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 $\mathrm{var}(\ln G_{AB}) \sim L^μ$, and similarly for $\mathcal{C}(r)$, with $μ\approx 0.5$. At the transition point, the distribution function of $G_{AB}$ becomes scale-invariant and $\mathcal{C}(r)$ exhibits multifractal statistics, $\overline{\mathcal{C}^{q}(r)}\sim r^{-q(d+1) - Δ_{q}}$. 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.

💡 Research Summary

In this work the authors investigate statistical fluctuations of quantum‑trajectory ensembles in a two‑dimensional free‑fermion lattice subject to local projective measurements of the on‑site charge. The central observables are the particle‑number covariance between two spatially separated regions, (G_{AB}), and the two‑point density correlation function, (\mathcal{C}(r)). By mapping the monitored dynamics onto a replica non‑linear sigma model (NLSM) they establish a deep analogy with Anderson localization in one higher dimension, albeit with a different replica limit, symmetry class, geometry and boundary conditions.

The model consists of a nearest‑neighbour hopping Hamiltonian on a square lattice (set (J=1)) together with stochastic projective measurements of (\hat n_x) at rate (\gamma). The state remains Gaussian, allowing a full description by the correlation matrix. The system is prepared in a half‑filled Slater determinant and evolved to a steady state independent of the initial condition. For each quantum trajectory the authors compute (G_{AB}) for two macroscopic regions of size (L/4\times L) separated by a distance of order (L), and the maximally separated density correlator (C_L).

In the diffusive (low‑measurement) regime ((\gamma=0.5,1.5)) the distribution (P(G_{AB})) is essentially Gaussian with a variance (\mathrm{var}(G_{AB})\approx8.6\times10^{-3}) that does not depend on system size or measurement strength. This reproduces the well‑known “universal conductance fluctuations” of mesoscopic physics. The normalized correlator (z=C_L/\langle C_L\rangle) follows a Porter‑Thomas law for weak monitoring, confirming analytical predictions from the NLSM.

In the localized (high‑measurement) regime ((\gamma=4.5)) the typical value of the covariance decays exponentially, (\langle\ln G_{AB}\rangle\sim -L/(4\ell_{\text{typ}}^{\text{loc}})), while the variance of (\ln G_{AB}) grows as a power law (\mathrm{var}(\ln G_{AB})\propto L^{\mu}) with (\mu\approx0.58). An analogous scaling is observed for (\ln \mathcal{C}(r)) with (\mu\approx0.5). These findings mirror the log‑normal conductance statistics of three‑dimensional Anderson localization and suggest a possible connection to the KPZ growth exponent in 2+1 dimensions.

At the critical measurement rate (\gamma_c\simeq2.93) the distribution of (\ln G_{AB}) becomes scale‑invariant, i.e. independent of (L). More strikingly, the moments of the density correlator exhibit multifractal scaling, \