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 \({\rm d}\) 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 \({\rm d}>1\) from a \(\textit{delocalized}\) to a \(\textit{localized}\) phase as the measurement strength \(\gamma\) is increased beyond a critical value \(\gamma_{c}\). In the language of surface growth, the delocalized phase corresponds to a \(\textit{smooth}\) phase while the localized phase corresponds to a \(\textit{rough}\) phase. We support our numerical results with perturbative renormalization group (RG) computations which are in qualitative agreement at one-loop order.