Random matrix prediction of average entanglement entropy in non-Abelian symmetry sectors

TL;DR

This paper analytically derives the average entanglement entropy of Haar-random states in quantum systems with SU(2) symmetry, revealing a volume law with finite-size corrections due to angular momentum coupling.

Contribution

It extends Page's entanglement results to non-Abelian SU(2) symmetry sectors, providing explicit formulas including finite-size corrections.

Findings

Average entanglement entropy follows a volume law with a 0.5 log V correction.

Explicit analytical expressions for entropy in SU(2) symmetric sectors.

Finite-size effects are characterized by angular momentum coupling contributions.

Abstract

We study the average bipartite entanglement entropy of Haar-random pure states in quantum many-body systems with global $\mathrm{SU}(2)$ symmetry, constrained to fixed total spin $J$ and magnetization $J_z = 0$. Focusing on spin-$\tfrac12$ lattices and subsystem fractions $f < \frac{1}{2}$, we derive a asymptotic expression for the average entanglement entropy up to constant order in the system volume $V$. In addition to the expected leading volume law term, we prove the existence of a $\frac{1}{2}\log V$ finite-size correction resulting from the scaling of the Clebsch-Gordon coefficients and compute explicitly the $O(1)$ contribution reflecting angular-momentum coupling within magnetization blocks. Our analysis uses features of random matrix ensembles and provides a fully analytical treatment for arbitrary spin densities, thereby extending Page type results to non-Abelian sectors and clarifying how $\mathrm{SU}(2)$ symmetry shapes average entanglement.