Phase diagram at finite temperature and quark density in the strong coupling limit of lattice QCD for color SU(3)

We study the phase diagram of quark matter at finite temperature ($T$) and chemical potential ($\mu$) in the strong coupling limit of lattice QCD for color SU(3). We derive an analytical expression of the effective free energy as a function of $T$ and $\mu$, including baryon effects. The finite temperature effects are evaluated by integrating over the temporal link variable exactly in the Polyakov gauge with an antiperiodic boundary condition for fermions. The obtained phase diagram shows the first and the second order phase transition at low and high temperatures, respectively, and those are separated by the tricritical point in the chiral limit. Baryon has effects to reduce the effective free energy and to extend the hadron phase to a larger $\mu$ direction at low temperatures.

Figure 1

Parameter range which satisfies the conditions of stability and high $T$ chiral symmetry restoration. The solid dot represents the parameter set, $\alpha =0.2$, ${\alpha }^2+{\gamma }^2=1/2-0$, which we adopt in the later discussion, and the square shows the parameter set without baryon effects.

Figure 2

Effective potential as a function of $\sigma$. Upper, middle, and lower panels show the effective free energies ${\mathcal{F}}_{\mathrm{eff}}$ in the unit of $T_c$ at $\mu =0$, $T=T_{\mathrm{TCP}}$, and $T=0$, respectively. Dots represent the equilibrium points, and dashed lines connect these points.

Figure 3

Phase structure in the strong coupling limit lattice QCD with $N_c=3$. The solid and upper thick dashed lines show the first and second order phase boundary, respectively. The dashed line shows the boundary on which the effective free energy curvature becomes zero at $\sigma =0$. The dot represents the tricritical point. Chemical potential and temperature are shown in the unit of $T_c$.

Figure 4

Chiral condensate $\sigma$ as a function of chemical potential and temperature. Thick lines show the phase boundary, and the dot indicates the tricritical point. Chemical potential and temperature are shown in the unit of $T_c$.

Figure 5

Approximate chiral condensate $\sigma$ and the chiral condensate $-\partial {\mathcal{F}}_{\mathrm{eff}}/\partial m_0$ as a function of temperature (upper panel) and chemical potential (lower panel). Solid and dashed lines show $\sigma$ and $-⟨\overline{\chi }\chi ⟩$, respectively. Chemical potential and temperature are shown in the unit of $T_c$.

Figure 6

Parameter dependence of ${\mu }_c(0)$ in vacuum (thick solid line), $T_{\mathrm{TCP}}$ (long-dashed line), ${\mu }_{\mathrm{TCP}}$ (short-dashed line), and $\sigma$ (dotted line) in the upper graph. In the lower graph, thick solid line and long-dashed line represents the parameter dependence of $b_{\sigma }$ and $g_{\sigma }$, respectively.

Figure 7

Parameter dependence of the effective free energy in vacuum in the chiral limit, $(T,\mu ,m_0)=(0,0,0)$. Long-dashed, thick solid, dash-dotted, and dotted lines show the effective free energy in the present work, ${\mathcal{F}}_{\mathrm{eff}}^{(Tb)}$, with parameters $\alpha =0.1$, 0.2, 0.5, and 0.6, respectively. Short-dashed line shows the effective free energy without baryon effects, ${\mathcal{F}}_{\mathrm{eff}}^{(T)}$.

Figure 8

Parameter dependence of the phase boundary. Long-dashed, thick solid, dash-dotted, and dotted lines show the phase boundary in ${\mathcal{F}}_{\mathrm{eff}}$ with $\alpha =0.1$, 0.2, 0.5, and 0.6, respectively. Short-dashed line shows the boundary in ${\mathcal{F}}_{\mathrm{eff}}^{(T)}$, without baryon effects. Thin solid line indicates $c_2=0$, where $c_2$ is the quadratic coefficient of $\sigma$.

Figure 9

Comparison of effective free energies in different treatments. Thin solid, dotted, long-dashed, and short-dashed lines show the effective free energies ${\mathcal{F}}_{\mathrm{eff}}^{(0)}$, ${\mathcal{F}}_{\mathrm{eff}}^{(0b)}$, ${\mathcal{F}}_{\mathrm{eff}}^{(0bv)}$, and ${\mathcal{F}}_{\mathrm{eff}}^{(T)}$, respectively. For ${\mathcal{F}}_{\mathrm{eff}}^{(0bv)}$ (long-dashed), we show the results with two parameters; $\alpha =\sqrt{1/2-{\gamma }^2}=0.2$ (thick) and $\gamma =2$ (thin). The thick solid line indicates the effective free energy in a finite temperature treatment with baryon effects (${\mathcal{F}}_{\mathrm{eff}}^{(Tb)}$, present work). We show the results in vacuum in the chiral limit, $(T,\mu ,m_0)=(0,0,0)$.

Figure 10

Expected phase diagram evolution from the strong coupling limit in the chiral limit with one species of staggered fermion towards the real world.