  We construct Brownian Sachdev-Ye-Kitaev (SYK) chains subjected to continuous
monitoring and explore possible entanglement phase transitions therein. We
analytically derive the effective action in the large-$N$ limit and show that
an entanglement transition is caused by the symmetry breaking in the enlarged
replica space. In the noninteracting case with SYK$_2$ chains, the model
features a continuous $O(2)$ symmetry between two replicas and a transition
corresponding to spontaneous breaking of that symmetry upon varying the
measurement rate. In the symmetry broken phase at low measurement rate, the
emergent replica criticality associated with the Goldstone mode leads to a
log-scaling entanglement entropy that can be attributed to the free energy of
vortices. In the symmetric phase at higher measurement rate, the entanglement
entropy obeys area-law scaling. In the interacting case, the continuous $O(2)$
symmetry is explicitly lowered to a discrete $C_4$ symmetry, giving rise to
volume-law entanglement entropy in the symmetry-broken phase due to the
enhanced linear free energy cost of domain walls compared to vortices. The
interacting transition is described by $C_4$ symmetry breaking. We also verify
the large-$N$ critical exponents by numerically solving the Schwinger-Dyson
equation.
