# The chiral SYK model

## TL;DR

This paper introduces a 1+1D chiral SYK model with all-to-all interactions, showing exact solvability at large N, integrability at small N, and a transition to chaos as N increases, revealing rich quantum dynamics and chaos signatures.

## Contribution

It generalizes the SYK model to a chiral 1+1D setting, providing exact solutions, integrability results, and insights into the transition from integrability to chaos with increasing N.

## Key findings

- Exact Schwinger-Dyson solutions at large N
- Integrability for N ≤ 6 via bosonization
- Chaos signatures in OTOCs and Lyapunov exponents

## Abstract

We study the generalization of the Sachdev-Ye-Kitaev (SYK) model to a $1+1$ dimensional chiral SYK model of $N$ flavors of right-moving chiral Majorana fermions with all-to-all random 4-fermion interactions. The interactions in this model are exactly marginal, leading to an exact scaling symmetry. We show the Schwinger-Dyson equation of this model in the large $N$ limit is exactly solvable. In addition, we show this model is integrable for small $N\le6$ by bosonization. Surprisingly, the two point function in the large $N$ limit has exactly the same form as that for $N=4$, although the four point functions of the two cases are quite different. The ground state entropy in the large $N$ limit is the same as that of $N$ free chiral Majorana fermions, leading to a zero ground state entropy density. The OTOC of the model in the large $N$ limit exhibits a non-trivial spacetime structure reminscent of that found by Gu and Kitaev for generic SYK-like models. Specifically we find a Lyapunov regime inside an asymmetric butterfly cone, which are signatures of quantum chaos, and that the maximal velocity dependent Lyapunov exponent approaches the chaos bound $2\pi/\beta$ as the interaction strength approaches its physical upper bound. Finally, the model is integrable for (at least) $N\le6$ but chaotic in the large $N$ limit, leading us to conjecture that there is a transition from integrability to chaos as $N$ increases past a critical value.