We demonstrate that the complex Langevin method (CLM) enables calculations in QCD at finite density in a parameter regime in which conventional methods, such as the density of states method and the Taylor expansion method, are not applicable due to the severe sign problem. Here we use the plaquette gauge action with $β= 5.7$ and four-flavor staggered fermions with degenerate quark mass $m a = 0.01$ and nonzero quark chemical potential $μ$. We confirm that a sufficient condition for correct convergence is satisfied for $μ/T = 5.2 - 7.2$ on a $8^3 \times 16$ lattice and $μ/T = 1.6 - 9.6$ on a $16^3 \times 32$ lattice. In particular, the expectation value of the quark number is found to have a plateau with respect to $μ$ with the height of 24 for both lattices. This plateau can be understood from the Fermi distribution of quarks, and its height coincides with the degrees of freedom of a single quark with zero momentum, which is 3 (color) $\times$ 4 (flavor) $\times$ 2 (spin) $=24$. Our results may be viewed as the first step towards the formation of the Fermi sphere, which plays a crucial role in color superconductivity conjectured from effective theories.
We present lattice simulations on the phase diagram of Quantum Chromodynamics (QCD) with two light quark flavours at finite chemical potential $μ$. To circumvent the sign problem we use the complex Langevin method. In this study, we have carried out finite density lattice computations for a pion mass of $\sim 480$ MeV. We report on the pressure, energy and entropy equations of state in ab-initio lattice QCD calculations, as well as the observation of the Silver Blaze phenomenon.
We discuss the flavor number dependence of QCD at low temperature and high density by the complex Langevin method. In our previous work, the complex Langevin method is confirmed to satisfy the criterion for correct convergence in certain regions, such as $μ_{\rm q} / T = 5.2-7.2$ on $8^3 \times 16$ and $μ_{\rm q} / T = 1.6-9.6$ on $16^3 \times 32$ using $N_{\rm f} = 4$ staggered fermion at $β= 5.7$. We extend this study to more realistic flavor cases, $N_{\rm f} = 2, 2 + 1, 3$, using Wilson fermions. We present the flavor number dependence of the validity regions of the complex Langevin method…
In this review we present the current state-of-the-art on complex Langevin simulations and their implications for the QCD phase diagram. After a short summary of the complex Langevin method, we present and discuss recent developments. Here we focus on the explicit computation of boundary terms, which provide an observable that can be used to check one of the criteria of correctness explicitly. We also present the method of Dynamic Stabilization and elaborate on recent results for fully dynamical QCD.
We explore the QCD phase diagram at finite density with four-flavor staggered fermions using the complex Langevin method, which is a promising approach to overcome the sign problem. In our previous work on an $8^3 \times 16$ lattice at $β= 5.7$ with the quark mass $m = 0.01$, we have found that the baryon number density has a clear plateau as a function of the chemical potential. In this study, we use a $16^3 \times 32$ lattice to reduce finite volume effects and find that the plateau structure survives. Moreover, the number of quarks in the plateau region turns out to be 24, which is exactly …
Lattice simulations of non-zero density QCD introduce the so-called sign problem (complex or negative probabilities), which invalidates importance sampling methods. To circumvent this, we use the Complex Langevin Equation (CLE), to measure the boundary terms and then compare these results with the ones gotten from reweighting, confirming the expectations from previous studies. We also investigate boundary terms in simulations using CLE with dynamic stabilization and compare this, to results calculated with reweighting.
We use complex Langevin simulations to explore the QCD phase diagram over a large range of chemical potentials and temperatures. For our simulations, we use two flavours of dynamical Wilson fermions with a pion mass of approximately $480\,$MeV with a spatial volume of $24^3$. Here we report on consistency checks at zero chemical potentials and present our results for the fermion density and the Polyakov loop. We find that at the lowest temperature the fermion density remains zero until $m_N/3$, in line with the expectations from the Silver Blaze phenomenon.
We employ the Complex Langevin method for simulation of complex-valued actions. First, we show how to test for convergence of the method by explicitely computing boundary terms and demonstrate this in a model. Then we investigate the deconfinement phase transition of QCD with $N_f=2$ Wilson-fermions using the Complex Langevin Method and. We give preliminary results for the transition temperatures up to $μ/T_c(μ=0)\approx 5$ and compute the curvature coefficient $κ_2$.
Exploring the phase diagram of QCD at finite density is a challenging problem since first-principle calculations based on standard Monte Carlo methods suffer from the sign problem. As a promising approach to this issue, the complex Langevin method (CLM) has been pursued intensively.In this work, we investigate the applicability of the CLM in the vicinity of the deconfinement phase transition using the four-flavor staggered fermions. In particular, we look for a signal of the expected first order phase transition within the validity region of the CLM.
We study the deconfinement transition line in QCD for quark chemical potentials up to $μ_q \sim 5 T$ ($μ_B \sim 15 T$). To circumvent the sign problem we use the complex Langevin equation with gauge cooling. The plaquette gauge action is used with two flavors of naive Wilson fermions at a relatively heavy pion mass of roughly 1.3 GeV. A quadratic dependence describes the transition line well on the whole chemical potential range.
Monte Carlo studies of QCD at finite density suffer from the sign problem, which becomes easily uncontrollable as the chemical potential $μ$ is increased even for a moderate lattice size. In this work we make an attempt to approach the high density low temperature region by the complex Langevin method (CLM) using four-flavor staggered fermions with reasonably small quark mass on a $8^3 \times 16$ lattice. Unlike the previous work on a $4^3 \times 8$ lattice, the criterion for correct convergence is satisfied within a wide range of $μ$ without using the deformation technique. In particular, the…
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 $μ/T$ regime with moderate quark mass using four-flavor staggered fermions on a $4^3\times 8$ lattice. While a straightforward application faces with the singular…
We study the use of the complex-Langevin equation (CLE) to simulate lattice QCD at a finite chemical potential ($μ$) for quark-number, which has a complex fermion determinant that prevents the use of standard simulation methods based on importance sampling. Recent enhancements to the CLE specific to lattice QCD inhibit runaway solutions which had foiled earlier attempts to use it for such simulations. However, it is not guaranteed to produce correct results. Our goal is to determine under what conditions the CLE yields correct values for the observables of interest. Zero temperature simulation…
We continue our simulations of lattice QCD at finite quark-number chemical potential, $μ$, using the complex-Langevin equation (CLE) with gauge-cooling and adaptive updating. The CLE is used because QCD at finite finite $μ$ has a complex fermion determinant, which prevents use of standard simulation methods. Simulations using the standard lattice action show a transition from hadronic to nuclear matter for $μ< m_π/2$ rather than the expected $μ\approx m_N/3$. This suggests that the CLE is being influenced by the phase-quenched theory, which has a transition at $μ= m_π/2$. We are therefore perf…
We simulate lattice QCD at finite quark-number chemical potential, $μ$, using the complex-Langevin equation (CLE) with gauge-cooling and adaptive updating to prevent instabilities. The CLE is used because QCD at finite $μ$ has a complex fermion determinant which precludes the use of standard simulation methods based on importance sampling. Since, even when CLE simulations converge, they are not guaranteed to produce correct results except under very stringent conditions, which lattice QCD at finite $μ$ does not obey, we need extensive testing to determine under what conditions it produces reli…
We present updated results of dynamic stabilisation (DS) applied to complex Langevin simulations of QCD in the heavy-dense limit and with staggered quarks. We show that DS is able to keep the unitarity norm sufficiently small, which leads to excellent agreements with Monte-Carlo simulations, when the latter is applicable.
We use complex Langevin simulations to study the QCD phase diagram with two light quark flavours. In this study, we use Wilson fermions with an intermediate pion mass of $\sim480\,$MeV. By studying thermodynamic quantities, in particular at lower temperatures, we are able to describe the equation of state.
The complex Langevin (CL) method is a promising approach to overcome the sign problem, which emerges in real-time formulations of quantum field theories. Over the past decade, stabilization techniques for CL have been developed with important applications in finite density QCD. However, they are insufficient for SU($N_c$) gauge theories on a Schwinger-Keldysh time contour that is required for a real-time formulation. In these proceedings we revise the discretization of the real-time CL equations and introduce a novel anisotropic kernel that enables CL simulations on discretized time contours. …
