The effective central charge of the measurement induced phase transition

Recently, there has been a growing interest in understanding the measurement driven entanglement transition in a variety of random circuit models. Two models that have received a significant amount of attention are the random Haar circuit and the stabilizer circuit. These two models lie in different regimes of computational complexity with the stabilizer circuit being able to be simulated efficiently on a classical computer and the Haar circuit requiring the details of the full Hilbert space. Surprisingly, both of these models seem to share many of the same critical properties with each other and with percolation suggesting they might belong to the same universality class. In this talk, we will introduce a method to calculate the effective central charge of the logarithmic conformal field theory at the critical point. What we find is clear evidence that separates the two circuit models and percolation into three distinct universality classes. This approach is extended to compute the leading Lyapunov exponents of a transfer matrix composed of unitary gates and projective measurements that we use to calculate the scaling dimensions of additional operators in the field theory at the transition.