructed from gradients and Hessians of the action. Testing in one-dimensional PT-symmetric models demonstrates 0.2-3\% accuracy in reproducing input temperatures. Crucially, configurational temperature detects algorithmic errors -- including noise mis-scaling, step-size artifacts, and incomplete thermalization -- significantly more sensitively than existing drift-based or operator-based criteria. The method relies on the derivatives of the local action, making it applicable to general lattice theories, though computational cost requires consideration in higher dimensions. Our results suggest configurational temperature as a valuable addition to CLM diagnostics, complementing existing tools with potential applications from supersymmetric matrix models to lattice QCD at finite density.

Figures (5)

• Figure 1: Langevin time history of the observables $G_k$, $k = 1, 2, 3, 4$, for the PT-symmetric model with $\delta = 1$ and coupling $g = 1.0$.
• Figure 2: The estimated configurational temperature $\beta_M$ and the error in estimated temperature, $(\beta - \beta_M)/\beta$) as functions of input $\beta$. The data are for the PT-symmetric model given in Eq. \ref{['eqn:lat-scalar-pt-symm-pot']} for $\delta = 1, 2$ and coupling $g = 1.0$. The systematic growth of the relative error with $\beta$ reflects ${\cal O}(1/N_\tau)$ finite-size corrections, demonstrating the estimator's sensitivity to lattice artifacts.
• Figure 3: 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 4: 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 5: Decay of the drift terms. The simulations are for the PT-symmetric model with $\delta = 1$ (left) and $\delta = 2$ (right) and coupling $g = 1.0$.