Symmetry classes, many-body zero modes, and supersymmetry in the complex Sachdev-Ye-Kitaev model

TL;DR

This paper classifies the complex Sachdev-Ye-Kitaev model's symmetry classes within the Altland-Zirnbauer framework, revealing that chiral symmetry induces many-body zero modes and supersymmetry without fine-tuning, with implications for long-time dynamics.

Contribution

It embeds the cSYK model's symmetry classes into the Altland-Zirnbauer framework and uncovers supersymmetry arising from chiral symmetry in charge-conserving fermionic systems.

Findings

Identification of symmetry classes within the Altland-Zirnbauer framework.

Existence of many-body zero modes for odd fermion number.

Long-time correlation plateaus indicating supersymmetry.

Abstract

The complex Sachdev-Ye-Kitaev (cSYK) model is a charge-conserving model of randomly interacting fermions. The interaction term can be chosen such that the model exhibits chiral symmetry. Then, depending on the charge sector and the number of interacting fermions, level spacing statistics suggests a fourfold categorization of the model into the three Wigner-Dyson symmetry classes. In this work, inspired by previous findings for the Majorana Sachdev-Ye-Kitaev model, we embed the symmetry classes of the cSYK model in the Altland-Zirnbauer framework and identify consequences of chiral symmetry originating from correlations across different charge sectors. In particular, we show that for an odd number of fermions, the model hosts exact many-body zero modes that can be combined into a generalized fermion that does not affect the system's energy. This fermion directly leads to quantum-mechanical supersymmetry that, unlike explicitly supersymmetric cSYK constructions, does not require fine-tuned couplings, but only chiral symmetry. Signatures of the generalized fermion, and thus supersymmetry, include the long-time plateau in time-dependent correlation functions of fermion-parity-odd observables: The plateau may take nonzero value only for certain combinations of the fermion structure of the observable and the system's symmetry class. We illustrate our findings through exact diagonalization simulations of the system's dynamics.