in the real QCD case with $N_c=3$, we normally call the chiral symmetric and confined phase as the quarkyonic phase as shown in the PNJL model [@McLerran:2008ua; @Sakai:2012ika].

In this paper, we make a step towards investigating the interplay between the deconfinement phase transition and chiral phase transition in a quenched dynamical holographic QCD model [@Li:2012ay; @Li:2013oda; @Li:2014dsa]. In this quenched dynamical holographic QCD model, the dilaton background describes the gluodynamics and the flavor/meson background describes the chiral dynamics, respectively, thus one can simultaneously realize the confinement/deconfinement phase transition and chiral symmetry breaking/restoration phase transition at finite temperature and chemical potential. However, it is worthy of mentioning that in the quenched dynamical holographic QCD model, the flavor background is added on the dilaton background as a probe, and the full QCD dynamics including the backreaction from the flavor background on the dilaton background or gluodynamics background has not been self-consistently solved yet, which is left for future work.

To extend the quenched dynamical holographic QCD model to finite chemical potential, the quark chemical potential is introduced by a $U(1)$ field in the Einstein-Dilaton-Maxwell framework. Except to fix the chemical potential dependence of the flavor background, one has also to fix 