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

Abstract

We show that operator dynamics in U(1) symmetric systems are constrained by two independent conserved charges and construct a non-Abelian operator size basis that respects both, enabling a symmetry-resolved characterization of operator growth. The non-Abelian operator size depends on the operator's nonlocal structure and is organized by an SU(2) algebra. Operators associated with large total angular momentum are relatively simple, while those with small angular momentum are more complex. Operator growth is thus characterized by a reduction in angular momentum and can be probed using out-of-time-ordered correlators. Using the charge-conserving Brownian Sachdev-Ye-Kitaev model, we derive an exact classical master equation that governs the size distribution, the distribution of an operator expanded in this basis, for arbitrary system sizes. The resulting dynamics reveal that the size distribution follows a chi-squared form, with the two conserved charges jointly determining the overall time scale and the shape of the distribution. In particular, single-particle operators retain a divergent peak at large angular momentum throughout the time evolution.

Figures (4)

• Figure 1: Sketch of the symmetry resolved operator growth in U(1) symmetric systems within each charge sector. Operator growth is captured by the reduction of total angular momentum from the near maximal value $N/2$ down to $|L_z|$.
• Figure 2: Numerical simulations of the master equation for system sizes $N = 200$--$4000$, with darker curves indicating larger $N$. The shifted time $\tau = t - t^*$ is held fixed as $N$ increases. The large $N$ distributions exhibit excellent agreement with the chi-squared form in Eq. \ref{['eq:chi']}. Panels (a), (b), and (c) show results for $C = 1$, $2$, and $10$, respectively. All simulations use $l_z = 0.3$ and $L_0 = (N - C)/2$.
• Figure 3: Full operator basis for N = 2 sites, organized by charge sectors $(Q_L, Q_R)$ and total spin $L$ within each block.
• Figure 4: Comparison of operator time evolution under the complex Brownian SYK Hamiltonian and the master equation for $N = 5$. Disorder averaging is performed over 15 independent realizations of complex random couplings.