# On the replica structure of Sachdev-Ye-Kitaev model

## TL;DR

This paper examines the structure of the Sachdev-Ye-Kitaev model, focusing on whether replica off-diagonal solutions exist, and finds numerical evidence supporting the dominance of replica-diagonal solutions over off-diagonal ones.

## Contribution

It provides a detailed analysis comparing replica-diagonal and off-diagonal solutions, concluding that the former accurately describes the model's long-time behavior.

## Key findings

- Numerical results agree with replica-diagonal saddle point predictions.
- Replica-off-diagonal solutions are unlikely to contribute significantly at short time scales.
- The structure of the Schwarzian action differs qualitatively between the two scenarios.

## Abstract

We investigate existence of replica off-diagonal solutions in the field-theoretical description of Sachdev-Ye-Kitaev model. To this end we evaluate a set of local and non-local dynamic correlation functions in the long time limit. We argue that the structure of the soft-mode Schwarzian action is qualitatively different in replica-diagonal vs. replica-off-diagonal scenarios, leading to distinct long-time predictions for the correlation functions. We then evaluate the corresponding correlation functions numerically and compare the simulations with analytical predictions of replica-diagonal and replica-off-diagonal calculations. We conclude that all our numerical results are in a quantitative agreement with the theory based on the replica-diagonal saddle point plus Schwarzian and massive Gaussian fluctuations (the latter do contain replica off-diagonal components). This seems to exclude any contributions from replica-off-diagonal saddle points, at least on the time scales shorter than the inverse many-body level spacing.