Measurement-induced criticality and entanglement clusters: A study of one-dimensional and two-dimensional Clifford circuits

Entanglement transitions in quantum dynamics present a novel class of phase transitions in nonequilibrium systems. When a many-body quantum system undergoes unitary evolution interspersed with monitored random measurements, the steady state can exhibit a phase transition between volume- and area-law entanglement. There is a correspondence between measurement-induced transitions in nonunitary quantum circuits in d spatial dimensions and classical statistical mechanical models in d + 1 dimensions. In certain limits these models map to percolation, but there is analytical and numerical evidence to suggest that away from these limits the universality class should generically be distinct from percolation. Intriguingly, despite these arguments, numerics on 1 + 1D qubit circuits give bulk exponents which are nonetheless close to those of 2D percolation, with some possible differences in surface behavior. In the first part of this work we explore the critical properties of 2 + 1D Clifford circuits. In the bulk, we find many properties suggested by the percolation picture, including several matching bulk exponents, and an inverse power law for the critical entanglement growth, S(t, L) similar to L(1 - a/t), which saturates to an area law. We then utilize a graph-state-based algorithm to analyze in 1 + 1D and 2 + 1D the critical properties of entanglement clusters in the steady state. We show that in a model with a simple geometric map to percolation-the projective transverse field Ising model-these entanglement clusters are governed by percolation surface exponents. However, in the Clifford models we find large deviations in the cluster exponents from those of surface percolation, highlighting the breakdown of any possible geometric map to percolation. Given the evidence for deviations from the percolation universality class, our results raise the question of why nonetheless many bulk properties behave similarly to those of percolation.