Non-Abelian operator size distribution and its dynamics in charge-conserving many-body systems

In this work, we introduce a non-Abelian operator size basis that respects conserved quantities to characterize operator growth in systems with U(1) symmetry. Unlike symmetry-free cases, where operator size is conventionally defined by its support, operator size in this setting depends on the operator's non-local structure and is organized through an SU(2) algebra. The growth of operators is captured by the reduction of total angular momentum to a value determined by the conserved charge and can be probed via a specific out-of-time-ordered correlator. We derive an exact classical master equation for the size distribution using the Brownian SYK model for arbitrary system size. This master equation can be solved numerically for large systems and analytically in the large-N limit. We show that the operator size follows a chi-squared distribution of a time-dependent variable. The resulting dynamics reveal how the conserved charge influences operator growth, affecting both the overall timescale and the shape of the distribution throughout the evolution.