been proposed that the chiral symmetry and deconfinement phase transition can be splitting and there would be a quarkyonic phase [@McLerran:2007qj], where the chiral symmetry is restored but still in confinement. The free energy of a heavy test quark added to the system is $e^{-\beta F_{q}} = \frac{1}{N_{c}} <L>$ and the baryon number density is $<N_B> \sim e^{-\beta(M_B-\mu_B)}$ with $\beta$ the inverse of temperature. The deconfinement temperature is around $T_d \sim \Lambda_{QCD}$.When the baryon number chemical potential is small compared to the baryon mass, i.e. $\mu_B \ll M_B $ , the number of baryons $<N_B> \sim e^{- \kappa_1 N_c}$ tends to zero at large $N_c$ for temperatures of $T \sim \Lambda_{QCD}$ and with $M_{B} \sim N_c$. Here, $\kappa_1$ is a number of order one. When $\mu_B \geq M_B$, the baryons begin to populate the system and the baryon number density becomes nonvanishing. Ref.[@McLerran:2007qj] argues that there are at least three phases in the QCD phase diagram at large $N_c$: the deconfinement phase with $T>T_d$, and in the region of $T<T_d$, there will be the mesonic phase which is confined and has vanishing baryon number density and the quarkyonic phase which has finite baryon number density and is confined [@McLerran:2008ua]. However, 