Mesoscopic Fluctuations and Multifractality at and across Measurement-Induced Phase Transition

TL;DR

This paper investigates fluctuations and multifractality in quantum trajectories across measurement-induced phase transitions, revealing analogies to Anderson localization and characterizing critical multifractal behavior.

Contribution

It introduces a mesoscopic framework for understanding fluctuations and multifractality in monitored quantum systems at phase transitions, with novel analysis of distribution functions and multifractal spectra.

Findings

In the delocalized phase, fluctuations are nearly Gaussian with variance of order unity.

In the localized phase, the distribution of $G_{AB}$ broadens with system size, showing scale-dependent variance.

At the transition, $G_{AB}$ becomes scale-invariant and $ ext{C}(r)$ exhibits multifractal statistics.

Abstract

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 $\mathrm{var}(\ln G_{AB}) \sim L^\mu$, and similarly for $\mathcal{C}(r)$, with $\mu \approx 0.5$. At the transition point, the distribution function of $G_{AB}$ becomes scale-invariant and $\mathcal{C}(r)$ exhibits multifractal statistics, $\overline{\mathcal{C}^{q}(r)}\sim r^{-q(d+1) - \Delta_{q}}$. We characterize the spectrum of multifractal dimensions $\Delta_q$. Our findings lay the groundwork for mesoscopic theory of monitored systems, paving the way for various extensions.