Renormalization group for measurement and entanglement phase transitions

We analyze the renormalization-group (RG) flows of two effective Lagrangians, one for measurement induced transitions of monitored quantum systems and one for entanglement transitions in random tensor networks. These Lagrangians, previously proposed on grounds of replica symmetry, are derived in a controlled regime for an illustrative family of tensor networks. They have different forms in the two cases, and involve distinct replica limits. The perturbative RG is controlled by working close to a critical dimensionality, ${d_c=6}$ for measurements and ${d_c=10}$ for random tensors, where interactions become marginal. The resulting RG flows are surprising in several ways. They indicate that in high dimensions $d>d_c$ there are at least two (stable) universality classes for each kind of transition, separated by a nontrivial tricritical point. In each case one of the two stable fixed points is Gaussian, while the other is nonperturbative. In lower dimensions, $d<d_c$, the flow always runs to the nonperturbative regime. This picture clarifies the "mean-field theory" of these problems, including the phase diagram of all-to-all quantum circuits. It suggests a way of reconciling exact results on tree tensor networks with field theory. Most surprisingly, the perturbation theory for the random tensor network (which also applies to a version of the measurement transition with "forced" measurements) formally possesses a dimensional reduction property analogous to that of the random-field Ising model. When only the leading interactions are retained, perturbative calculations in $d$ dimensions reduce to those in a simple scalar field theory in ${d-4}$ dimensions. We show that this holds to all orders by writing the action in a superspace formulation.