Replica Transition at Measurement-Altered Ising Criticality

Performing measurements on gapless quantum ground states can give rise to new universal phenomena, whose origins remain poorly understood even for the paradigmatic Ising critical point. Replica methods offer a natural framework to study such post-measurement ensembles, with the replica limit Q to 1 describing physical Born-averaged observables. Here we show that, for the monitored critical Ising chain, the replicated post-measurement ensemble of states itself undergoes a phase transition precisely at the physically relevant replica limit Q to 1, preventing standard replica treatments. We use boundary conformal field theory (BCFT) methods to explain the underlying mechanism of this transition. Our methods yield a perturbative analytic calculation of the universal O(1) constant in the Shannon entropy of the measurement record, previously known only numerically. We also apply our tools to study measurement-averaged spin correlators, providing a unified description of the universal properties of measurement outcomes and post-measurement states in the critical Ising chain.