Dual gravities from entanglement entropy

Abstract

Applying a rule-based holographic method, we investigate the reconstruction of dual gravity theories from the quantum field theory (QFT) data, specifically entanglement entropy. We first derive a three-dimensional black hole geometry from the entanglement entropy of a two-dimensional thermal system. Using the reconstructed solution, we extract various thermodynamic quantities with small numerical errors. Moreover, we explore how to reconstruct the dual gravity theory beyond the geometry itself. For an undeformed conformal field theory (CFT), we show that the dual gravity theory can be constructed analytically from the analytic form of the entanglement entropy. In particular, we demonstrate how to reconstruct the analytic dual geometry by applying the Abel transformation. Finally, we investigate the numerical reconstruction of the dual gravity theory from numerical entanglement entropy data for a relevantly deformed CFT. After reconstructing the dual gravity, we show that additional information about the renormalization group (RG) flow, for instance, the $\b$-function and the $c$-function, can be extracted for the considered relevantly deformed CFT.

Figures (5)

• Figure 1: (a) The entanglement entropy relying on the subsystem size. We evaluate the entanglement entropy holographically for $M=q_0 =1/2$ with $R=G=z_h=1$, $\epsilon= 10^{-3}$, and $N=20$. (b) When the entanglement entropy is given without knowing the values of $M$ and $\rho_0$, we reconstruct the dual black hole geometry from the entanglement entropy as the input data.
• Figure 2: The input data for the reconstruction process. (a) The entanglement entropy $S_E(\ell)$ as a function of subsystem size $\ell$, generated from the theoretical model. (b) The entropic C-function $d S_E / d \ln \ell$. The monotonic decrease from the UV (small $\ell$) to the IR (large $\ell$) confirms the validity of the c-theorem and the existence of a proper RG flow.
• Figure 3: The numerical reconstruction results of the metric. (a) The reconstructed warp factor $A(r)$ in the range $r < 15$. (b) The derivative $\dot{A} = d A/dr$, showing the stability of the reconstruction near the AdS limit ($\dot{A} \approx 1$).
• Figure 4: The reconstruction results. (a) The profile of the bulk scalar field $\phi(r)$ along the radial coordinate. (b) The reconstructed scalar potential $V(\phi)$ (red solid line) compared with the theoretical prediction (blue dashed line). The reconstruction matches the theoretical model with high precision, demonstrating the robustness of our numerical method.
• Figure 5: (a) The $\beta$-function as a function of the coupling constant $\phi$. (b) The holographic $c$-function as a function of the coupling constant $\phi$.