  The entanglement entropy (EE) of the ground state of a one-dimensional
Hamiltonian at criticality has a universal logarithmic scaling with a prefactor
given by the central charge $c$ of the underlying 1+1d conformal field theory.
When the system is probed by measurements, the entanglement in the critical
ground state is inevitably affected due to wavefunction collapse. In this
paper, we study the effect of weak measurements on the entanglement scaling in
the ground state of the one-dimensional critical transverse-field Ising model.
For the measurements of the spins along their transverse spin axis, we identify
interesting post-measurement states associated with spatially uniform
measurement outcomes. The EE in these states still satisfies the logarithmic
scaling but with an alternative prefactor given by the effective central charge
$c_{\text{eff}}$. We derive the analytical expression of $c_{\text{eff}}$ as a
function of the measurement strength. Using numerical simulations, we show that
for the EE averaged over all post-measurement states based on their Born-rule
probabilities, the numerically extracted effective central charge appears to be
independent of the measurement strength, contrary to the usual expectation that
local and non-overlapping measurements reduce the entanglement in the system.
We also examine the behavior of the average EE under (biased) forced
measurements where the measurement outcomes are sampled with a pre-determined
probability distribution without inter-site correlations. In particular, we
find an optimal probability distribution that can serve as a mean-field
approximation to the Born-rule probabilities and lead to the same
$c_{\text{eff}}$ behavior. The effects of the measurements along the
longitudinal spin axis and the post-measurement correlation functions are also
discussed.
