is nonzero $\langle L(T)\rangle  \neq 0$ in the deconfined phase, which indicates the center symmetry is broken. From the holographic dictionary, the expectation value of the Polyakov loop is related to the Nambu-Goto action $S_{NG}$ for the string world sheet [@Witten1998Anti] $$S_{NG} = \frac{1}{2\pi\alpha_p}\int d^2\eta\sqrt{{\rm det}(g^s_{\mu\nu}\mathcal{X}^\mu_a\mathcal{X}^\nu_b)}$$ in the following way $$\langle L(T)\rangle = \int D \mathcal{X} e^{-S_{NG}},$$ where $\frac{1}{2\pi\alpha_p}$ is the string tension, $g^s_{\mu\nu}$ is the metric in the string frame, and $\mathcal{X}^\mu_a$ is the embedding function of the worldsheet in the target space, $\mu,\nu$ are the five dimensional space-time indices and $a,b$ represent the worldsheet coordinates. From the metric of , we have $$\begin{aligned}
S_{NG} = \frac{1}{2\pi\alpha_p}\int_{0}^{z_h} dz \frac{e^{2A_s}}{z^2}\sqrt{1+f(z)(\vec{x}^\prime)^2},\end{aligned}$$ with $g_p = \frac{1}{2\alpha_p}$ the redefinition of the string tension. The prime denotes the derivative with respect to $z$. Then, the equation of motion for $\vec{x}$ can be derived as $$\begin{aligned}
[{\frac{e^{2A_s}}{z^2}f(z)\vec{x}^\prime/\sqrt{1+f(z)(\vec{x}^\prime)^2}}]^\prime = 0.\end{aligned}$$

Substituting the $\vec{x}^\prime$ into the action $S_{NG}$, the minimal world sheet can be obtained as $$\begin{aligned}
S_0 = c_p +S^{\prime}_0 = c_p +\frac{g_p}{\pi T}\int^{z_h}_{0} dz(\frac{e^{2A_s}}{z^2}-\frac{1}{z^2}),\end{aligned}$$ where $c_p$ is a normalization constant. Finally, we can get the expectation value of the Polyakov loop: $$\begin{aligned}
\langle L(T)\rangle = \omega e^{-S_0} = e^{C_p - S^\prime_0},\end{aligned}$$ with $C_p$ another normalization 