Code and Data for "Topology and Spectrum in Measurement-Induced Phase Transitions"

Competition among repetitive measurements of noncommuting observables and unitary dynamics can give rise to a wide variety of entanglement phases. Here, we characterize topological phases in monitored quantum systems by their spectrum and many-body topological invariants. We analyze (1+1)-dimensional monitored circuits for Majorana fermions, which have topological and trivial area-law entangled phases and a critical phase with sub-volume-law entanglement, through the Lyapunov spectrum. We uncover the presence (absence) of edge-localized zero modes inside the bulk gap in the topological (trivial) area-law phase and a bulk gapless spectrum in the critical phase. Furthermore, by suitably exploiting the fermion parity with twisted measurement outcomes at the boundary, we construct a topological invariant that sharply distinguishes the two area-law phases and dynamically characterizes the critical phase. Our work thus paves the way to extend the bulk-edge correspondence for topological phases from equilibrium to monitored quantum dynamics.非对易可观测量（noncommuting observables）的重复测量与幺正演化（unitary dynamics）之间的竞争，可催生种类丰富的纠缠相。本研究借助能谱与多体拓扑不变量，对受监测量子系统中的拓扑相进行表征。我们通过李雅普诺夫能谱（Lyapunov spectrum），针对马约拉纳费米子（Majorana fermions）的(1+1)维受监测电路展开分析，该系统包含拓扑与平庸面积律纠缠相，以及具备亚体积律纠缠特性的临界相。我们发现：在拓扑（平庸）面积律相的体能隙内，存在（不存在）边缘局域零模；而临界相则呈现体能谱无隙的特征。进一步地，我们通过巧妙利用边界处带有扭曲测量结果的费米宇称（fermion parity），构造出一种拓扑不变量，该不变量可清晰区分两类面积律相，并实现对临界相的动态表征。因此，本研究为将拓扑相的体-边对应关系（bulk-edge correspondence）从平衡态拓展至受监测量子动力学领域铺平了道路。