Entanglement phase transitions in quantum chaotic systems subject to projective measurements and in random tensor networks have emerged as a new class of critical points separating phases with different entanglement scaling. We propose a mean-field theory of such transitions by studying the entanglement properties of random tree tensor networks. As a function of bond dimension, we find a phase transition separating area-law from logarithmic scaling of the entanglement entropy. Using a mapping onto a replica statistical mechanics model defined on a Cayley tree and the cavity method, we analyze the scaling properties of such transitions. Our approach provides a tractable, mean-field-like example of an entanglement transition. We verify our predictions numerically by computing directly the entanglement of random tree tensor network states.

• Received 19 March 2020
• Revised 12 June 2020
• Accepted 27 July 2020

DOI:https://doi.org/10.1103/PhysRevB.102.064202