[Paper Review] Entanglement and Chaos in De Sitter Holography: An SYK Example

Leonard Susskind|arXiv (Cornell University)|Sep 28, 2021

Black Holes and Theoretical Physics29 references4 citations

TL;DR

This paper proposes that a specific limit of the Sachdev-Ye-Kitaev (SYK) model—where the interaction rank $ q = N^p $ in the large $ N $ limit—exhibits hyperfast scrambling and complexity growth, making it a concrete, computable holographic dual for de Sitter space. The model satisfies criteria for de Sitter holography, including scrambling on the horizon scale and exponential volume growth, suggesting a radical conjecture that this SYK limit may realize de Sitter spacetime via quantum information principles.

ABSTRACT

Entanglement, chaos, and complexity are as important for de Sitter space as for AdS and for black holes. There are similarities and great differences between AdS and dS in how these concepts are manifested in the space-time geometry. In the first part of this paper the Ryu-Takayanagi prescription, the theory of fast scrambling, and the holographic complexity correspondence are reformulated for de Sitter space. Criteria are proposed for a holographic model to describe de Sitter space. The criteria can be summarized by the requirement that scrambling and complexity growth must be "hyperfast." In the later part of the paper I show that a certain limit of SYK is a concrete, computable, holographic model of de Sitter space. Calculations are described which support the conjecture.

Motivation & Objective

• To identify the necessary conditions for a quantum system to be a holographic dual of de Sitter space, particularly focusing on entanglement, chaos, and complexity.
• To reformulate the Ryu-Takayanagi, fast-scrambling, and complexity growth prescriptions for de Sitter space, replacing AdS boundary conditions with the static patch horizon.
• To demonstrate that a specific limit of the SYK model—where $ q = N^p $—satisfies the criteria for hyperfast scrambling and complexity growth, suggesting it as a candidate for de Sitter holography.
• To argue that hyperfast complexity growth in this SYK limit corresponds to the exponential volume growth in de Sitter spacetime, linking quantum information dynamics to cosmological geometry.

Proposed method

• Reformulate the Ryu-Takayanagi and HRT entanglement entropy prescriptions for de Sitter space by replacing the AdS boundary with the static patch horizon.
• Define 'hyperfast scrambling' as scrambling on a timescale of order the de Sitter horizon scale $ R $, contrasting with the $ R \log S $ scale in AdS, implying non-k-local Hamiltonians.
• Introduce a criterion that hyperfast complexity growth must match the exponential volume growth in de Sitter spacetime, dual to the time evolution of complexity in the boundary theory.
• Analyze the SYK model in the limit $ q = N^p $, showing that the complexity grows as $ \mathcal{C}(\omega) = 2^{N^p} \omega $, exponentially faster than in standard fast-scrambling systems.
• Use the dimensionless time $ \omega = \mathcal{T} t $, where $ \mathcal{T} $ is a temperature-like parameter, to express complexity growth in a form comparable to bulk volume evolution.
• Compare the complexity growth rate in the hyperfast limit to the standard $ \mathcal{C} \sim N\omega $, showing an exponential enhancement consistent with de Sitter's exponential expansion.

Experimental results

Research questions

• RQ1 Can the Ryu-Takayanagi and HRT entanglement entropy prescriptions be generalized to de Sitter space, with the horizon as the boundary?
• RQ2 What dynamical criteria must a holographic model satisfy to describe de Sitter space, particularly regarding scrambling and complexity?
• RQ3 Does the SYK model in the limit $ q = N^p $ exhibit hyperfast scrambling and complexity growth consistent with de Sitter spacetime?
• RQ4 Is there a bulk geometric interpretation of hyperfast complexity growth in the boundary theory, such as exponential volume growth between horizons?
• RQ5 Can a non-k-local Hamiltonian in the SYK model lead to a dual description of de Sitter space via quantum information principles?

Key findings

• The hyperfast limit of the SYK model, with $ q = N^p $, exhibits complexity growth as $ \mathcal{C}(\omega) = 2^{N^p} \omega $, which is exponentially faster than the standard $ \mathcal{C} \sim N\omega $, satisfying the hyperfast criterion.
• The scrambling time in this limit is of order the de Sitter horizon scale $ R $, consistent with the decay time of quasinormal modes, indicating hyperfast scrambling.
• The time to reach maximum complexity is $ \omega \sim 2^N $, which is exponentially long, suggesting that classical general relativity breaks down only after such a time.
• The complexity growth rate in the hyperfast limit is infinitely faster than in standard fast-scrambling systems in the large $ N $ limit, matching the exponential volume growth in de Sitter spacetime.
• The model supports the conjecture that a limit of SYK is a concrete, computable, and holographically dual description of de Sitter space, with the stretched horizon as the boundary.
• The analysis suggests that the $ \Lambda $-deformation in $ \bar{T}T $-deformed CFTs may lead to similar hyperfast behavior, potentially realizing de Sitter-like phases through non-k-local couplings.