Average pure-state entanglement entropy in spin systems with SU(2) symmetry

We study the effect that the SU(2) symmetry, and the rich Hilbert space structure that it generates in lattice spin systems, has on the average entanglement entropy of highly excited eigenstates of local Hamiltonians and of random pure states. Focusing on the zero total magnetization sector ($J_z=0$) for different fixed total spin $J$, we argue that the average entanglement entropy of highly excited eigenstates of quantum-chaotic Hamiltonians and of random pure states has a leading volume-law term whose coefficient $s_A$ depends on the spin density $j=J\mathrm{/}(\mathfrak{j}L)$, with $s_A(j\to 0)=\ln (2\mathfrak{j}+1)$ and $s_A(j\to 1)=0$, where $\mathfrak{j}$ is the microscopic spin. We provide numerical evidence that $s_A$ is smaller in highly excited eigenstates of integrable interacting Hamiltonians, which lends support to the expectation that the average eigenstate entanglement entropy can be used as a diagnostic of quantum chaos and integrability for Hamiltonians with non-Abelian symmetries. In the context of Hamiltonian eigenstates we consider spins $\mathfrak{j}=\frac{1}{2}$ and $1$, while for our calculations based on random pure states we focus on the spin $\mathfrak{j}=\frac{1}{2}$ case.