  We investigate entanglement phase transitions from volume-law to area-law
entanglement in a quantum many-body state under continuous position measurement
on the basis of the quantum trajectory approach. We find the signatures of the
transitions as peak structures in the mutual information as a function of
measurement strength, as previously reported for random unitary circuits with
projective measurements. At the transition points, the entanglement entropy
scales logarithmically and various physical quantities scale algebraically,
implying emergent conformal criticality, for both integrable and nonintegrable
one-dimensional interacting Hamiltonians; however, such transitions have been
argued to be absent in noninteracting regimes in some previous studies. With
the aid of $U(1)$ symmetry in our model, the measurement-induced criticality
exhibits a spectral signature resembling a Tomonaga-Luttinger liquid theory
from symmetry-resolved entanglement. These intriguing critical phenomena are
unique to steady-state regimes of the conditional dynamics at the
single-trajectory level, and are absent in the unconditional dynamics obeying
the Lindblad master equation, in which the system ends up with the featureless,
infinite-temperature mixed state. We also propose a possible experimental setup
to test the predicted entanglement transition based on the subsystem
particle-number fluctuations. This quantity should readily be measured by the
current techniques of quantum gas microscopy and is in practice easier to
obtain than the entanglement entropy itself.
