Entanglement Phases in large-N hybrid Brownian circuits with long-range couplings*

We study measurement-induced phases and transitions in tractable large-N models, including a Brownian qubit model and a Brownian SYK model, in the presence of long-range couplings which decay as a power law, with α being the power-law exponent. In one dimension and in interacting models, the long-range coupling is irrelevant for α>3/2, thus the volume-law and area-law entanglement phases and the phase transition remain similar to the short-ranged case. For α<3/2 the long-range coupling is relevant and leads to a nontrivial dynamical exponent at the measurement-induced phase transition. More interestingly, for α<1 the entanglement pattern receives a sub-volume correction for both area-law and volume-law phases. The volume-law phase with such a sub-volume correction realizes a novel quantum error correcting code whose code distance scales as L^2-2α. We extend the calculation in a quadratic Brownian SYK model to study the phase diagram of the long-range free fermion model under monitoring, and we find that two distinct fractal entangled phases emerge when α is sufficiently low.