[Paper Review] Extending complex Langevin simulations to full QCD at nonzero density

Dénes Sexty|arXiv (Cornell University)|Oct 23, 2013

High-Energy Particle Collisions Research20 references3 citations

TL;DR

This paper extends complex Langevin simulations to full QCD at nonzero baryon density using gauge cooling to stabilize the algorithm, enabling reliable simulations with light quarks up to saturation density. The method successfully reproduces expected physical behavior—such as Polyakov loop suppression and chiral condensate evolution—demonstrating consistency with HQCD at high masses and validating the approach for sign-problem-free simulations of dense QCD.

ABSTRACT

Simulations of full QCD at nonzero baryon density using light quark masses are presented. The sign problem is evaded by the usage of the complex Langevin equation. The simulations are stabilized by the gauge cooling procedure for small lattice spacings. The method allows simulations at high densities, up to the saturation. The sign average is measured in the full as well as the phasequenched theory. Results are compared to the HQCD approach, in which the spatial hopping terms of fermionic variables are dropped, and good agreement is found at large masses.

Motivation & Objective

• To overcome the sign problem in full QCD at nonzero baryon density using complex Langevin dynamics.
• To stabilize complex Langevin simulations for light quark masses using gauge cooling, particularly at small lattice spacings.
• To validate the method by comparing observables like the Polyakov loop and fermion density with the HQCD approximation at high quark masses.
• To demonstrate the method's reliability up to the saturation limit, where all fermionic states are filled.
• To provide a sign-problem-free alternative to reweighting and other methods limited to small chemical potentials.

Proposed method

• Uses the complex Langevin equation with complexified gauge variables in SL(3,ℂ) to sample the path integral with a complex action.
• Applies the gauge cooling procedure to localize the distribution of link variables on the complex plane, preventing instability and ensuring convergence.
• Employs the bilinear noise scheme for efficient computation of the fermionic drift force in the Langevin update.
• Uses the effective action S_eff[U] = S_g[U] + ln det M(μ,U) to compute the drift term, with left derivatives defined via group derivatives.
• Performs simulations with unimproved staggered fermions and the Wilson plaquette gauge action on small lattices to reduce computational cost.
• Compares results in full QCD with those in the phase-quenched theory and the HQCD approximation to assess reliability and sign-problem severity.

Experimental results

Research questions

• RQ1 Can complex Langevin dynamics with gauge cooling produce reliable results for full QCD with light quarks at nonzero chemical potential?
• RQ2 How does the average phase factor of the fermion determinant behave across chemical potential, and what does it reveal about the severity of the sign problem?
• RQ3 To what extent does the HQCD approximation agree with full QCD in the large quark mass and high density regime?
• RQ4 Does the complex Langevin method correctly reproduce the physical behavior of the Polyakov loop and chiral condensate at high densities?
• RQ5 Can the method reach the saturation limit, where all fermionic states are filled, and does it show the expected restoration of Z3 symmetry?

Key findings

• The complex Langevin method with gauge cooling successfully simulates full QCD with light quarks at nonzero baryon density, up to the saturation point.
• The average phase factor of the determinant drops to near zero at high chemical potentials, indicating a severe sign problem that invalidates reweighting.
• Fermion density increases immediately at high temperature, reflecting plasma-like behavior rather than the Silver Blaze onset at low temperatures.
• The Polyakov loop decreases and eventually vanishes at saturation, consistent with Z3 symmetry restoration and larger fluctuations.
• The inverse Polyakov loop peaks before the Polyakov loop, as expected in the dense regime.
• Good quantitative agreement is found between full QCD and HQCD at large quark masses (m=4), supporting the reliability of the complex Langevin results.