Supersymmetric SYK models

TL;DR

This paper introduces supersymmetric generalizations of the SYK model, analyzing their supersymmetry properties, spectra, and low-energy effective actions, revealing differences between ${ m N}=1$ and ${ m N}=2$ cases.

Contribution

It presents the first detailed study of supersymmetric SYK models, including their large N behavior, spectrum, and supersymmetry breaking patterns.

Findings

${ m N}=1$ model has unbroken supersymmetry at large N but breaks non-perturbatively.

${ m N}=2$ model preserves supersymmetry and matches the Witten index with entropy.

Both models exhibit supersymmetric Schwarzian actions at low energies.

Abstract

We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev model. These are quantum mechanical models involving $N$ Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The ${\cal N}=1$ model with a single supercharge has unbroken supersymmetry at large $N$, but non-perturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large $N$ equation, and also by performing numerical computations for small values of $N$. We also compute the large $N$ spectrum of "singlet" operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an ${\cal N}=2$ version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large $N$ computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.