  We develop an analytical approach to the study of one-dimensional free
fermions subject to random projective measurements of local site occupation
numbers, based on the Keldysh path-integral formalism and replica trick. In the
limit of rare measurements, $\gamma / J \ll 1$ (where $\gamma$ is measurement
rate per site and $J$ is hopping constant in the tight-binding model), we
derive a non-linear sigma model (NLSM) as an effective field theory of the
problem. Its replica-symmetric sector is described by a $U(2) / U(1) \times
U(1) \simeq S_2$ sigma model with diffusive behavior, and the
replica-asymmetric sector is a two-dimensional NLSM defined on $SU(R)$ manifold
with the replica limit $R \to 1$. On the Gaussian level, valid in the limit
$\gamma / J \to 0$, this model predicts a logarithmic behavior for the second
cumulant of number of particles in a subsystem and for the entanglement
entropy. However, the one-loop renormalization group analysis allows us to
demonstrate that this logarithmic growth saturates at a finite value $\sim (J /
\gamma)^2$ even for rare measurements, which corresponds to the area-law phase.
This implies the absence of a measurement-induced entanglement phase transition
for free fermions. The crossover between logarithmic growth and saturation,
however, happens at exponentially large scale, $\ln l_\text{corr} \sim J /
\gamma$. This makes this crossover very sharp as a function of the measurement
frequency $\gamma / J$, which can be easily confused with a transition from the
logarithmic to area law in finite-size numerical calculations. We have
performed a careful numerical analysis, which supports our analytical
predictions.
