Higher Dimensional Generalizations of the SYK Model

TL;DR

This work generalizes the SYK framework to a 1+1D lattice by introducing a low-pass, momentum-dependent interaction that acts on a chiral subset of low-momentum fermions, yielding a relativistic UV regime and an IR SYK-like regime whose scaling is tunable by the filter shape. By solving the Schwinger-Dyson equations in the continuum limit, the authors derive a family of IR scaling dimensions Δ determined by the filter exponent γ, leading to a dynamical exponent z that encodes hyperscaling in the IR. They also explore a probe-type extension with external fermions coupled to the SYK core, showing how the probe acquires nontrivial self-energy corrections while the core remains largely undeformed at leading order. Together, these results offer a controlled setting to study higher-dimensional generalizations of SYK and their holographic interpretations, including potential connections to AdS/CFT and chaos propagation in extended systems.

Abstract

We discuss a 1+1 dimensional generalization of the Sachdev-Ye-Kitaev model. The model contains $N$ Majorana fermions at each lattice site with a nearest-neighbour hopping term. The SYK random interaction is restricted to low momentum fermions of definite chirality within each lattice site. This gives rise to an ordinary 1+1 field theory above some energy scale and a low energy SYK-like behavior. We exhibit a class of low-pass filters which give rise to a rich variety of hyperscaling behaviour in the IR. We also discuss another set of generalizations which describes probing an SYK system with an external fermion, together with the new scaling behavior they exhibit in the IR.

Figures (2)

• Figure 1: Diagrams contributing to the $\eta$ propagator.
• Figure 2: Diagram contributing to the $\rho$ propagator.