[Paper Review] A Note on Sachdev-Ye-Kitaev Like Model without Random Coupling

Takahiro Nishinaka, S. TERASHIMA|arXiv (Cornell University)|Nov 30, 2016

Quantum Chromodynamics and Particle Interactions28 references15 citations

TL;DR

This paper proposes a quantum mechanical model without quenched disorder that reproduces the large-N limit of the Sachdev-Ye-Kitaev (SYK) model by replacing random couplings with dynamical, nearly massless scalar fields. In the small mass limit, the fermionic correlation functions match those of the SYK model with a temperature-dependent coupling, offering a field-theoretic realization of SYK dynamics without quenched randomness.

ABSTRACT

We study a description of the large N limit of the Sachdev-Ye-Kitaev (SYK) model in terms of quantum mechanics without quenched disorder. Instead of random couplings, we introduce massive scalar fields coupled to fermions, and study a small mass limit of the theory. We show that, under a certain condition, the correlation functions of fermions reproduce those of the SYK model with a temperature dependent coupling constant in the large N limit. We also discuss a supersymmetric generalization of our quantum mechanical model. As a byproduct, we develop an efficient way of estimating the large N behavior of correlators in the SYK model.

Motivation & Objective

• To eliminate quenched disorder in the SYK model while preserving its large-N correlation structure.
• To provide a field-theoretic realization of the SYK model's dynamics without relying on random couplings.
• To clarify the physical origin of the SYK model's chaotic and black hole-like properties by replacing quenched disorder with dynamical fields.
• To develop a systematic method for estimating large-N correlators in the SYK model using the proposed framework.
• To extend the construction to a supersymmetric version that reproduces the supersymmetric SYK model in the large-N, massless limit.

Proposed method

• Introduce massive scalar fields coupled to Majorana fermions, with the scalar fields treated as dynamical degrees of freedom instead of random couplings.
• Take the small mass limit of the scalar fields to recover SYK-like behavior in the large-N limit.
• Use path-integral formulation with a Lagrangian that includes kinetic terms for the scalar fields and a quartic interaction with fermions.
• Derive effective couplings by integrating out the scalar fields, showing that the effective coupling becomes temperature-dependent.
• Apply the same construction to the supersymmetric SYK model by introducing superfields for the couplings and their fermionic partners.
• Demonstrate that in the massless limit, the fermionic partners of the scalar couplings decouple, reducing the model to the original supersymmetric SYK model.

Experimental results

Research questions

• RQ1 Can the large-N limit of the SYK model be reproduced without quenched disorder by promoting coupling constants to dynamical fields?
• RQ2 What is the effective coupling structure in the resulting model, and how does it depend on temperature?
• RQ3 How do the correlation functions of the proposed model compare to those of the original SYK model in the large-N limit?
• RQ4 Can the construction be generalized to the supersymmetric SYK model, and what is the role of the superpartners in the large-N limit?
• RQ5 What is the physical interpretation of the nearly massless scalar fields in the context of black hole or quantum gravity models?

Key findings

• The proposed model reproduces the two- and four-point fermionic correlation functions of the SYK model in the large-N limit, with the coupling replaced by a temperature-dependent effective coupling.
• The effective coupling constant in the model is found to be proportional to the inverse of the scalar field mass, leading to a temperature-dependent coupling in the small mass limit.
• The eight-point correlation function in the model matches the SYK result up to subleading corrections in the large-N limit.
• In the supersymmetric generalization, the fermionic partners of the dynamical scalar fields decouple in the massless limit, reducing the model to the original supersymmetric SYK model.
• The path-integral formulation of the model is shown to be equivalent to a previously proposed model with harmonic oscillator degrees of freedom, confirming consistency.
• The model provides a systematic method for estimating large-N correlators in the SYK model by treating the coupling as a dynamical field rather than a quenched average.