deconfinement transition temperature is always a crossover and its critical temperature depends weakly on the chemical potential. This finding is consistent with that in the PNJL model [@Li:2018ygx].

![The deconfinement phase transition line in the $(T,\mu)$ plane in the quenched dynamical holographic QCD model.[]{data-label="Tmu1"}](Tmu1.eps){width="6.5cm" height="4.5cm"}

Chiral phase transition and the quarkyonic phase {#chiral}
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In previous sections, we have fixed the chemical potential dependence of the dilaton background from the equation of state and baryon number susceptibilities, and we have investigated the deconfinement phase transition in the $(T,\mu)$ plane. In this section, we will discuss the chiral phase transition at finite temperature and chemical potential.

Building $V(\chi, F^2)$
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We will consider the action part in Eq.(\[SMatter\]), which includes the dynamics of the scalar field $X$, corresponding to the chiral condensate $\sigma\equiv\langle\bar{q}q\rangle$. When the scalar field $X$ obtains a vacuum expectation value $X_0$, the $SU(2)_L \times SU(2)_R$ symmetry of the matter sector $S_M$ would be broken. We consider the two-flavor case $N_f = 2$ with $m_ u = m_d$, and we set $X_0(z) = \chi(z)I_2/2$ with $I_2$ the $2 \times 2$ identity matrix. From Eq.(\[SMatter\]), the action reduces to the form of $$\begin{aligned}
S_\chi = - \int d^5x \sqrt{-g^s}e^{-\Phi}[\frac{1}{2}g^{zz}\chi^{\prime2}+V(\chi,F^2)],\end{aligned}$$ where 