[Paper Review] Holographic thermalization of mutual and tripartite information in 2d CFTs

Alice Bernamonti, Neil B. Copland+2 more|arXiv (Cornell University)|Dec 4, 2012

Black Holes and Theoretical Physics14 references4 citations

TL;DR

This paper investigates the holographic thermalization of mutual and tripartite information in 2D conformal field theories (CFTs) using a time-dependent AdS3-Vaidya model with a shell of null dust injecting energy. It shows that tripartite information evolves non-trivially over time, remains non-positive during thermalization, and confirms monogamy of mutual information in strongly coupled systems, contrasting with time-independent results in free quantum quench models.

ABSTRACT

We discuss the concepts of mutual and tripartite information by referring to simple intuitive examples, and describe their use as probes of thermalization in holographic models. We then review our computation of these quantities in a simple time dependent model, in which energy injection in a strongly coupled field theory is modeled by a shell of null dust falling into 3d anti-de Sitter space. We complete those results with a discussion of the possible equilibrium phases of the tripartite information.

Motivation & Objective

• To study the time evolution of mutual and tripartite information as probes of thermalization in strongly coupled 2D CFTs.
• To analyze how non-local entanglement measures like tripartite information reveal dynamics beyond free quasi-particle descriptions.
• To compare holographic thermalization in a strongly coupled, long-range correlated system with the time-independent results from global quantum quenches in weakly correlated systems.
• To determine the equilibrium phases of tripartite information in the (ℓ, d) parameter space and identify phase boundaries in the thermal and vacuum limits.
• To verify that the holographic prescription satisfies strong subadditivity and that the null energy condition underlies monogamy of mutual information.

Proposed method

• Model energy injection via a thin shell of null dust in 3D AdS space, leading to the AdS3-Vaidya metric with a step function θ(v) marking black hole formation.
• Compute mutual and tripartite information using the holographic entanglement entropy prescription, where entanglement entropy is proportional to the length of the minimal surface in the bulk.
• Use the formula $ I_3(A,B,C) = I(A,B) + I(A,C) - I(A,B igcup C) $ to define tripartite information, ensuring symmetry and sign variability.
• Derive time-dependent expressions for $ I_3 $ in the non-equilibrium regime, showing its evolution from vacuum to thermal values.
• Analyze the equilibrium limit by taking $ r_H \to 0 $ for vacuum and $ r_H \to \infty $ for thermal phases, obtaining phase diagrams in the (ℓ, d) plane.
• Identify three distinct phases for thermal and vacuum tripartite information separated by critical distances $ d_{\text{1th}}, d_{\text{2th}} $ and $ d_{\text{1v}}, d_{\text{2v}} $, with explicit logarithmic expressions in terms of hyperbolic functions.

Experimental results

Research questions

• RQ1 How does tripartite information evolve during the thermalization of a strongly coupled 2D CFT under energy injection?
• RQ2 What is the role of long-range correlations in the time dependence of tripartite information, and how does it differ from free quantum quench models?
• RQ3 What are the equilibrium phases of tripartite information in the (ℓ, d) plane, and how do they depend on the black hole temperature (r_H)?
• RQ4 Does the holographic computation of tripartite information satisfy strong subadditivity, and what does this imply for the validity of the entanglement prescription?
• RQ5 Under what conditions does the mutual information become monogamous, and how is this related to the null energy condition in the bulk?

Key findings

• The tripartite information remains non-positive throughout the thermalization process, indicating monogamy of mutual information in the strongly coupled regime.
• In the non-equilibrium regime, tripartite information is time-dependent, contrasting sharply with the time-independent result in global quantum quenches, signaling the breakdown of a free quasi-particle picture.
• The equilibrium tripartite information exhibits three distinct phases in the (ℓ, d) plane, separated by critical distances $ d_{\text{1th}} $ and $ d_{\text{2th}} $, with explicit expressions involving hyperbolic functions of $ r_H $.
• In the vacuum limit ($ r_H \to 0 $), the tripartite information takes a logarithmic form depending on $ \ell $ and $ d $, with phase boundaries at $ d_{\text{1v}} = (\sqrt{2}-1)\ell $ and $ d_{\text{2v}} = \ell/2 $.
• The phase structure of tripartite information is richer than that of mutual information, introducing a new intermediate phase bounded by $ d_{\text{2th}} $, which does not appear in the mutual information.
• The results confirm that the holographic entanglement prescription satisfies strong subadditivity, and the null energy condition is necessary for monogamy, supporting the consistency of the framework.