Chiral phase transition in two-flavor QCD from an imaginary chemical potential

We investigate the order of the finite temperature chiral symmetry restoration transition for QCD with two massless fermions, by using a novel method, based on simulating imaginary values of the quark chemical potential $\mu =i{\mu }_i$, ${\mu }_i\in \mathbb{R}$. Our method exploits the fact that, for low enough quark mass $m$ and large enough chemical potential ${\mu }_i$, the chiral transition is decidedly first order, then turning into crossover at a critical mass $m_c(\mu )$. It is thus possible to determine the critical line in the $m-{\mu }^2$ plane, which can be safely extrapolated to the chiral limit by taking advantage of the known tricritical indices governing its shape. We test this method with standard staggered fermions and the result of our simulations is that $m_c(\mu =0)$ is positive, so that the phase transition at zero density is definitely first order in the chiral limit, on our coarse $N_t=4$ lattices with $a\simeq 0.3\mathrm{fm}$.

Figure 1

Schematic behavior of the order of the finite temperature phase transition for $N_f=2+1$ QCD as a function of the up/down and strange quark masses ($m_{u,d}$, $m_s$), at zero baryon chemical potential. $O(4)$ and $Z_2$ denote transitions of the second order in the universality class of the $3d$ $O(4)$ and Ising models, respectively.

Figure 2

Possible scenarios for the chiral phase transition as function of the pion mass.

Figure 3

Generic phase diagram as a function of imaginary chemical potential and temperature. Solid lines are first order Roberge-Weiss transitions. The behavior along dashed lines depends on the number of flavors and the quark masses.

Figure 4

Schematic picture of the phase diagram for $N_f=2$ and $N_f=3$ at the Roberge-Weiss point ${\mu }_i/T=\pi /3$. Solid lines denote first order transitions, the dashed line indicates a second order transition of the $3d$ Ising universality class and the two dots represent the tricritical points, at which the order of the transition changes.

Figure 5

The behavior of the order of the finite temperature transition in $N_f=2+1$ QCD at the Roberge-Weiss point ${\mu }_i/T=\pi /3$. The regions denoted by $1^{\mathrm{st}}$ and $Z_2$ represent, respectively, regions where first order transitions and second order transitions of the universality class of the $3d$ Ising model take place. The solid lines are lines of tricritical points and the dashed line is the line $m_s=m_{u,d}$ corresponding to $N_f=3$ QCD. The dots represent the tricritical points found in previous $N_f=2$ ([34, 35]) and $N_f=3$ ([36]) simulations.

Figure 6

The same as in Fig. 1, with the quark chemical potential $\mu$ as an additional parameter. The critical boundary lines sweep out surfaces as $\mu$ is turned on. At $\mu /T=i\pi /3$ (value denoted by $RW$), as well as at $m_{u,d}=0$, the critical surfaces terminate in tricritical lines (represented by bold dashed lines), which fix the curvature of the surfaces through tricritical scaling.

Figure 7

The $N_f=2$ (i.e., $m_s=\infty$) section of Fig. 6. The two tricritical points at $m_{u,d}=0$ and $\mu =i\pi T/3$ are connected by a $Z_2$ critical line. As in Fig. 6 this picture is plotted assuming the $N_f=2$ finite temperature transition to be second order in the chiral limit.

Figure 8

Binder cumulant for fixed quark mass ($a{m}_{u,d}=0.005$) as a function of imaginary chemical potential and volume. The intersection signals the critical point.

Figure 9

Data corresponding to the calculated critical points, the line is a fit according to tricritical scaling Eq. (4).