On the validity of the complex Langevin method near the deconfining phase transition in QCD at finite density

TL;DR

This work tests the practical validity of the complex Langevin method for finite-density QCD near the deconfinement transition on a $24^3 \times 12$ lattice with four-flavor staggered fermions. By analyzing drift-term histograms and employing gauge cooling, the authors find that CL remains valid across a broad region of the deconfined phase up to $\mu/T = 4.8$, but begins to fail as the system approaches the phase boundary due to a singular drift associated with near-zero Dirac eigenvalues. The findings are interpreted through a generalized Banks-Casher relation at finite density, linking the breakdown to the accumulation of near-zero modes and chiral symmetry restoration. The results extend prior low-temperature studies and delineate the practical limits of CL for finite-density QCD in the vicinity of deconfinement.

Abstract

In our previous paper [JHEP 10 (2020) 144], we found that the complex Langevin (CL) method works for QCD at finite density on the $16^3 \times 32$ lattice in the low-temperature high-density regime within the range $μ/ T = 1.6 - 9.6$ with $μ$ and $T$ being the quark chemical potential and the temperature, which enabled us to see a clear trend towards the formation of the Fermi sphere. Here we investigate the validity of the CL method on the $24^3 \times 12$ lattice in the deconfined phase near the deconfinement phase transition. As before, we use four-flavor staggered fermions and judge the validity using the criterion based on the probability distribution of the drift term. The spatial extent is $L = (1.3 - 2.7 {\rm ~fm} )> Λ_{\rm LQCD}^{-1} \sim 1{\rm ~fm}$, in contrast to our previous study with $L < Λ_{\rm LQCD}^{-1}$. We find that the CL method works in a broad region up to $μ/ T = 4.8$, while it starts to fail as we approach the phase boundary due to the singular drift problem, which can be understood qualitatively by extending the Banks-Casher relation to the case at finite density.

Figures (13)

• Figure 1: The drift histogram for the gauge part (Left) and the fermionic part (Right) is shown for $\mu / T = 0.6$ with $5.2 \le \beta \le 5.5$.
• Figure 2: Histogram of the fermionic drift is shown in the log-log scale (Left) and the semi-log scale (Right) for $\mu/T=0.6$ at $5.2 \le \beta \le 5.5$. The solid lines represent the fits to a power law $x^{-p}$ (Left) and an exponential function $e^{-x/\alpha}$ (Right). Different colors indicate the results with a different fitting range.
• Figure 3: The validity region of the CL method is shown in the $(\mu,T)$-plane, where the dashed lines correspond to $\mu / T = 0.6, 1.2, 2.4, 3.6$ and $4.8$. The circles (crosses) represent the parameter points on which the distribution of the drift term can (not) be fitted by an exponential function. The numbers close to these symbols represent the power obtained by fitting the tail of the distribution to a power law. The plus symbols indicate that the CG method does not converge at these points. The diamonds represent the critical temperature Engels:1996agFodor:2015doa obtained at two different values of $\mu$ assuming that the string tension is given by $\sqrt{\sigma} = 440$ MeV. The diagonal and horizontal solid blue lines correspond to the parameter sets used in Ref. Fodor:2015doa and Ref. Sexty:2019vqx, respectively.
• Figure 4: The eigenvalue distribution of $M^\dagger M$, where $M$ is the Dirac operator defined in \ref{['Mmat']}, is plotted for $\mu/T = 0.6$ at $\beta=5.2, 5.3, 5.4$ and $5.5$.
• Figure 5: The Langevin-time history of the unitarity norm \ref{['def-unitarity-norm']} is plotted for $\mu / T = 0.6$.