Novel measurement-induced behavior in a gapless parent of the 1D cluster state

The 1D cluster state model provides an exactly solvable realization of a Z₂ x Z₂ symmetry-protected topological (SPT) phase. Projectively measuring half of the sites has been shown to convert its ground state, after a decoding protocol, into a long-range-ordered GHZ state. We introduce a gapless parent Hamiltonian for the SPT and consider the analogous fate of the ground state under such projective measurements. Our parent Hamiltonian constitutes a variant of the cluster state model endowed with additional spatial symmetries, and maps to a spin-1 XY model following a duality transformation. For measurements yielding the most likely outcome, we show that the post-measurement state simultaneously exhibits long-range order, power-law correlations, and area-law entanglement---similar to behavior recently found for quantum critical states subjected to weak measurements. We explain this coexistence through both perturbative and field-theoretic approaches. We further demonstrate that the same decoding protocol for the descendant SPT phase reveals power-law correlations distinct from those in the pre-measured state.