is the determinant of metric $g_{\mu\nu}$, $G_5$ is the 5D Newton constant, $\Phi$ is the dilaton field, and $X$ is the bulk scalar field which corresponds to $\bar{q}q$ condensate of QCD. $V_G$ represents the dilaton potential, and $V_X$ is the bulk scalar potential coupled with the strength tensor of gauge field. The leading term in $V_X$ is the mass term $m_{5}^{2}XX^{\dagger}$, which can be determined as $m_{5}^{2} = -3$ from the AdS/CFT prescription $m_{5}^{2} = (\Delta - p)(\Delta + p - 4)$ by taking $\Delta = 3, p = 0$ [@Witten1998Anti; @BallonBayona:2007vp]. $h(\Phi)$ is a gauge kinetic function constraining the $\mu$ dependence of the system and will be fixed by fitting the lattice data on baryon number susceptibilities. $F_{\mu\nu}$ are the strength tensor of gauge field dual to the baryon number current. If $F_{\mu\nu}=0$, the system is reduced to zero chemical potential case, and $F_{\mu\nu}\neq0$ corresponds to finite baryon number chemical potential case of the system.

To consider the gravity dual of QCD at finite temperature and baryon number density, we can take the following metric ansatz in the string frame: $$\begin{aligned}
ds^2=\frac{e^{2A_s(z)}}{z^2}[-f(z)dt^2+\frac{1}{f(z)}dz^2+d\vec{x}^2]. \label{metrics}\end{aligned}$$ As discussed in Ref.[@Li:2011hp], it is more convenient to work out thermodynamics in the Einstein frame, 