标题： 介观涨落和测量诱导相变及其跨相变的多重分形性 显示英文标题

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

摘要： 我们研究了在二维自由费米子模型中，对局域电荷进行投影监测时，在测量诱导相变的量子轨迹系综中的统计涨落。 我们的可观测量是空间分离区域之间的粒子数协方差，$G_{AB}$，以及两点密度关联函数，$\mathcal{C}(r)$。 我们的结果与安德森局域表现出显著的类比，其中$G_{AB}$对应于两终端电导，$\mathcal{C}(r)$对应于两点电导，尽管具有不同的复制品极限和非常规的对称性类、几何结构和边界条件。 在非局域化相中，$G_{AB}$表现出“普遍”的、近似高斯的涨落，其方差为单位量级。 在局域相中，我们发现$G_{AB}$的分布很广，其中$\overline{-\ln G_{AB}} \sim L $（$L$为系统大小）和方差$\mathrm{var}(\ln G_{AB}) \sim L^\mu$，同样对于$\mathcal{C}(r)$，有$\mu \approx 0.5$。 在临界点，$G_{AB}$的分布函数变为尺度不变，并且$\mathcal{C}(r)$展现出多分形统计特性，$\overline{\mathcal{C}^{q}(r)}\sim r^{-q(d+1) - \Delta_{q}}$。我们表征了多分形维数的谱$\Delta_q$。我们的发现为受监控系统的介观理论奠定了基础，为各种扩展铺平了道路。 显示英文摘要

摘要： 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.