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 $\overline{−ln⁡𝐺𝐴⁢𝐵}$ ∼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, $\overline{𝒞^𝑞⁡(𝑟)}$∼𝑟$^{−𝑞⁢(𝑑+1)−Δ_𝑞}$
. We characterize the spectrum of multifractal dimensions Δ$_𝑞$. ... mehrOur findings lay the groundwork for mesoscopic theory of monitored systems, paving the way for various extensions.