Exploring the Infrared Landscape of the SYK Model

Abstract

We analyse a class of SYK models whose Hamiltonian is the sum of two SYK Hamiltonians with different numbers of fermions $q, \tilde q$ in each interaction. We consider both Euclidean and Lorentzian probes of the quantum system in the large $N$ limit. In the strong coupling phase, the entropy provides a diagnostic of the thermal renormalisation group flow. Under certain conditions, two parametrically separated regimes of near-conformal behaviour emerge. The first reproduces the standard linear-in-temperature scaling characteristic of the single SYK model. The system then flows to another near-fixed point whose entropy scaling depends on the ratio $n = q/\tilde q$. For $n<3/2$, the entropy exhibits anomalous, stronger-than-linear scaling in temperature. At $n=3/2$, there is an additional logarithmic enhancement. Using conformal perturbation theory, we argue that in the infrared regime of the SYK model, there may exist disordered conformal operators with dimensions $1 < \Delta \leq 3/2$. In Lorentzian signature, we study the out-of-time-ordered correlator and show that these deformed theories exhibit near-maximal chaos in both regimes (when they exist). We comment on the relation between the anomalous scalings found here and those observed in certain near-extremal black holes in two and higher dimensions.