# Randomness and Chaos in Qubit Models

## TL;DR

This paper investigates how introducing randomness affects spectral properties in quantum models, revealing that certain spectral behaviors indicate the presence of quantum chaos and randomness, supported by numerical analysis of various models.

## Contribution

It demonstrates that the dip-ramp-plateau spectral form factor behavior signals random eigenvectors in quantum many-body systems, supported by analysis of SYK and Ising models.

## Key findings

- Spectral form factors show dip-ramp-plateau in SYK$_2$ at high temperatures.
- Consistent results in Gaussian random transverse Ising model.
- Late-time observable expectations match thermal ensemble in a random qubit model.

## Abstract

We introduce randomness into a class of integrable models and study the spectral form factor as a diagnostic to distinguish between randomness and chaos. Spectral form factors exhibit a characteristic dip-ramp-plateau behavior in the $N>2$ SYK$_2$ model at high temperatures that is absent in the $N=2$ SYK$_2$ model. Our results suggest that this dip-ramp-plateau behavior implies the existence of random eigenvectors in a quantum many-body system. To further support this observation, we examine the Gaussian random transverse Ising model and obtain consistent results without suffering from small $N$ issues. Finally, we demonstrate numerically that expectation values of observables computed in a random quantum state at late times are equivalent to the expectation values computed in the thermal ensemble in a Gaussian random one-qubit model.