Measurement-induced phase transition in a single-body tight-binding model

We study the statistical properties of a single free quantum particle evolving coherently on a discrete lattice in $\mathit{\text{d}}$ spatial dimensions where every lattice site is additionally subject to continuous measurement of the occupation number. Our numerical results indicate that the system undergoes a measurement-induced phase transition (MiPT) for $\mathit{\text{d}}>1$ from a delocalized to a localized phase as the measurement strength $\ensuremath{\gamma}$ is increased beyond a critical value ${\ensuremath{\gamma}}_{c}$. In the language of surface growth, the delocalized phase corresponds to a smooth phase while the localized phase corresponds to a rough phase. We support our numerical results with perturbative renormalization group (RG) computations which are in qualitative agreement at one-loop order.