An SYK-Like Model Without Disorder (1610.09758v2)

Abstract: Making use of known facts about "tensor models," it is possible to construct a quantum system without quenched disorder that has the same large $n$ limit for its correlation functions and thermodynamics as the SYK model. This might be useful in further probes of this approach to holographic duality.

• The paper demonstrates that a carefully constructed tensor model replicates the SYK model's leading-order diagrammatic structure without relying on quenched disorder.
• It reveals that treating coupling constants as slow quantum variables allows for a novel analysis of gauge symmetries and large N limits.
• The findings establish a new framework for exploring holographic dualities and quantum gravity, simplifying the connection between quantum systems and gravitational theories.

An SYK-Like Model Without Disorder: An Overview

The paper under discussion, "An SYK-Like Model Without Disorder" by Edward Witten, explores a quantum mechanical model that mirrors the Sachdev-Ye-Kitaev (SYK) model, specifically at large $N$, without the necessity of quenched disorder. This is achieved through the construction of tensor models, potentially offering novel insights into approaches to holographic duality.

Core Concept and Model Construction

The SYK model, a well-studied quantum model of fermions, is characterized by random couplings and has large $N$ limit solutions that facilitate exploration of emergent holographic spacetime. Traditional approaches rely on randomness (quenched disorder) to realize these properties, which poses theoretical challenges, particularly in applications to holographic dual theories and black hole physics.

Witten's approach circumvents the reliance on quenched disorder by formulating a tensor model. Such models are designed to replicate the diagrammatic dominance characteristic of the SYK model's large $N$ limit through known techniques in tensor field theory. Specifically, this paper examines "melonic graphs" in tensor models, which govern the same large $N$ Feynman diagrams as the SYK model. These tensor models maintain certain symmetries that standard quenched-disorder applications do not inherently possess.

Key Results and Techniques

The implementation of a tensor model without disorder posits that coupling constants can be treated as quantum variables with slow dynamics, differing fundamentally from static randomness. In this setup, the number of fermion fields is matched with configurations of a product of symmetry groups, allowing for the construction of large $N$ limits without overwhelming thermodynamic entropy effects typical of quenched settings.

The presented tensor model demonstrates equivalence to the SYK model at the level of leading-order Feynman diagrams, confirming the theoretical viability of such an approach. These equivalences are rigorously shown through analyses of strand resolutions and graph theoretic reductions to familiar matrix model forms.

Implications

This research has significant implications for theoretical physics, particularly in the landscape of quantum gravity and holography. By eliminating quenched disorder, the paper introduces tensor models as potential frameworks more closely aligned with physical quantum systems. This may enhance understanding in the duality between quantum mechanical systems and higher-dimensional gravitational theories.

The constructed tensor models also present a canonical way to analyze gauge symmetries and dualities, possibly leading to new models beyond traditional SYK settings that might apply to broader classes of quantum phenomena and holographic theories.

Future Directions

The paper opens avenues for future investigations in several directions:

1. Refinement of Tensor Models: Further exploration into variations of the tensor models could illuminate more intricate properties or a wider array of SYK-like behaviors in higher dimensions.
2. Application to Holographic Dualities: Integrating this approach into explicit holographic conjectures and models, beyond the two-dimension emergent theories, could offer deeper understanding and validation of space-time and quantum entanglement concepts.
3. Exploration of Symmetric Properties: The potential to refine interpretations of symmetry (global and gauge) within these models invites in-depth exploration, with implications for understanding quantum information scrambling and chaos.

This work represents a disciplined advancement in modeling complex quantum systems, contributing to the overarching theory of holographic duality, and establishes a framework for eliminating disorder while maintaining theoretical integrity and applicability.