Near Conformal Perturbation Theory in SYK Type Models

TL;DR

This paper develops a systematic perturbation theory framework to analyze the low energy dynamics and correlation functions of SYK models, emphasizing the role of conformal symmetry breaking and the Schwarzian action.

Contribution

It introduces a novel perturbation scheme for SYK models that accurately captures the effects of conformal symmetry breaking on correlation functions and the effective Schwarzian action.

Findings

Confirmed agreement with explicit symmetry breaking calculations

Derived $O(1)$ corrections to bi-local correlation functions

Clarified the origin of subleading contributions in the large $q$ limit

Abstract

We present a systematic procedure to extract the dynamics of the low energy soft mode in SYK type models with a single energy scale $J$ and emergent reparametrization symmetry in the IR. This is given in the framework of the perturbation theory scheme of arXiv:1608.07567 based on specific (off-shell) breaking of conformal invariance in the UV, adjusted to yield the exact large-$N$ saddle point. While this breaking formally vanishes on-shell, it has a non-trivial effect on correlation functions and the effective action. In particular, it leads to the Schwarzian action with a specific coupling to bi-local matter. The method is applied to the evaluation of $O(1)$ corrections to the correlation function of bi-locals. As a byproduct we confirm precise agreement with the explicit, symmetry breaking procedure. We provide a verification in the large $q$ limit (Liouville theory), where the correlators can be calculated exactly at all length scales. In this case, our scheme illuminates how the enhanced $O(J)$ and the subleading $O(1)$ contributions originate from the Schwarzian dynamics of the soft mode and its interaction with $h=2$ (bi-local) matter.