Dirac spectrum in complex Langevin simulations of QCD

QCD at nonzero baryon chemical potential suffers from the sign problem, due to the complex quark determinant. Complex Langevin dynamics can provide a solution, provided certain conditions are met. One of these conditions, holomorphicity of the Langevin drift, is absent in QCD since zeroes of the determinant result in a meromorphic drift. We first derive how poles in the drift affect the formal justification of the approach and then explore the various possibilities in simple models. The lessons from these are subsequently applied to both heavy dense QCD and full QCD, and we find that the results obtained show a consistent picture. We conclude that with careful monitoring, the method can be justified a posteriori, even in the presence of meromorphicity.

mentioning

confidence: 99%

Complex Langevin dynamics and zeroes of the fermion determinant

Aarts

Seiler

Sexty

et al. 2017

“…The quark contribution leads to poles in the drift, namely where det M = 0 and M −1 does not exist. In some cases this affects the results negatively [38,40], but in HDQCD this is not the case, as far as is understood [18,39]. In order to avoid numerical instabilities and regulate large values of the drift, it is necessary to change the Langevin stepsize ε adaptively [16], based on the absolute value of the drift term K a x,ν .…”

Section: Complex Langevin Equation and Gauge Coolingmentioning

confidence: 99%

The QCD phase diagram in the limit of heavy quarks using complex Langevin dynamics

Aarts

Attanasio

Jäger

et al. 2016

Complex Langevin simulations allow numerical studies of theories that exhibit a sign problem, such as QCD, and are thereby potentially suitable to determine the QCD phase diagram from first principles. Here we study QCD in the limit of heavy quarks for a wide range of temperatures and chemical potentials. Our results include an analysis of the adaptive gauge cooling technique, which prevents large excursions into the non-compact directions of the SL(3, C) manifold. We find that such excursions may appear spontaneously and change the statistical distribution of physical observables, which leads to disagreement with known results. Results whose excursions are sufficiently small are used to map the boundary line between confined and deconfined quark phases.

“…On the other hand, the chiral condensate, which is a holomorphic quantity, can be expressed in terms of the eigenvalue distribution [37]. This implies that the non-universal eigenvalue distribution still has a universal property, which does not depend on the norm used for gauge cooling as long as the CLM is working.…”

Section: Introductionmentioning

confidence: 99%

Gauge cooling for the singular-drift problem in the complex Langevin method — a test in Random Matrix Theory for finite density QCD

Nagata

Nishimura

Shimasaki

2016

Recently, the complex Langevin method has been applied successfully to finite density QCD either in the deconfinement phase or in the heavy dense limit with the aid of a new technique called the gauge cooling. In the confinement phase with light quarks, however, convergence to wrong limits occurs due to the singularity in the drift term caused by small eigenvalues of the Dirac operator including the mass term. We propose that this singular-drift problem should also be overcome by the gauge cooling with different criteria for choosing the complexified gauge transformation. The idea is tested in chiral Random Matrix Theory for finite density QCD, where exact results are reproduced at zero temperature with light quarks. It is shown that the gauge cooling indeed changes drastically the eigenvalue distribution of the Dirac operator measured during the Langevin process. Despite its non-holomorphic nature, this eigenvalue distribution has a universal diverging behavior at the origin in the chiral limit due to a generalized Banks-Casher relation as we confirm explicitly.