Strong-to-weak Symmetry Breaking and Entanglement Transitions

When interacting with an environment, the entanglement within quantum many-body systems is rapidly transferred to the entanglement between the system and the bath. For systems with a large local Hilbert space dimension, this leads to a first-order entanglement transition for the reduced density matrix of the system. On the other hand, recent studies have introduced a new paradigm for classifying density matrices, with particular focus on scenarios where a strongly symmetric density matrix undergoes spontaneous symmetry breaking to a weak symmetry phase. This is typically characterized by a finite R\'enyi-2 correlator or a finite Wightman correlator. In this work, we study the entanglement transition from the perspective of strong-to-weak symmetry breaking, using solvable complex Brownian SYK models. We perform analytical calculations for both the early-time and late-time saddles. The results show that while the R\'enyi-2 correlator indicates a transition from symmetric to symmetry-broken phase, the Wightman correlator becomes finite even in the early-time saddle due to the single-replica limit, demonstrating that the first-order transition occurs between a near-symmetric phase and a deeply symmetry-broken phase in the sense of Wightman correlator. Our results provide a novel viewpoint on the entanglement transition under symmetry constraints and can be readily generalized to systems with repeated measurements.