SYK Correlators for All Energies

TL;DR

This work analyzes the SYK model in the large-q, large-N limit and shows that the theory reduces to a broken Liouville description, enabling universal closed-form expressions for two- and four-point correlators at arbitrary temperature. By introducing twisted bi-local boundary conditions, the authors derive all-energy two-point functions and obtain a connected four-point function driven by the twisted two-point data, with a finite-domain symmetry reduction. They demonstrate that time-ordered correlators are governed by finite-temperature OPEs into the identity and Hamiltonian, while out-of-time-ordered correlators scramble with a Lyapunov exponent λ_L = 2πv/β, and crucially the thermalization and scrambling rates coincide for all energies. The results unify high- and low-energy behaviors via a single parameter v(βJ) and reinforce the view that simple, epidemic-like operator-growth structures underlie the complex internal dynamics of SYK at large N and large q.

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.