# The tenfold way and many-body zero modes in the Sachdev-Ye-Kitaev model

## TL;DR

This paper classifies the SYK model into the tenfold Altland-Zirnbauer symmetry classes, revealing how symmetry-dependent zero modes influence spectral and dynamical properties, thus extending understanding of many-body quantum chaos.

## Contribution

It demonstrates the emergence of the tenfold symmetry classification in the SYK model and analyzes the impact of zero modes on spectral features and dynamics.

## Key findings

- Zero modes appear in most symmetry classes and affect spectral functions.
- Signatures include zero-energy peaks or holes in spectral functions.
- Zero modes influence long-time behavior of out-of-time-order correlators.

## Abstract

The Sachdev-Ye-Kitaev (SYK) model, in its simplest form, describes $k$ Majorana fermions with random all-to-all four-body interactions. We consider the SYK model in the framework of a many-body Altland-Zirnbauer classification that sees the system as belonging to one of eight (real) symmetry classes depending on the value of $k\mod 8$. We show that, depending on the symmetry class, the system may support exact many-body zero modes with the symmetries also dictating whether these may have a nonzero contribution to Majorana fermions, i.e., single-particle weight. These zero modes appear in all but two of the symmetry classes. When present, they leave clear signatures in physical observables that go beyond the threefold (Wigner-Dyson) possibilities for level spacing statistics studied earlier. Signatures we discover include a zero-energy peak or hole in the single-particle spectral function, depending on whether symmetries allow or forbid zero modes to have single-particle weight. The zero modes are also shown to influence the many-body dynamics, where signatures include a nonzero long-time limit for the out-of-time-order correlation function. Furthermore, we show that the extension of the four-body SYK model by quadratic terms can be interpreted as realizing the remaining two complex symmetry classes; we thus demonstrate how the entire tenfold Altland-Zirnbauer classification may emerge in the SYK model.