It is well known that investigating QCD at finite density by standard Monte Carlo methods is extremely difficult due to the sign problem. Some years ago, the complex Langevin method with gauge cooling was shown to work at high temperature, i.e., in the deconfined phase. The same method was also applied to QCD in the so-called heavy dense limit in the whole temperature region. In this paper, we attempt to apply this method to the large <math><mi>μ</mi><mo>/</mo><mi>T</mi></math> regime with moderate quark mass using four-flavor staggered fermions on a <math><msup><mn>4</mn><mn>3</mn></msup><mo>×</mo><mn>8</mn></math> lattice. While a straightforward application faces the singular-drift problem, which spoils the validity of the method, we overcome this problem by the deformation technique proposed earlier. Explicit results for the quark number density and the chiral condensate obtained in this way for <math><mn>3.2</mn><mo>≤</mo><mi>μ</mi><mo>/</mo><mi>T</mi><mo>≤</mo><mn>5.6</mn></math> are compared with the results for the phase-quenched model obtained by the standard rational hybrid Monte Carlo calculation. This reveals a clear difference, which is qualitatively consistent with the silver blaze phenomenon.

Japan

Keitaro Nagata

Jun Nishimura

Shinji Shimasaki

Complex Langevin calculations in finite density QCD at large <math><mi>μ</mi><mo>/</mo><mi>T</mi></math> with the deformation technique

Recently the complex Langevin method (CLM) has been attracting attention as a solution to the sign problem, which occurs in Monte Carlo calculations when the effective Boltzmann weight is not real positive. An undesirable feature of the method, however, was that it can happen in some parameter regions that the method yields wrong results even if the Langevin process reaches equilibrium without any problem. In our previous work, we proposed a practical criterion for correct convergence based on the probability distribution of the drift term that appears in the complex Langevin equation. Here we demonstrate the usefulness of this criterion in two solvable theories with many dynamical degrees of freedom, i.e., two-dimensional Yang-Mills theory with a complex coupling constant and the chiral Random Matrix Theory for finite density QCD, which were studied by the CLM before. Our criterion can indeed tell the parameter regions in which the CLM gives correct results.

Testing the criterion for correct convergence in the complex Langevin method

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.

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

We investigate the quantum entanglement entropy for the four-dimensional Euclidean SU(3) gauge theory. We present the first non-perturbative calculation of the entropic -function ( ) of SU(3) gauge theory in lattice Monte Carlo simulation using the replica method. For  fm, where is the length of the subspace, the entropic -function is almost constant, indicating conformally invariant dynamics. The value of the constant agrees with that perturbatively obtained from free gluons, with 20% discrepancy. When is close to the scale, the entropic -function decreases smoothly, and it is consistent with zero within error bars at  fm.

Etsuko Itou

Yoshiyuki Nakagawa

Russia

Atsushi Nakamura

V.I. Zakharov

Entanglement in four-dimensional SU(3) gauge theory

We propose a new method to construct canonical partition functions of quantum chromodynamics (QCD) from net number distributions, such as net baryon, net charge, and net strangeness, by using only the CP symmetry. To demonstrate the method, we apply it to the net-proton number distribution recently measured at the Relativistic Heavy Ion Collider. We show that both and the canonical partition functions are determined by using the CP invariance . Comparing obtained from the present analysis for the net-proton distribution and that obtained from a thermal statistical model, we find remarkable agreement for a wide range of beam energies. Constructing a grand canonical partition function , we study moments and Lee–Yang zeros for RHIC data, and discuss the possible regions of a phase-transition line in QCD. This is the first Lee–Yang zero diagram obtained for RHIC data, which helps us to see contributions of large net-proton data for exploring the QCD phase diagram. We also calculate by the lattice QCD simulations, and find a clear indication of a Roberge–Weiss phase transition in the quark–gluon plasma phase. The method does not rely on Taylor expansions, which prevent us going to large .

Probing QCD phase structure using baryon multiplicity distribution

Recently, there has been remarkable progress in the complex Langevin method, which aims to solve the complex action problem by complexifying the dynamical variables in the original path integral. In particular, a new technique, called gauge cooling, has been introduced and the full QCD simulation at finite density has been made possible in the high-temperature (deconfined) phase or with heavy quarks. Here we provide an explicit justification of the complex Langevin method including the gauge cooling procedure. We first show that the gauge cooling can be formulated in the form of a modified complex Langevin equation involving a complexified gauge transformation, which is chosen appropriately as a function of the configuration before cooling. The probability distribution of the complexified dynamical variables is modified accordingly. However, this modification is shown not to affect the Fokker–Planck equation for the corresponding complex weight as long as observables are restricted to gauge-invariant ones. Thus we demonstrate explicitly that gauge cooling can be used as a viable technique to satisfy the convergence conditions for the complex Langevin method. We also discuss “gauge cooling” in 0D systems such as vector models or matrix models.