Phase diagram of chiral 2-flavor QCD based on an effective approach

TL;DR

The paper uses a 3d $O(4)$ non-linear sigma model as an effective, sign-problem-free surrogate for chiral 2-flavor QCD to map the finite-$\mu_B$ phase diagram. By formulating a lattice action that includes a topological term corresponding to baryon number and performing cluster-based Monte Carlo simulations, the authors determine the critical line $T_c(\mu_B)$ and find it decreases monotonically from $T_c(0) \approx 132$ MeV to $T_c(\mu_B\approx309\,\mathrm{MeV}) \approx 106$ MeV, with no Critical Endpoint observed up to this limit. Thermodynamic observables show consistent second-order behavior along the critical line, and finite-size extrapolations yield robust estimates of the infinite-volume critical temperature. Converting to physical units via $\beta_{\rm c,lat}=0.9359(1)$ and $T_c\approx132$ MeV places the CEP, if it exists, beyond $\mu_B\approx309$ MeV and below $T<106$ MeV in this effective framework. These results provide bounds and qualitative insights for the QCD phase diagram and demonstrate the viability of the 3d $O(4)$ approach for exploring finite-density physics in the chiral limit.

Abstract

Despite intense experimental and theoretical research, the QCD phase diagram at finite baryon density remains to a large extent unexplored. From the theoretical side, the obvious non-perturbative approach is lattice QCD simulations, which are, however, obstructed by a severe sign problem. Here we employ the O(4) non-linear $σ$-model as an effective theory for 2-flavor QCD in the chiral limit. The identical pattern of spontaneous symmetry breaking indicates that they belong to the same universality class. We assume high temperature dimensional reduction to the 3d O(4) model, with topological charge taking the role of the baryon number, along the lines of Skyrme's model. In this effective formulation, the baryonic chemical potential $μ_{B}$ can be included in the lattice formulation without causing any sign problem in Monte Carlo simulations. This allows us to pin down the critical line, i.e. the critical temperature $T_{\rm c}(μ_{B})$, which decreases monotonically for increasing $μ_{B}$. In the range $0 < μ_{B} \lesssim 309~{\rm MeV}$ and $132~{\rm MeV} \gtrsim T_{\rm c} \gtrsim 106~{\rm MeV}$, we do not find a Critical Endpoint (CEP), although there are hints for it to be in the vicinity of the maximal $μ_{B}$-value that we could explore.

Figures (11)

• Figure 1: Left: Illustration of the division of a lattice unit cube into six tetrahedra (with distinct colors and line-types), as used in this work. Right: A schematic sketch of a spherical tetrahedron on the sphere $S^{3}$, with edges $e_{1}, \dots ,e_{6}$.
• Figure 2: The auto-correlation time with respect to the energy, $\tau_{H}$ (top), the magnetization, $\tau_{M}$ (center), and the topological charge, $\tau_{Q}$ (bottom), all in units of multi-cluster update steps ("sweeps"). We show the exponential auto-correlation time obtained by fits of good qualities, as the small uncertainties confirm. (For $\mu_{B,{\rm lat}} <1$, $\tau_Q$ is of ${\cal O}(1)$ sweep.) The peak locations provide a first hint for the critical value of $\beta_{\rm lat}$, and the peak heights show that the simulations become rapidly more demanding for increasing $\mu_{B,{\rm lat}}$.
• Figure 3: The dynamical critical exponent, based on relation (\ref{['tauxiz']}), with respect to the energy, $z_{H}$, the magnetization, $z_{M}$, and the topological charge, $z_{Q}$. (They are obtained for $\beta_{\rm lat}$ in the vicinity of its critical value, to be discussed in Section \ref{['critical']}.) The $z$-values are very low at $\mu_{B,{\rm lat}} \approx 0$, which confirms that the critical slowing down is mild in this case, thanks to the cluster algorithm. However, when the baryonic chemical potential increases to $\mu_{B,{\rm lat}} \approx 1$, the dynamical critical exponents also attain values around 1.
• Figure 4: Energy density $\epsilon = \langle H \rangle /V$ (top), magnetization density $m = \langle | \vec{M} | \rangle / V$, $\vec{M} = \sum_{x} \vec{e}_{x}$ (center) and topological density $q = \langle Q \rangle /V$ (bottom), for $\mu_{B,{\rm lat}} \in [0, 2.5]$, in the lattice volume $V = 20^{3}$. We see that increasing $\mu_{B,{\rm lat}}$ moves the interval of maximal slope --- which is the vicinity of the phase transition --- to larger values of $\beta_{\rm lat}$, and this slope becomes steeper. In particular at $\mu_{B,{\rm lat}} = 2.5$ one might wonder whether the phase transition is still of second order, or if this would be a jump in infinite volume, i.e. a first order phase transition.
• Figure 5: Results for the (standard) correlation length $\xi$, and for the second moment correlation length $\xi_{2}$, in the volume $V=20^{3}$. The hints for a phase transition of Figure \ref{['EMagTopDensity']} are substantiated, now with evidence for the transition to be of second order, and with consistent estimates for $\beta_{\rm c,lat} (\mu_{B,{\rm lat}})$.