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

TL;DR

This paper introduces a non-Abelian operator size basis for charge-conserving many-body systems, revealing how operator growth is constrained by conserved charges and characterized by an SU(2) algebra, with exact results derived from a Brownian SYK model.

Contribution

It constructs a symmetry-resolved operator size basis respecting non-Abelian charges and derives an exact classical master equation for size distribution in a charge-conserving model.

Findings

Operator size distribution follows a chi-squared form.

Large angular momentum operators are simpler, small are more complex.

Single-particle operators maintain a divergent peak at large angular momentum.

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.