  We derive field theory descriptions for measurement-induced phase transitions
in free fermion systems. We focus on a multi-flavor Majorana chain, undergoing
Hamiltonian evolution with continuous monitoring of local fermion parity
operators. Using the replica trick, we map the dynamics to the imaginary time
evolution of an effective spin chain, and use the number of flavors as a large
parameter for a controlled derivation of the effective field theory. This is a
nonlinear sigma model for an orthogonal $N\times N$ matrix, in the replica
limit $N\to 1$. (On a boundary of the phase diagram, another sigma model with
higher symmetry applies.) Together with known results for the
renormalization-group beta function, this derivation establishes the existence
of stable phases -- nontrivially entangled and disentangled respectively -- in
the physically-relevant replica limit $N\to 1$. In the nontrivial phase, an
asymptotically exact calculation shows that the bipartite entanglement entropy
for a system of size $L$ scales as $(\log L)^2$, in contrast to findings in
previously-studied models. Varying the relative strength of Hamiltonian
evolution and monitoring, as well as a dimerization parameter, the model's
phase diagram contains transitions out of the nontrivial phase, which we map to
vortex-unbinding transitions in the sigma model, and also contains separate
critical points on the measurement-only axis. We highlight the close analogies
as well as the differences with the replica approach to Anderson transitions in
disordered systems.
