Mesoscopic fluctuations and multifractality at and across measurement-induced phase transitions

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 A B , and the two-point density correlation function, C ( r ) . Our results exhibit a remarkable analogy to Anderson localization, with G A B corresponding to two-terminal conductance and C ( r ) to two-point conductance, albeit with different replica limits and unconventional symmetry class, geometry, and boundary conditions. In the delocalized phase, G A B exhibits “universal,” nearly Gaussian, fluctuations with variance of order unity. In the localized phase, we find a broad distribution of G A B with − ln G A B ¯ ∼ L (where L is the system size) and the variance var ( ln G A B ) ∼ L μ , and similarly for C ( r ) , with μ ≈ 0.5 . At the transition point, the distribution function of G A B becomes scale invariant and C ( r ) exhibits multifractal statistics, C q ( r ) ¯ ∼ r − q ( d + 1 ) − Δ q . We characterize the spectrum of multifractal dimensions Δ q . Our findings lay the groundwork for mesoscopic theory of monitored systems, paving the way for various extensions.