Boundary transitions from a single round of measurements on gapless quantum states

Measurements can qualitatively alter correlations and entanglement emerging in gapless quantum matter. We show how a single round of measurements on gapless quantum systems can, upon rotating the measurement basis, induce non-trivial transitions separating regimes displaying universal characteristics governed by distinct boundary conformal field theories. We develop the theory of such `measurement-induced boundary transitions' by investigating a gapless parent of the one-dimensional cluster state, obtained by appropriately symmetrizing a commuting projector Hamiltonian for the latter. Projective measurements on the cluster state are known to convert the wavefunction, after post-selection or decoding, into a long-range-ordered Greenberger-Horne-Zeilinger (GHZ) state. Similar measurements applied to the gapless parent (i) generate long-range order coexisting with power-law correlations when post-selecting for uniform outcomes, and (ii) yield power-law correlations distinct from those in the pre-measurement state upon decoding. In the post-selection scenario, rotating the measurement basis preserves long-range order up until a critical tilt angle marking a measurement-induced boundary transition to a power-law-ordered regime. Such a transition -- which does not exist in the descendant cluster state -- establishes new connections between measurement effects on many-body states and non-trivial renormalization-group flows. We extend our analysis to tricritical Ising and three-state Potts critical theories, which also display measurement-induced boundary transitions, and propose general criteria for their existence in other settings.