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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

The phase diagram of QCD at finite density and temperature is speculated to have a very rich structure. This is not only interesting from theoretical viewpoints but is also relevant to the physics related to heavy-ion collision experiments and the interior structure of neutron stars. However, the speculated phase structure still remains elusive mainly because first-principle calculations based on lattice QCD are extremely difficult at finite density due to the complex fermion determinant, which causes the so-called sign problem.

As a promising solution to this problem, the complex Langevin method (CLM)

However, it is known that the method yields wrong results in some cases even if the stochastic process reaches equilibrium without any problem. This issue was discussed theoretically for the first time in Refs.

There are actually two cases that can lead to the frequent appearance of large drifts. One is the case in which the dynamical variables make frequent excursions in the imaginary directions during the stochastic process, which is referred to as the excursion problem

The gauge cooling made it possible to apply the CLM to finite density QCD in the deconfined phase

This paper is organized as follows. In Sec.

Our calculation is based on lattice QCD on a four-dimensional Euclidean periodic lattice defined by the partition function

In this work, we use the unimproved staggered fermions, for which the fermion matrix

The observables we consider in this paper are the baryon number density

In this section, we explain how we apply the CLM

To calculate the vacuum expectation value (VEV) of a gauge-invariant observable

As is mentioned in the Introduction, the proof of

It is known that the slow fall-off of the drift distribution, which invalidates

To solve the excursion problem, one can use the gauge cooling, which amounts to making a complexified gauge transformation after each Langevin step in such a way that the complexified link variables come closer to a unitary configuration

At large

In the case at hand, we introduce a deformation parameter

For

We probe the drift distribution at each

In this section, we show our results for finite density QCD obtained by the CLM as explained in the previous section. We use a

First, we check the validity of the CLM by probing the probability distribution of the drift term, which is shown in Fig.

The probability distribution of

The power-law tail of the probability distribution for

The eigenvalue distribution of the fermion matrix is shown for

In Fig.

The baryon number density (top) and the chiral condensate (bottom) obtained by the CLM are plotted against

The baryon number density (top) and the chiral condensate (bottom) obtained by the CLM are plotted against

When we performed complex Langevin simulations

In Fig.

The extrapolated values of the baryon number density (top) and the chiral condensate (bottom) obtained from Fig.

Note that the value of

In full QCD at zero temperature in the infinite volume limit, physical observables are independent of

We have made an attempt to extend the success of the CLM in investigating finite density QCD in the deconfined phase or in the heavy dense limit to the large

By comparing the results of the CLM with those obtained by the RHMC calculations in the phase-quenched model, we observe that the onset of the baryon number density in the full model occurs at larger

The authors would like to thank Y. Ito, T. Kaneko, H. Matsufuru, K. Moritake, A. Tsuchiya, and S. Tsutsui for valuable discussions. Computations were carried out on Cray XC40 at YITP in Kyoto University, SX-ACE at CMC and RCNP in Osaka University and PC clusters at KEK. K. N. and J. N. were supported in part by Grant-in-Aid for Scientific Research (Grants No. 26800154 and No. 16H03988, respectively) from Japan Society for the Promotion of Science. S. S. was supported by the MEXT-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” (Grant No. S1511006).

To calculate the drift term

The idea is to replace the trace in

The above procedure is exact if we take an average over infinitely many

See Refs.

In this Appendix, we present our results for the Polyakov line defined by

The Polyakov line obtained by the CLM is plotted against

In the phase-quenched model, the expectation value

For this to be true, it has long been considered that all the eigenvalues of the Fokker-Planck Hamiltonian should have a positive real part. However, it was argued in Ref.

This technique was also used to solve the singular-drift problem in the chiral random matrix theory

Preliminary results are presented in Lattice 2017