Measurement-Induced Phase Glasses

We study a spin-S chain evolved by weak measurements of the relative phase (planar spin orientation) across nearest neighbor bonds. Non-commutativity of the phase measurements on adjacent bonds and the fuzziness of the phase as an observable at finite S generate quantum fluctuations in the measurement only dynamics, which are suppressed at large S. Using a spin-coherent state path integral we show that these monitored dynamics exhibit a Kosterlitz–Thouless transition in individual quantum trajectories from a trivial area-law phase at small spin size to a phase glass with subsystem entanglement scaling as log(L) for $S>S_c$. The phase glass is characterized by a power-law decay of Edwards-Anderson spin correlation and complements the charge-sharp and charge-fuzzy phases previously established in dynamics using only charge measurements and unitaries. A similar transition occurs in bosonic chains, where it is tuned by the averaged site occupation. We discuss possible approaches to observe this transition experimentally.