A mean-field description of strong-to-weak symmetry breaking in the monitored three-dimensional Bose-Hubbard model

Strong-to-weak spontaneous symmetry breaking has emerged as a novel form of ordering in monitored and open quantum systems, yet its characterization has so far primarily relied on nonlocal diagnostics. Here, we develop a Gutzwiller mean-field framework for monitored bosonic lattice systems, enabling the direct simulation of stochastic measurement dynamics in three spatial dimensions. Applying this approach to the monitored Bose-Hubbard model with local density measurements and Lindbladian dissipation, we identify strong-to-weak symmetry breaking through a trajectory-averaged local order parameter. We find that this local order parameter becomes critical near the same measurement strength as the charge-sharpening transition and exhibits Lorentz invariance with a correlation-length exponent, νsimeq 1.2, comparable to that of the charge-sharpening transition, suggesting that the two phenomena may originate from a common underlying critical point. Our work establishes a local characterization of strong-to-weak symmetry breaking, reveals its connection to charge sharpening, and provides concrete predictions for future experiments on the monitored Bose-Hubbard model.