For quantum systems at equilibrium, quantum criticality can often be understood following the principle of symmetry and topology. In contrast, quantum systems out of equilibrium can exhibit different dynamical phases and criticality that are fundamentally distinguishable only by their internal entanglement dynamics and scaling. Recently, entanglement phases and entanglement criticality in dynamical systems have attracted lots of attention. In particular, many entanglement critical points have been found and explored numerically. Due to the exotic nature of these numerically-observed entanglement phases, the analytical understandings of many of them remains elusive. In this talk, I will present two examples of entanglement quantum criticality can be studied analytically. The first example concerns with random unitary circuits with projective measurements. A pure initial state can evolve in the long time limit into a state with either area-law or volume-law entanglement entropy, which corresponds to the two distinct dynamical entanglement phases. We show that the critical point between the two dynamical phases can be understood using an exact mapping between the random circuits and a statistical mechanics model. In a certain limit, the universality class of the entanglement transition in this system is identified with the percolation transition. In the second example, I will describe a critical entanglement phase in the system of random Gaussian non-unitary circuits. The random circuit under consideration can be identified as a Haar-random ensemble of Gaussian tensor networks. I will present our numerical observation of the critical entanglement phases. I will also present the analytical understanding of this critical entanglement phase which is found in non-unitary circuit dynamics without any symmetry, via the theory of (unitary) disordered metal in the symmetry class DIII.

# Entanglement Criticality in random quantum circuits

Host: Yong Baek Kim