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⁢𝐵 ∼𝐿 (where 𝐿 is the system size) and the variance var⁡ (ln⁡𝐺👳⁢👴) ∼𝐿^𝜇, and similarly for 𝒞⁡ (𝑟), with 𝜇≈0. 5. At the transition point, the distribution function of 𝐺👳⁢👴 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.