  Interspersing unitary dynamics with local measurements results in
measurement-induced phases and transitions in many-body quantum systems. When
the evolution is driven by a local Hamiltonian, two types of transitions have
been observed, characterized by an abrupt change in the system size scaling of
entanglement entropy. The critical point separates the strongly monitored
area-law phase from a volume law or a sub-extensive, typically logarithmic-like
one at low measurement rates. Identifying the key ingredients responsible for
the entanglement scaling in the weakly monitored phase is the key purpose of
this work. For this purpose, we consider prototypical one-dimensional spin
chains with local monitoring featuring the presence/absence of U(1) symmetry,
integrability, and interactions. Using exact numerical methods, the system
sizes studied reveal that the presence of interaction is always correlated to a
volume-law weakly monitored phase. In contrast, non-interacting systems present
sub-extensive scaling of entanglement. Other characteristics, namely
integrability or U(1) symmetry, do not play a role in the character of the
entanglement phase.
