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

TL;DR

We address the problem of computing the average entanglement entropy of Haar-random pure states constrained to non-Abelian SU(2) symmetry sectors with fixed total spin $J$ and magnetization $J_z=0$ on a spin-1/2 lattice, for subsystem fraction $f<1/2$. The authors develop an analytic framework that maps the fixed-trace reduced density matrix to a coupled Wishart ensemble and uses Wigner’s moment method within a planar-diagram approximation, augmented by a dimensional-selection principle that isolates dominant spin-sector contributions. They obtain a closed-form asymptotic expression for the average entropy: $oxed{ \\langle S_A\\rangle = V f \\\beta(j) + frac{1}{2}\\log V + ext{const}(f,j) + o(1) }$, where the leading $O(V)$ term is set by the density of states through $\\beta(j)$, a universal $\\tfrac{1}{2}\\log V$ correction arises from the scaling of Clebsch–Gordan coefficients, and an $O(1)$ constant encodes angular-momentum coupling within the magnetization blocks. The analysis disentangles a non-Abelian contribution, CG terms, and an Abelian-like correction, showing how SU(2) symmetry shapes the entanglement structure and providing a framework potentially extensible to SU($N$) or other symmetry groups.

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.

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