Complex Langevin in low-dimensional QCD: the good and the not-so-good

Bloch, Jacques; Mahr, Johannes; Schmalzbauer, Sebastian

doi:10.48550/arxiv.1508.05252

Cited by 12 publications

(20 citation statements)

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“…Interestingly this case is also covered by the action (7,11). The only difference lies in the scaling laws imposed during the first limiting transition (18).…”

Section: A Harmonic Oscillatormentioning

confidence: 99%

“…Therefore, the general positive distribution (11) in 2N real variables describes a harmonic oscillator if we identify…”

Section: A Harmonic Oscillatormentioning

confidence: 99%

“…This is the main modification compared to the free particle case. With the first limit taken along the trajectory (20) the action (11) provides a positive representation for Minkowski path integral of a one-dimensional harmonic oscillator.…”

Section: A Harmonic Oscillatormentioning

confidence: 99%

“…This was caused by reported lately progress in simulating lattice QCD at finite chemical potential [5,6]. At the same time the old problems [7,8,9], the approach suffered with, resurfaced again [10,11] only to emphasize the difficulty with theoretical foundations of the method [12,13].…”

Section: Introduction and A Single Integralmentioning

confidence: 99%

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Beyond complex Langevin equations: from simple examples to positive representation of Feynman path integrals directly in the Minkowski time

Wosiek

2016

J. High Energ. Phys.

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A positive representation for an arbitrary complex, gaussian weight is derived and used to construct a statistical formulation of gaussian path integrals directly in the Minkowski time. The positivity of Minkowski weights is achieved by doubling the number of real variables. The continuum limit of the new representation exists only if some of the additional couplings tend to infinity and are tuned in a specific way. The construction is then successfully applied to three quantum mechanical examples including a particle in a constant magnetic field -a simplest prototype of a Wilson line. Further generalizations are shortly discussed and an intriguing interpretation of new variables is alluded to.

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“…Interestingly this case is also covered by the action (7,11). The only difference lies in the scaling laws imposed during the first limiting transition (18).…”

Section: A Harmonic Oscillatormentioning

confidence: 99%

“…Therefore, the general positive distribution (11) in 2N real variables describes a harmonic oscillator if we identify…”

Section: A Harmonic Oscillatormentioning

confidence: 99%

Section: A Harmonic Oscillatormentioning

confidence: 99%

Section: Introduction and A Single Integralmentioning

confidence: 99%

See 2 more Smart Citations

Beyond complex Langevin equations: from simple examples to positive representation of Feynman path integrals directly in the Minkowski time

Wosiek

2016

J. High Energ. Phys.

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“…Recently an explicit justification of the CLM including the gauge cooling procedure was given by the present authors [9]. Since the advent of this new technique, the CLM has been attracting much attention; see e.g., [10,11,12,13,14,15].…”

Section: Introductionmentioning

confidence: 97%

Testing a generalized cooling procedure in the complex Langevin simulation of chiral Random Matrix Theory

Nagata¹,

Nishimura²,

Shimasaki³

2016

Proceedings of the 33rd International Symposium on Lattice Field Theory — PoS(LATTICE 2015)

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The complex Langevin method has been attracting much attention as a solution to the sign problem since the method was shown to work in finite density QCD in the deconfined phase by using the so-called gauge cooling procedure. Whether it works also in the confined phase with light quarks is still an open question, though. In order to shed light on this question, we apply the method to the chiral Random Matrix Theory, which describes the epsilon regime of finite density QCD. Earlier works reported that a naive implementation of the method fails to reproduce the known exact results and that the problem can be solved by choosing a suitable coordinate. In this work we stick to the naive implementation, and show that a generalized gauge cooling procedure can be used to avoid the problem.

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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

J. High Energ. Phys.

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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.

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Complex Langevin in low-dimensional QCD: the good and the not-so-good

Cited by 12 publications

References 0 publications

Beyond complex Langevin equations: from simple examples to positive representation of Feynman path integrals directly in the Minkowski time

Beyond complex Langevin equations: from simple examples to positive representation of Feynman path integrals directly in the Minkowski time

Testing a generalized cooling procedure in the complex Langevin simulation of chiral Random Matrix Theory

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

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