A Note on Sachdev-Ye-Kitaev Like Model without Random Coupling

TL;DR

The paper presents a disorder-free construction of a SYK-like large-N theory by promoting random couplings to dynamical, massive scalar fields and analyzing a small-mass, large-N limit. It demonstrates that fermionic correlators reproduce SYK results with a temperature-dependent effective coupling J_eff, and extends the framework to a SUSY generalization, offering an efficient means to estimate large-N correlators. Finite-temperature analysis reveals a temperature-dependent mapping to the SYK coupling and preserves Schwarzian-dominated dynamics under appropriate conditions. The work connects to other disorder-free SYK models and provides insight into how disorder averaging can emerge from dynamical fields, with potential implications for holography and black hole physics.

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.

Figures (8)

• Figure 1: Two expressions for external lines in the diagram. The shaded region is arbitrary. Left: The conventional one. Right: Our new expression for the same diagram in terms of dots. Each dot expresses two external fermions in a pair in the sense explained in the main text. Since we take the sum over $i_k$, we usually omit the indices in the diagram.
• Figure 2: Two expressions for an interaction vertex. Left: The conventional one. The dashed line stands for the disorder average associated with $J_{ijk\ell}$. Right: Another expression for the left picture in terms of a "bundle." The red circle in the middle stands for a bundle corresponding to the disorder average in the left picture. We say two fermion lines are connected to each other when they are involved in the same bundle.
• Figure 3: "Untying" a bundle corresponds to replacing the left picture with right.
• Figure 4: A diagram whose contribution is of $\mathcal{O}(N)$ splits into four disconnected parts when a bundle in it is untied. Here the shaded regions are arbitrary.
• Figure 5: "Cutting" a bundle in the diagram implies replacing the left picture with the right. This corresponds to cutting the dashed line in fig. \ref{['fig:vertex']}. Note that this "cutting" is different from "untying" a bundle discussed in fig. \ref{['fig:untying']}.