  We explore a class of random tensor network models with "stabilizer" local
tensors which we name Random Stabilizer Tensor Networks (RSTNs). For RSTNs
defined on a two-dimensional square lattice, we perform extensive numerical
studies of entanglement phase transitions between volume-law and area-law
entangled phases of the one-dimensional boundary states. These transitions
occur when either (a) the bond dimension $D$ of the constituent tensors is
varied, or (b) the tensor network is subject to random breaking of bulk bonds,
implemented by forced measurements. In the absence of broken bonds, we find
that the RSTN supports a volume-law entangled boundary state with bond
dimension $D\geq3$ where $D$ is a prime number, and an area-law entangled
boundary state for $D=2$. Upon breaking bonds at random in the bulk with
probability $p$, there exists a critical measurement rate $p_c$ for each $D\geq
3$ above which the boundary state becomes area-law entangled. To explore the
conformal invariance at these entanglement transitions for different prime $D$,
we consider tensor networks on a finite rectangular geometry with a variety of
boundary conditions, and extract universal operator scaling dimensions via
extensive numerical calculations of the entanglement entropy, mutual
information and mutual negativity at their respective critical points. Our
results at large $D$ approach known universal data of percolation conformal
field theory, while showing clear discrepancies at smaller $D$, suggesting a
distinct entanglement transition universality class for each prime $D$. We
further study universal entanglement properties in the volume-law phase and
demonstrate quantitative agreement with the recently proposed description in
terms of a directed polymer in a random environment.
