Critical ground state under weak measurements is not critical

Critical systems host nontrivial entanglement structure that is generally sensitive to additional couplings. In the present work, we study effect of weak measurements on the entanglement structure of massless free fermions on their critical ground state. While the power-law decaying correlation and logarithmic growing entanglement has been observed as typical signatures of emergent criticality, the conformal symmetry is lost and the entanglement Hamiltonian generally becomes gapped for arbitrary small measurement strength. We analytically reveal these features by considering a field-theory calculation at continuum, which allows us to establish a mapping between the entanglement Hamiltonian before and after weak measurements.