Fluctuations in the delocalization level of eigenstates and thermalization

Using full exact diagonalization, we study the transition to chaos in one-dimensional gapless bosonic and fermionic systems. We show that the strength of the integrability-breaking term required for the emergence of a Wigner-Dyson distribution and for reduced fluctuations of delocalization measures decreases with system size. We discuss how these results relate to the onset of thermalization in isolated quantum many-body systems.