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

TL;DR

This paper explores how holographic duality and MERA tensor networks can model quantum correlations and mutual information between disjoint blocks in a critical system, revealing a phase transition linked to black hole geometries.

Contribution

It demonstrates that the causal structure of MERA networks reflects a transition analogous to black hole formation in AdS space, connecting quantum correlations to holographic geometries.

Findings

Identifies a phase transition in mutual information at a critical conformal ratio.

Shows the transition can be modeled by an AdS black hole geometry.

Provides an entropic explanation for the instability in quantum correlations.

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.