Comparison of complex Langevin and mean field methods applied to effective Polyakov line models

TL;DR

This work analyzes two effective Polyakov line actions derived from SU(3) gauge–matter systems at finite chemical potential to address the sign problem. By applying both complex Langevin dynamics and mean field theory, it shows near-perfect agreement when both methods are valid, and identifies branch-cut crossings in logarithmic measure terms as the source of discrepancies. The results reveal a first-order–like transition at moderate $\mu$ for a lighter gauge–Higgs scenario and confirm reliable agreement between methods in several cases, while also highlighting limitations of complex Langevin at large $\mu$. The findings support the surprising accuracy of mean field predictions for these nonlocal SU(3) spin models and point to potential remedies for CL failures, such as evolving SU(3) links rather than angles. Overall, the study strengthens confidence in using PLA-based approaches with complementary numerical methods to explore finite-density gauge theories.

Abstract

Effective Polyakov line models, derived from SU(3) gauge-matter systems at finite chemical potential, have a sign problem. In this article I solve two such models, derived from SU(3) gauge-Higgs and heavy quark theories by the relative weights method, over a range of chemical potentials where the sign problem is severe. Two values of the gauge-Higgs coupling are considered, corresponding to a heavier and a lighter scalar particle. Each model is solved via the complex Langevin method, following the approach of Aarts and James, and also by a mean field technique. It is shown that where the results of mean field and complex Langevin agree, they agree almost perfectly. Where the results of the two methods diverge, it is found that the complex Langevin evolution has a branch cut crossing problem, associated with a logarithm in the action, that was pointed out by Mollgaard and Splittorff.

Figures (12)

• Figure 1: Comparison of Polyakov lines $\langle \text{Tr}(U) \rangle, \langle \text{Tr}(U^\dagger) \rangle$ and number density vs. $\mu/T$, computed via complex Langevin and mean field techniques, in gauge-Higgs theory at $\kappa=3.8$.
• Figure 2: An estimate of $\langle \exp[iS_I] \rangle_{pq}$ vs. $\mu/T$ in the phase-quenched version of gauge-Higgs theory at $\kappa=3.8$, obtained from the second order cumulant. $S_I$ is the imaginary part of the action.
• Figure 3: Argument of the logarithm for gauge-Higgs theory at $\beta=5.6,~\kappa=3.8$, at chemical potential $\mu/T=5.0$, evaluated at each Langevin time step. Values near the negative real axis are a negligible fraction of the sample.
• Figure 4: Comparison of Polyakov lines $\langle \text{Tr}(U) \rangle, \langle \text{Tr}(U^\dagger) \rangle$ and number density vs. $\mu/T$, computed via complex Langevin and mean field techniques in the heavy quark model. Note the saturation at high $\mu/T$ at density=3.
• Figure 5: An estimate of $\langle \exp[iS_I] \rangle_{pq}$ vs. $\mu/T$ in the phase-quenched version of the heavy quark model, obtained from the second order cumulant. $S_I$ is the imaginary part of the action.