later calculation we will take a specific group of parameters $\lambda_2 =-25$, $\lambda_4 =200$, $\lambda_6 =-\frac{1000}{3}$ to show the qualitative results and the approximate quantitative results to get a close CEP position of PNJL model. But we emphasize that the exact value of this parameters could be fixed in a full back-reaction scenario and the position of CEP could be predicted in that model, which will be left for the future study.

Interplay of chiral and deconfinement phase transition: quarkyonic phase
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According to the above discussion, finally, we will take the parameters $\lambda_0=0, \lambda_2 =-25$, $\lambda_4 =200$, $\lambda_6 =-\frac{1000}{3}, v_4 =2$, $v_6 =100$ in our calculation. Taking the physical quark mass $m_q=5\rm{MeV}$, we could solve out chiral condensate $\sigma$ as function of $T$ and $\mu$. We find that at small chemical potential, the phase transition is of crossover type, like the dashed line for $\mu=0$ in Fig.\[sigma\]. At large chemical potential, it is found that the phase transition turns to first order one, with multi-valued region appears, like the solid line for $\mu=0.3\rm{GeV}$ in Fig.\[sigma\]. In between the two cases, there is a critical point, where the phase transition turns to second order type, like the dotdashed line with 