Dynamics of spectral correlations in the entanglement Hamiltonian of the Aubry-Andr\'e-Harper model

TL;DR

This paper investigates how spectral correlations in the entanglement Hamiltonian of the Aubry-Andre9-Harper model evolve over time, revealing phase-dependent behaviors and correlations with entanglement entropy.

Contribution

It provides a systematic analysis of spectral correlation dynamics in the entanglement Hamiltonian across delocalized and localized phases, highlighting three distinct timescales in the evolution.

Findings

Spectral form factor evolves in three intervals in delocalized and transition phases.

Number of timescales remains three unless in localized phase or with maximally entangled initial states.

Strong correlation between entanglement entropy and the SFF ramp length.

Abstract

We numerically study the evolution of spectral correlations in the entanglement Hamiltonian (EH) of non-interacting fermions in the Aubry-Andr\'e-Harper (AAH) model. We analyze the time evolution of the EH spectrum in a nonequilibrium setting by studying several quantities: spectral distribution, level statistics, entanglement entropy, and spectral form factor (SFF) in the context of the delocalization-localization transition in the AAH model. It is observed that the SFF of the entanglement spectrum in the delocalized phase and at the phase-transition point evolves in three-time intervals. We make a systematic study of the emergence of these three timescales for various initial states and find that the number of time intervals remains three unless the Hamiltonian is tuned in the localized phase or when the initial state is maximally entangled, then there is a featureless time evolution. We find a broad direct correlation between the entanglement entropy and the length of the ramp of the SFF. We also find that in the delocalized phase the spectral correlations are stronger in the center of the spectrum and grow progressively weaker as more and more of the spectrum is considered.