Exploring the phase diagram of QCD with complex Langevin simulations

TL;DR

The paper tackles the sign problem in lattice QCD at finite chemical potential by applying Complex Langevin dynamics on the extended SL$(3,\mathbb{C})$ manifold, focusing on heavy dense QCD (HDQCD) where the fermion determinant simplifies in terms of Polyakov loops. By performing extensive scans over $N_\tau$ (temperature) and $\mu$ (density) on a fixed $8^3\times N_\tau$ lattice with $N_f=2$, and using adaptive step-size and gauge cooling, the authors map the phase diagram via the fermion density and Polyakov loop observables, along with their susceptibilities. They observe an onset at $\mu_c \simeq m_q=-\ln(2\kappa)$, a sharper transition at lower $T$, and a lattice saturation at $n_{\text{sat}}=12$, with a distinct phase boundary identified, albeit with a lattice artifact at $\mu/m_q \approx 1$. The work demonstrates a viable, first-principles pathway to the QCD phase diagram at finite density in the HDQCD limit and lays groundwork for extensions to full QCD with dynamical fermions and more precise determinations of transition orders.

Abstract

Simulations of QCD with a finite chemical potential typically lead to a severe sign problem, prohibiting any standard Monte Carlo approach. Complex Langevin simulations provide an alternative to sample path integrals with oscillating weight factors and therefore potentially enable the determination of the phase diagram of QCD. Here we present results for QCD in the limit of heavy quarks and show evidence that the phase diagram can be mapped out by direct simulation. We apply adaptive step-size scaling and adaptive gauge cooling to ensure the convergence of these simulations.

Figures (6)

• Figure 1: A possible sketch of the QCD phase diagram.
• Figure 2: Left: The real part of the Polyakov loop $P$ as function of the Langevin time $t$. Right: The unitarity norm, i.e. the distance of the gauge links from the SU$(3)$ manifold (below) and the average number of gauge cooling steps in a small Langevin time interval used between each Langevin update (above) as a function of Langevin time.
• Figure 3: Strategy for determining the phase diagram of heavy dense QCD. The red and blue lines indicate scans for fixed $\mu$ or fixed temperature $T$, at a given $\beta$ and $\kappa$.
• Figure 4: The fermion density as a function of the chemical potential $\mu$. The expected critical chemical potention $\mu_c = m_q = - \ln(2 \kappa)$ is shown as well.
• Figure 5: The real part of the Polyakov loop as function of the temperature and chemical potential for fixed $\beta$ and $\kappa$. The black points are the results of complex Langevin simulations for a given value of $\mu$ and $T$. The surface is the result of a cubic interpolation, in which the value of the Polyakov loop is encoded in the colour.