Measurement induced criticality in \texorpdfstring{$\mathbb{Z}_2$}{} symmetric quantum automaton circuits

We study entanglement dynamics in hybrid $\mathbb{Z}_2$ symmetric quantum automaton circuits subject to local composite measurements. We show that there exists an entanglement phase transition from a volume law phase to a critical phase by varying the measurement rate $p$. By analyzing the underlying classical bit string dynamics, we demonstrate that the critical point belongs to parity conserving universality class. We further show that the critical phase with $p>p_c$ is related to the diffusion-annihilation process and is protected by the $\mathbb{Z}_2$ symmetric measurement. We give an interpretation of the entanglement entropy in terms of a two-species particle model and identify the coefficient in front of the critical logarithmic entanglement scaling as the local persistent coefficient. The critical behavior observed at $p\geq p_c$ and the associated dynamical exponents are also confirmed in the purification dynamics.