Topology and Spectrum in Measurement-Induced Phase Transitions

TL;DR

This paper introduces a Lyapunov-based framework to analyze topological properties and phase transitions in monitored quantum systems, connecting spectrum analysis with topological invariants in measurement-induced phases.

Contribution

It develops a general method to characterize topological features in monitored quantum systems using Lyapunov analysis and many-body invariants, extending bulk-edge correspondence.

Findings

Identifies edge-localized zero modes in topological phases.

Distinguishes phases using a constructed topological invariant.

Analyzes spectrum and entanglement in monitored Majorana circuits.

Abstract

Competition among repetitive measurements of noncommuting observables and unitary dynamics can give rise to a wide variety of entanglement phases. Here, we propose a general framework based on Lyapunov analysis to characterize topological properties in monitored quantum systems through their spectrum and many-body topological invariants. We illustrate this framework by analyzing (1+1)-dimensional monitored circuits for Majorana fermions, which are known to exhibit topological and trivial area-law entangled phases as well as a critical phase with sub-volume-law entanglement. Through the Lyapunov analysis, we identify the presence (absence) of edge-localized zero modes inside the bulk gap in the topological (trivial) 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 distinguishes the two area-law phases and dynamically characterizes the critical phase. Our framework thus provides a general route to extend the notion of bulk-edge correspondence to monitored quantum dynamics.