A Generalization of Sachdev-Ye-Kitaev

TL;DR

This paper extends the SYK model to multiple fermion flavors, analyzing its infrared behavior, operator spectrum, and chaos properties, revealing richer symmetry and spectrum features compared to the original model.

Contribution

It introduces a multi-flavor generalization of the SYK model, computes its infrared dimensions, spectrum, and demonstrates the persistence of maximal chaos and conformal symmetry breaking.

Findings

Presence of a dimension-two operator indicating conformal symmetry breaking.

Richer spectrum due to the global symmetry group structure.

Analysis of large q limit and exact solution at q=2 for finite N.

Abstract

The SYK model: fermions with a $q$-body, Gaussian-random, all-to-all interaction, is the first of a fascinating new class of solvable large $N$ models. We generalize SYK to include $f$ flavors of fermions, each occupying $N_a$ sites and appearing with a $q_a$ order in the interaction. Like SYK, this entire class of models generically has an infrared fixed point. We compute the infrared dimensions of the fermions, and the spectrum of singlet bilinear operators. We show that there is always a dimension-two operator in the spectrum, which implies that, like in SYK, there is breaking of conformal invariance and maximal chaos in the infrared four-point function of the generalized model. After a disorder average, the generalized model has a global $O(N_1) \times O(N_2) \times \ldots\times O(N_f)$ symmetry: a subgroup of the $O(N)$ symmetry of SYK; thereby giving a richer spectrum. We also elucidate aspects of the large $q$ limit and the OPE, and solve $q=2$ SYK at finite $N$.