Spectral form factor in chaotic, localized, and integrable open quantum many-body systems

TL;DR

This paper investigates the spectral statistics of open quantum many-body systems using the dissipative spectral form factor, revealing universal random matrix theory behaviors in chaotic regimes and distinct signatures in non-chaotic and localized phases.

Contribution

It introduces the dissipative spectral form factor as a tool to identify quantum chaos in open systems and demonstrates RMT universality across various models, including systems without Hamiltonian dynamics.

Findings

Chaotic OQMBS exhibit quadratic ramp-plateau behavior consistent with Ginibre ensemble.

The many-body Thouless time scales with system size and marks the emergence of RMT behavior.

Non-chaotic and localized systems show spectral statistics distinct from RMT predictions.

Abstract

We numerically study the spectral statistics of open quantum many-body systems (OQMBS) as signatures of quantum chaos (or the lack thereof), using the dissipative spectral form factor (DSFF), a generalization of the spectral form factor to complex spectra. We show that the DSFF of chaotic OQMBS generically displays the $\textit{quadratic}$ ramp-plateau behaviour of the Ginibre ensemble from random matrix theory, in contrast to the linear ramp-plateau behaviour of the Gaussian ensemble in closed quantum systems. Furthermore, in the presence of many-body interactions, such RMT behaviour emerges only after a time scale $\tau_{\mathrm{dev}}$, which generally increases with system size for sufficiently large system size, and can be identified as the non-Hermitian analogue of the $\textit{many-body Thouless time}$. The universality of the random matrix theory behavior is demonstrated by surveying twelve models of OQMBS, including random Kraus circuits (quantum channels) and random Lindbladians (Liouvillians) in several symmetry classes, as well as Lindbladians of paradigmatic models such as the Sachdev-Ye-Kitaev (SYK), XXZ, and the transverse field Ising models. We devise an unfolding and filtering procedure to remove variations of the averaged density of states which would otherwise hide the universal RMT-like signatures in the DSFF for chaotic OQMBS. Beyond chaotic OQMBS, we study the spectral statistics of non-chaotic OQMBS, specifically the integrable XX model and a system in the many-body localized (MBL) regime in the presence of dissipation, which exhibit DSFF behaviours distinct from the ramp-plateau behaviour of random matrix theory. Lastly, we study the DSFF of Lindbladians with the Hamiltonian term set to zero, i.e. only the jump operators are present, and demonstrate that the results of RMT universality and scaling of many-body Thouless time survive even without coherent evolution.