Mesoscopic Fluctuations and Multifractality at and across Measurement-Induced Phase Transition

We explore statistical fluctuations over the ensemble of quantum trajectories in a model of two-dimensional free fermions subject to projective monitoring of local charge across the measurement-induced phase transition. Our observables are the particle-number covariance between spatially separated regions, $G_{AB}$, and the two-point density correlation function, $\mathcal{C}(r)$. Our results exhibit a remarkable analogy to Anderson localization, with $G_{AB}$ corresponding to two-terminal conductance and $\mathcal{C}(r)$ to two-point conductance, albeit with different replica limit and unconventional symmetry class, geometry, and boundary conditions. In the delocalized phase, $G_{AB}$ exhibits ``universal’', nearly Gaussian, fluctuations with variance of order unity. In the localized phase, we find a broad distribution of $G_{AB}$ with $\overline{-\ln G_{AB}}\sim L$ (where $L$ is the system size) and the variance $\mathrm{v}\mathrm{a}\mathrm{r}(\ln G_{AB})\sim L^{\mu }$, and similarly for $\mathcal{C}(r)$, with $\mu \approx 0.5$. At the transition point, the distribution function of $G_{AB}$ becomes scale-invariant and $\mathcal{C}(r)$ exhibits multifractal statistics, $\overline{{\mathcal{C}}^q(r)}\sim r^{-q(d+1)-{\mathrm{\Delta }}_q}$. We characterize the spectrum of multifractal dimensions ${\mathrm{\Delta }}_q$. Our findings lay the groundwork for mesoscopic theory of monitored systems, paving the way for various extensions.