Holographic View on Quantum Correlations and Mutual Information between Disjoint Blocks of a Quantum Critical System

TL;DR

This work investigates quantum correlations between two disjoint blocks in a quantum critical system through a holographic lens provided by MERA-induced AdS/CFT duality. By mapping MERA causal cone structure to an AdS$_{d+2}$ bulk with a black hole geometry and applying the Ryu-Takayanagi formula, it shows a phase transition in mutual information as the conformal cross-ratio $x$ crosses a critical value, validated explicitly in a BTZ AdS$_3$ setup. The analysis reveals an entropic mechanism behind the transition, and discusses how finite temperature and finite size affect the phenomenon, connecting MERA, holography, and entanglement in critical systems. These results illuminate how bulk geometries encode boundary correlations and offer a path toward optimizing tensor networks via holographic insights, with potential extensions to rotating BTZ and Lifshitz-like geometries.

Abstract

In (d+1) dimensional Multiscale Entanglement Renormalization Ansatz (MERA) networks, tensors are connected so as to reproduce the discrete, (d + 2) holographic geometry of Anti de Sitter space (AdSd+2) with the original system lying at the boundary. We analyze the MERA renormalization flow that arises when computing the quantum correlations between two disjoint blocks of a quantum critical system, to show that the structure of the causal cones characteristic of MERA, requires a transition between two different regimes attainable by changing the ratio between the size and the separation of the two disjoint blocks. We argue that this transition in the MERA causal developments of the blocks may be easily accounted by an AdSd+2 black hole geometry when the mutual information is computed using the Ryu-Takayanagi formula. As an explicit example, we use a BTZ AdS3 black hole to compute the MI and the quantum correlations between two disjoint intervals of a one dimensional boundary critical system. Our results for this low dimensional system not only show the existence of a phase transition emerging when the conformal four point ratio reaches a critical value but also provide an intuitive entropic argument accounting for the source of this instability. We discuss the robustness of this transition when finite temperature and finite size effects are taken into account.

Figures (5)

• Figure 1: Top: Schematic representation of MERA ${\mathcal{CC}}$ for two disjoint finite intervals $A$ and $B$ when the separation between them allows for overlapping after $w_{*} \sim \log {\mathit d}$ renormalization steps. The overlap occurs before the causal cones shrink to one (in our representation, when causal cones stabilize their width after $w_H = \log {\it l}$ renormalization steps). Bottom: Schematic representation of MERA ${\mathcal{CC}}$ for two disjoint finite intervals $A$ and $B$ when the separation between them does not allow for overlapping after $w^* \sim \log {\mathit d}$ renormalization steps. The curve $\gamma$ that goes through the links between the nodes of the MERA network is the minimal curve separating the ${\mathcal{CC}}(A)$ and ${\mathcal{CC}}(B)$ from the traced out sites in the MERA bulk geometry (\ref{['AdS']}).
• Figure 2: Geodesic used in computing the entanglement entropy of a region of length ${\mathit l}=\vert x_1-x_2\vert$ in the AdS Black Hole geometry. Left: $\gamma$ does not approach the horizon (dotted line). Right: $\gamma$ wraps around the horizon.
• Figure 3: Minimal curves used in the holographic computation of $S_{A \cup B}$ and $I_{(A:B)}$ for to disjoint intervals $A$ and $B$.
• Figure 4: Dependence of the transition point $x_0$ on $|\tau|= LT$i.e$1/|\tau| \propto {\mathit l}/L$ for the non rotating BTZ black hole (Eq (\ref{['mitheta']}), circles) and for the quasi-extremal rotating BTZ black hole (Eq (\ref{['mitheta2']}), squares).
• Figure 5: Characteristic length scales $z_R$ and $z_L$ of the rotating BTZ black hole background. The event horizon $z_+$ lies between $z_L<z_+<z_R$ (thin dotted line between $z_L$ and $z_R$). Top. Geodesic connecting the two closest boundary points of the intervals $A$ and $B$ used in the computation of eq. (\ref{['2pcspin']}). Bottom: Geodesic ${\mathit L}_{A \cup B}^{(con)}={\mathit L}(u_2, v_2) + {\mathit L}(v_1,u_2)$ used in the computation of eq. (\ref{['mibtz']}).