SYK Correlators for All Energies

TL;DR

This paper derives universal closed-form expressions for SYK model correlators at all energies, revealing insights into thermalization, scrambling, and the model's internal dynamics across temperature regimes.

Contribution

It provides novel, exact correlator formulas valid for all energies in the large q limit, bridging low and high-energy behaviors and confirming scrambling properties.

Findings

Correlators take a universal closed form at all energies.

Time-ordered correlators are determined by OPEs into identity and Hamiltonian.

Out-of-time-order correlators always scramble, confirming nontrivial dynamics.

Abstract

The Sachdev-Ye-Kitaev (SYK) model, a theory of N Majorana fermions with q-body interactions, becomes in the large q limit a conformally-broken Liouville field theory. Taking this limit preserves many interesting properties of the model, yet makes the theory as a whole much more tractable. Accordingly, we produce novel expressions for the two and four-point correlators at arbitrary temperature and find the surprising result they take a universal closed form. We note that these expressions correctly match onto and interpolate between previously-obtained low-energy results and simple high-energy perturbative checks. We find that the time-ordered four-point correlators are always determined by finite temperature OPEs into the identity and Hamiltonian, while the out-of-time-order four-point correlators remain nontrivial and always scramble. This has only been established in the conformal limit, so to find that it holds for large q at all temperatures/couplings is a nontrivial result. Finally, we determine the system's thermalization and scrambling rates and find that they always agree, regardless of temperature. This adds to the increasing body of evidence that there exists simple structures in large N internal dynamics, such as those formed by SYK's epidemic operator growth.