No measurement induced phase transition in the entanglement dynamics of monitored non-interacting one-dimensional fermions in a disordered or quasiperiodic potential

We show that the entanglement entropy (EE) of one-dimensional (1d) non-interacting fermions with $U(1)$ symmetry in the presence of a disordered or quasi-periodic potential in which the occupation number is being monitored by homodyne or projective protocols is always in an area-law phase so no measurement induced phase transition (MIPT) occurs. The reason for the previously claimed MIPT in these systems was a finite size effect related to the fact that the maximum lattice size $L \sim 500$ was of the order of the correlation length. By increasing the system size up to $L \leq 18000$, employing Graphics Processing Unit (GPU), and performing a careful finite size scaling analysis, we find that the critical monitoring strength is consistent with zero so no MIPT occurs. For the disordered case, these numerical results are fully supported by an analytical calculation based on mapping the problem onto a nonlinear sigma model (NLSM) that confirms the absence of the MIPT for any monitoring or disorder strength. The effect of disorder is captured by a change of symmetry, from BDI to AIII, which results in an enhanced correlation length in the weak disorder limit and, by an effective monitoring strength that increase linearly with disorder.