  We investigate the onset of thermalization and quantum chaos in finite
one-dimensional gapped systems of hard-core bosons. Integrability in these
systems is broken by next-nearest-neighbor repulsive interactions, which also
generate a superfluid to insulator transition. By employing full exact
diagonalization, we study chaos indicators and few-body observables. We show
that with increasing system size, chaotic behavior is seen over a broader range
of parameters and, in particular, deeper into the insulating phase.
Concomitantly, we observe that, as the system size increases, the eigenstate
thermalization hypothesis extends its range of validity inside the insulating
phase and is accompanied by the thermalization of the system.
