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, 𝐺𝐴⁢𝐵 , and the two-point density correlation function, 𝒞⁡(𝑟) . Our results exhibit a remarkable analogy to Anderson localization, with 𝐺𝐴⁢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, 𝐺𝐴⁢𝐵 exhibits “universal,” nearly Gaussian, fluctuations with variance of order unity. In the localized phase, we find a broad distribution of 𝐺𝐴⁢𝐵 with −ln⁡𝐺𝐴⁢𝐵̄ ∼𝐿 (where 𝐿 is the system size) and the variance var⁡(ln⁡𝐺𝐴⁢𝐵)∼𝐿^𝜇, and similarly for 𝒞⁡(𝑟), with 𝜇≈0.5. At the transition point, the distribution function of 𝐺𝐴⁢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.