Quantum measurement fundamentally alters the system by extracting information and collapsing superposition of quantum state, giving rise to rich and nontrivial dynamics. In recent years, quantum systems evolving under repeated measurements have been found to harbor a wide variety of nonequilibrium phase transitions, called the measurement-induced phase transitions. Among them, the sequence of non-commuting measurements can induce phase transitions reminiscent of topological phase transitions known in equilibrium quantum systems. Despite the growing interest, however, such measurement-induced topological transitions have not been characterized in terms of energy spectra or topological invariants, fundamental concepts well-established in equilibrium settings. This is because of the lack of an appropriate framework for defining these concepts in monitored systems.
In this study, the research team proposes a concrete framework to define both an effective energy spectrum and a topological invariant for quantum systems under measurements. The effective energy spectrum is defined through the Lyapunov analysis of the system’s evolution, while the topological invariant is constructed by implementing different boundary conditions via appropriate post-selection of inherently random measurement outcomes.
The results reveal that the topological phase hosts edge-localized zero-energy states originated from Majorana particles, enigmatic particles that were hypothesized to exist by Ettore Majorana in 1937. Furthermore, the proposed topological invariant not only clearly distinguishes the topological and trivial phases, but also dynamically captures the transition into an intermediate critical phase that is unique to monitored quantum systems.
Importantly, their framework can be naturally extended to general many-body quantum systems. As such, this work provides a significant step toward establishing the bulk-edge correspondence in nonequilibrium quantum systems under measurements, offering a new paradigm for understanding quantum phases beyond equilibrium.
Papers
Journal: Physical Review Letters
Title: Topology and Spectrum in Measurement-Induced Phase Transitions
Authors: Hisanori Oshima*, Ken Mochizuki, Ryusuke Hamazaki, Yohei Fuji
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