The description of the entanglement dynamics of monitored noninteracting fermions, including the existence of measurement-induced phase transitions (MIPTs), is a challenging problem with conflicting results in the literature. The mapping of the problem onto a nonlinear sigma model (NLSM) indicates that relatively large lattice sizes are required to determine the nature of the entanglement entropy in the thermodynamics limit. Here we address this problem numerically for monitored noninteracting fermions with $U\left(1\right)$ symmetry. The use of graphics processing unit techniques, even with outdated hardware, makes it possible to reach much larger lattice sizes $[L=16384$ and $160\times 160$ in one (1D) and two (2D) dimensions, respectively] than in previous studies, which enables us to characterize quantitatively the entanglement dynamics. In 1D, we show that to confirm the absence of a MIPT, for both projective and homodyne measurements, predicted by the NLSM, it is necessary to reach $L\sim 10000$. In 2D, also as predicted by the NLSM, we observe for both protocols a MIPT at a finite monitoring rate characterized by scale invariant mutual information. The critical monitoring strength depends on the protocol while the critical exponent $\nu \approx 1.3$ governing the approach to the MIPT is similar in both cases. These features are not correctly predicted by the NLSM. Our results pave the way for a fully quantitative description of the entanglement dynamics of monitoring quantum systems.

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