We investigate the interplay between unitary and nonunitary driven many-body dynam ics in ($1+1$)-dimensional quantum critical systems described by conformal field theory (CFT). By formulating a coherent state approach, we demonstrate that the growth of entanglement entropy and energy can be found analytically for a class of nonunitary driven CFTs, where the evolution alternates between real and imaginary time evolution, the latter corresponding to postselected weak measurements. We find that nonunitary evolution leads to the emergence of steady states at infinite times for the cases of periodic, quasiperiodic, and random drives. In a special class of drives, for mixed initial states, we uncover purification phase transitions that arise as a result of the competition between unitary evolution and weak measurements. We compare the CFT evolution with the corresponding nonunitary dynamics of critical lattice models, finding remarkable agreement.

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• Dynamical phase transitions
• Nonequilibrium statistical mechanics
• Floquet systems
• Non-Hermitian systems
• Nonequilibrium lattice models
• Quantum many-body systems
• Strongly correlated systems
• Conformal field theory
• Many-body techniques